j. differential geometry

64 (2003) 247-303

THE NORMALIZED MEAN CURVATURE FLOW FOR A SMALL BUBBLE IN A RIEMANNIAN MANIFOLD

NICHOLAS D. ALIKAKOS & ALEXANDRE FREIRE

Abstract

The evolution of an embedded surface under the normalized mean curvature flow is the result of a complicated interaction between the geometry of the evolving surface and the geometry of the ambient space, and is not well understood in the context of a general Riemannian manifold. In the present paper we identify a class of initial conditions, that we call “bubbles”, whose dynamics is primarily determined by the ambient space. A bubble is an embedded surface that is close to a small geodesic ball; we find that its shape is robust along the evolution. Moreover, under a relatively tight condition relating shape to size, we show that the velocity of the center of the bubble is given, to principal order, by the gradient of the scalar curvature. Finally under natural conditions of compactness and nondegeneracy we show that such solutions converge, as t tends to infinity, to surfaces of constant mean curvature.

0. Introduction

In this paper we consider the motion of hypersurfaces Σ(t) in a compact n-dimensional Riemannian manifold M by normalized mean curvature, that is, with normal velocity given by the equation:

⟨X_{t}, N̂⟩ = H^{Σ} − H, (0.1)

where N̂ denotes the unit outward normal, H is the mean curvature of Σ and H^{Σ} is the average mean curvature:

H^{Σ} = \frac{1}{A(Σ)} ∫_{Σ} Hdσ,

N.D.A. partially supported by a PENED 99/527 interdisciplinary grant in Materials; A.F. partially supported by NSF.

Received 05/23/2002.

with A(Σ) = ∫_{Σ} dσ the (n−1)-dimensional volume of Σ in the metric induced from M. Topologically, our hypersurfaces are embedded spheres, parametrized by a time-dependent smooth embedding X(t) : S → M from the (n−1)-sphere to M. It is easily verified that the left-hand side of (0.1) is, in fact, independent of parametrization.

The basic property of the law of evolution (0.1) is its isoperimetric nature. The volume V (Σ) of the region enclosed by Σ is preserved:

\frac{dV (Σ(t))}{dt} = ∫_{Σ}⟨X_{t}, N̂⟩dσ = ∫_{Σ}(H^{Σ} − H)dσ = 0,

while the area A(Σ) ((n−1)-dimensional volume) of Σ strictly decreases, unless H is constant:

(0.2)  \frac{dA(Σ(t))}{dt} = ∫_{Σ}⟨X_{t}, HN̂⟩dσ = ∫_{Σ}(H^{Σ} − H)Hdσ
= − ∫_{Σ}(H − H^{Σ})^{2}dσ.

In particular, it is natural to expect that the hypersurface will tend to move towards regions of larger scalar curvature.

The evolution (0.1) was originally considered by M. Gage (n = 2) [6] and G. Huisken [8] (n ≥ 3), when the ambient manifold M is euclidean space ℝ^{n}. Huisken proved that, in this case, if the initial hypersurface is smooth and uniformly convex, the evolution is defined for all time and the hypersurface remains convex and converges to a round sphere as t → ∞. More recently Escher and Simonett [5] were able to prove this for initial hypersurfaces close to spheres (including some non-convex ones), by a different method, closer in spirit to ours in the present paper.

Relative to the much better understood evolution by mean curvature, the normalized problem presents several difficulties. The main one is that, the law being non-local, the maximum principle for parabolic equations does not apply. One consequence of this is that initially embedded hypersurfaces may develop self-intersections. An example for curves was described by Owen and Sternberg [13], and considered in more detail by Mayer and Simonett [12]. Another consequence is that, for the evolution on manifolds (even of constant curvature), convexity of the initial hypersurface may not be preserved, as was pointed out by Huisken. In spite of this difficulty, Huisken and S.-T. Yau [9] were able to use (0.1) to construct foliations by spheres of constant mean curvature — the steady-states of (0.1) — of asymptotically flat ends of positive mass (for 3-manifolds of positive scalar curvature). Their motivation was to develop a coordinate-independent definition of ‘center of mass’ of an isolated gravitational system. The problem of existence of local foliations by hypersurfaces of constant mean curvature was also considered by R. Ye [14]. He proved such foliations exist in a neighborhood of any nondegenerate critical point of the scalar curvature; and, conversely, that such foliations are unique in a certain class, and exist only in a neighborhood of a critical point.

R. Ye [14] used a perturbation of geodesic spheres of small radius to construct the local foliation. For the dynamical problem (0.1), this points to a potential problem when trying to construct solutions by a perturbation argument: except for special cases (‘harmonic manifolds’, including constant curvature manifolds), geodesic spheres are not equilibria of (0.1); in fact, on a general manifold the existence of timeindependent solutions is a nontrivial issue. Nevertheless, we will show in this paper that if the initial hypersurface is sufficiently close to a small geodesic sphere (i.e., is a small ‘bubble’), the evolution is defined for all time and (under certain conditions) will converge to a leaf of one of these foliations, as t → ∞. Clearly to obtain such a result one must be able to track the evolution of the hypersurface ‘in the large’ on the ambient manifold M, a problem not considered by the authors above. Our idea is to ‘decouple’ the effect of the ambient from the effect of the geometry of the interface: while the hypersurfaces start and remain very close to small geodesic spheres, we can describe how an appropriately defined ‘barycenter’ moves on M (at least asymptotically in a perturbation parameter).

The usual line of argument in studying existence and development of singularities for the mean-curvature flow proceeds in intrinsic fashion, by studying the evolution of geometric quantities on Σ (such as the trace-free second fundamental form). This approach is very effective and geometric; but in order to follow the large-scale motion of Σ on M we are led, instead, to work directly with the parametrizations X(t), i.e., to study a motion in a manifold of embeddings. This introduces some technical difficulties-for example, there is no uniquely defined law of motion for embeddings corresponding to (0.1)-but, on the other hand, allows for the introduction of an infinite-dimensional dynamical systems approach to the problem. At least in the present ‘perturbation’ setting, we are able to draw on results from semigroup theory (‘maximal regularity’) and known results on infinite dimensional systems to give a fairly non-technical proof. It is our hope that such an approach will also prove fruitful for other geometric evolution equations.

The evolution we construct takes place in a submanifold of the manifold E of C^{2+α} ‘small quasispherical embeddings’ X : S → M, which are radial graphs over a small geodesic sphere in M with center ξ ∈ M, radius R > 0. To define such an embedding, we need a diffeomorphism F from S to the unit tangent sphere at ξ and a ‘shape function’ ψ : C^{2+α} → ℝ, which we take to be a C^{2+α} function on S, with zero average on S. Introducing two small scale parameters δ ∈ (0, δ_{0}), ϵ ∈ (0, ϵ_{0}), we let E = E_{δ_{0},ϵ_{0}} be the space of embeddings which can be written in the form:

X_{(R,ξ,F,ψ)}(u) = exp_{ξ}[δR(1 + ϵψ(u))F (u)],

where 0 < R < 1, ψ ∈ C^{2+α}(S) satisfies ∥ψ∥_{C^{2+α}} < 1 and ave_{S}[ψ] = 0 and F is a C^{2+α} diffeomorphism from S to the unit tangent sphere S_{ξ} ⊂ T_{ξ}M. We take δ_{0}, ϵ_{0} small enough that the open set int (X) bounded by the hypersurface Σ = image(X) (and containing ξ) is contained in a totally convex normal neighborhood of ξ, and is uniformly convex.

The same embedding X ∈ E can be written in the form X_{(R,ξ,F,ψ)} in different ways, parametrized by ξ ∈ int(X). We need a choice of ξ that is as canonical as possible, given X; that is, a ‘barycenter’ for X. While a general Riemannian notion of ‘barycenter’ exists (see [10]), here we find it useful to work with the notion of ‘analytic barycenter’, which appears already in [1]. This is the unique ξ ∈ int (X) for which X may be written as X_{(R,ξ,F,ψ)} , with ψ taken in the space:

K^{2+α} = C^{2+α}(S) ∩ C_{0}(S),
C_{0}(S) = {ψ ∈ C^{0}(S); ave_{S}[ψ] = 0 = ave_{S}[ψu^{i}], i = 1, … n}.

In Section 1 we prove (in Lemma 1.1) that for X in a neighborhood N of the submanifold of ‘standard parametrizations of geodesic spheres’ E_{0} ⊂ E, there is a unique ξ = B(X) with this property; N = N (δ_{1}, ϵ_{1}) is open in E_{δ_{1},ϵ_{1}}.

The existence of the analytic barycenter allows us to consider evolution equations on the submanifold N_{std} = N_{std}(δ_{1}, ϵ_{1}) ⊂ N, defined as the set of embeddings X ∈ N of the form:

X = X_{(R,ξ,e,ψ)} = exp_{ξ}[R(1 + ψ(·))e(·, ξ)],
ξ = B(X), ave_{S}[ψ] = ave_{S}[ψu^{i}] = 0, i = 1, …, n,

where e : S → S_{ξ} is an isometry, defined by an orthonornal frame at ξ. N_{std} is the image under a smooth injective map Φ of the manifold:

M_{0} = M_{0}(δ_{1}, ϵ_{1}) = (0, δ_{1}) × 𝔽M × K_{ϵ_{1}}^{2+α},

where 𝔽M denotes the orthonormal frame bundle of M and the last factor is the ϵ_{1}-ball in K^{2+α}. The next step is to find an evolution equation for (R, ξ, e, ψ) ∈ M_{0}, the solutions of which map under Φ to parametrized solutions of (0.1) in N_{std}. Since the barycenter B(X) depends on the parametrization X (and not just on its image Σ), we need to fix an equation of motion in the space of embeddings. As a first attempt, one might expect that the equations on M_{0} would be induced by the evolution on N:

X_{t} = (H^{Σ} − H)N̂. (0.3)

It turns out, however, that N_{std} is not invariant under (0.3), which therefore does not induce a system on M_{0}. Fortunately it is possible to compute a ‘tangential correction’ to (0.3) which does preserve N_{std}. This is explained in Section 1, where we find (in Lemma 1.7) a system on M_{0} whose solutions map to parametrized solutions of (0.1); and conversely, any motion Σ(t) of ‘small bubbles’ by normalized mean curvature can be parametrized by X_{(R,ξ,e,ψ)} ∈ N_{std}, so that (R, ξ, e, ψ)(t) is a solution of the system on M_{0}. For each choice of the scale parameters δ, ϵ, we obtain the system on M_{0}:

δR_{t} = ave_{S}[v_{N} − E] (0.4)

ξ_{t} = n ave_{S}[(v_{N} − E)e]
∇_{ξ_{t}}e = 0
δϵRψ_{t} = (v_{N} − E)_{K} − (δψ)ave_{S}[v_{N} − E].

Here v_{N} = (H^{Σ} − H)∥N∥ (where N is a particular normal vector to Σ) and E = E(v_{N}) corresponds to the tangential correction referred to above. v_{N} and E are computed at X_{(δR,ξ,e,ϵψ)}. (In particular, e is simply obtained by parallel transport of e(0) along ξ(t)). It is for this system on M_{0} that our main result is proved.

Given a totally convex open set U ⊂ M and positive numbers δ < δ_{1}, ϵ < ϵ_{1}, define the open subset of M_{0}:

O_{δ,ϵ}^{2+α}(U) = {(R, ξ, e, ψ) ∈ M_{0}; 0 < R < δ, (ξ, e) ∈ 𝔽U, ∥ψ∥_{C^{2+α}} < ϵ};

we define analogously the open subset O_{δ,ϵ}^{α}(U) of:

M^{α} = M^{α}(δ_{1}, ϵ_{1}) = (0, δ_{1}) × 𝔽M × K_{ϵ_{1}}^{α},

where K^{α} = C^{α} ∩ C_{0}(S). We find local solutions in the open subset:

W_{δ,ϵ}^{T}(U) = C^{0}([0, T], O_{δ,ϵ}^{2+α}(U)) ∩ C^{1}((0, T], O_{δ,ϵ}^{α}(U))

of the space:

W_{δ,ϵ}^{T} = C^{0}([0, T], M_{0}(δ, ϵ)) ∩ C^{1}((0, T], M^{α}(δ, ϵ)).

In general we denote N_{std}(δ, ϵ) = Φ(M_{0}(δ, ϵ)). Our main result follows.

Main Theorem.

(i) (Local existence) There exist constants δ_{2} ∈(0, δ_{1}/2), ϵ_{2} ∈ (0, ϵ_{1}/2) depending only on M, and T = T (δ_{2}, ϵ_{2}) > 0 so that for any Ψ(0) = (δR(0), ξ(0), e(0), ϵψ(0)) ∈ M_{0}(δ_{2}, ϵ_{2}) with ψ(0) ∈ C^{∞}(S), there exists a unique solution Ψ(t) = (δR(t), ξ(t), e(t), ϵψ(t)) of (0.4)_{δ,ϵ} in W_{δ_{2},ϵ_{2}}^{T}(U_{0}) (where U_{0} is a totally convex neighborhood of ξ(0)). The solution Ψ(t) is smooth for t > 0. The hypersurfaces Σ(t) parametrized by X(t) ∈ N_{std}:

X(t)(u) = exp_{ξ(t)}[δR(1 + ϵψ(t, u))e(t, u)]

are smooth for t ≥ 0 and satisfy Equation (0.1); in particular, we have smooth local solutions for any initial embedding X(0) ∈ N_{std}(δ_{2}, ϵ_{2}).

(ii) (Global existence) For a given Ψ ∈ M_{0}(δ_{2}, ϵ_{2}), define T_{Ψ}^{∗} as the supremum of all T > 0 such that the solution Ψ(t) found in (i) with initial value Ψ is in W_{δ_{2},ϵ_{2}}^{T}. There exist constants δ_{3} > 0, ϵ_{3} > 0 depending only on M, so that if Ψ is in M_{0}(δ, ϵ), with ψ(0) ∈ C^{∞}(S), δ < δ_{3}, ϵ < ϵ_{3} and δ^{2} ≪ ϵ, then T_{Ψ}^{∗} = ∞. Thus, the solution Ψ(t) defines a parametrization X(t) ∈ N_{std} of a global smooth solution Σ(t) of the normalized mean curvature flow (0.1). In particular, we have a global solution for arbitrary initial embedding X(0) ∈ N_{std}(δ, ϵ) under the same conditions on δ, ϵ, and the solution stays in N_{std}(δ_{2}, ϵ_{2}).

(iii) (Motion of the barycenter) In addition to the assumptions in (ii), suppose the scale parameters δ, ϵ satisfy:

δ^{2} ≪ ϵ ≪ δ^{3/2}.

Then the leading term in the equation of motion for ξ in (0.4)_{δ,ϵ} is:

ξ_{t} = c_{n}∇^{M}Scal(ξ)R^{2}δ^{2} + O(ϵ^{2}δ^{−1}),

where c_{n} = 2n/3(n + 2). Here ∇^{M}Scal denotes the gradient of scalar curvature.

(iv) (Asymptotic behavior) Any accumulation point along a sequence t_{n} → ∞ of a solution X(t) of (0.1) in N_{std}, X(t_{n}) → X_{∗} ∈ N_{std}, parametrizes a hypersurface Σ_{∗} of constant mean curvature. In addition, assume all critical points of the scalar curvature function Scal are nondegenerate. Then there are disjoint open neighborhoods V_{p} ⊂ M of the (finitely many) critical points p of Scal and constants δ_{4} > 0, ϵ_{4} > 0, so that any global solution Ψ(t) of (0.4) _{δ,ϵ} with δ < δ_{4}, ϵ < ϵ_{4}, ϵ ≪ δ^{3/2}, converges as t → ∞ to some Ψ_{∗} ∈ M_{0}(δ, ϵ). The embedding X_{∗} ∈ N_{std} corresponding to Ψ_{∗} has barycenter ξ_{∗} ∈ V_{p} for some critical point p of Scal, and parametrizes a hypersurface of constant mean curvature Σ_{∗} ⊂ V_{p}, which is the unique leaf of the local c.m.c foliation ([14]) at p enclosing the same volume as Σ(0).

In Parts (ii) and (iii) of the statement, δ^{2} ≪ ϵ means δ^{2} = ϵ^{γ} for some γ > 1; ϵ ≪ δ^{3/2} is understood analogously.

Local existence (Part (i)) is well-known for (0.1). We include a proof for the equivalent system (0.4) in the framework of semigroup theory, since this leads to the continuation criterion we use for global existence. In fact, neither the decomposition φ = R(1+ψ) of the ‘shape function’ nor detailed asymptotics are needed, and the proof we give (in Section 3, Lemma 3.2) works for more general initial data than stated in (i). We use results from ‘maximal regularity theory’ ([2], [3], [11]); all that is needed is to verify that the hypothesis of Theorem 2.14 in [3] are satisfied.

In Section 2 we develop the asymptotic expansions of equations (0.4) that are needed for global existence. The starting point are standard Taylor expansions of Jacobi fields in Riemannian normal coordinates, from which expansions for the mean curvature and its average (for a radial graph) follow easily. Another consequence is an a priori estimate for the radius R(t), which follows from the conservation of the enclosed volume.

After giving a proof of local existence (Lemma 3.2), we prove global existence in Section 3 by an argument involving the variation of constants representation formula and maximal regularity estimates. Here the scale parameters and asymptotics are needed, and the restriction δ^{2} ≪ ϵ is required. It guarantees that, in the equation for ψ_{t}, the ‘euclidean’ term δ^{−2}R^{−2}Aψ (where A is the linearized operator for euclidean ambient, Aψ = Δ_{S}ψ + (n − 1)ψ) dominates the largest Riemannian term, which is of order ϵ^{−1}. The fact that the spectrum of A is bounded above by a negative constant is crucial, and motivates our definition of analytic barycenter. A similar argument was used in [1].

We conclude in Section 4 with the proof of asymptotic convergence to a geodesic sphere. Since we assume the critical points of the scalar curvature function are nondegenerate, a small constant mean curvature sphere near a critical point must be a leaf of the local foliation at that critical point ([14]), and there is only one of those enclosing a given volume. We may then appeal to general results on infinite-dimensional dynamical gradient systems to conclude.

We close the introduction with a few heuristic remarks. If we consider the motion of small geodesic spheres on M as ‘approximate solutions’ of the flow, it is not hard to understand the leading term in the equation of motion for the barycenter from isoperimetric considerations. The (n-1)-dimensional area (‘perimeter’) A(ξ, R) of a geodesic sphere with center ξ, radius R, is given by the classical Riemannian formula:

A(ξ, R) = R^{n−1} ∫_{S} [1 − \frac{1}{6}Ric (ξ, u)R^{2} + O(R^{3})] du,

where Ric (ξ, u) denotes the Ricci curvature of M at ξ in the direction u, if we identify S with the unit tangent sphere at ξ. (This is just the Gauss curvature at ξ when n = 2.) Differentiating in ξ for fixed R, we obtain:

(0.6) ∂_{ξ}(A(ξ, R)) = −R^{n−1} (∫_{S} [\frac{1}{6}∇_{u}Ric (ξ, u)u^{i}R^{2} + O(R^{3})] du) e_{i}

= −\frac{2ω_{n−1}}{3(n + 2)}R^{n+1}∇^{M}Scal(ξ) + ⋯,

by the calculation in [14] (ω_{n−1} = vol (S^{n−1})).

Consider the space of embeddings H of S^{n−1} into M, endowed with the Hilbert manifold structure defined by the L^{2} inner product on T_{X}H = {vector fields along X}. From (0.2), we see that for the perimeter functional A(X):

grad_{H}A(X) = HN̂,

and for the enclosed volume functional V (X):

grad_{H}V (X) = N̂.

Denoting by H_{0} the ‘submanifold’ of embeddings enclosing a fixed volume, it follows formally that:

grad_{H_{0}}A(X) = (H − H^{Σ})N̂,

so that the evolution law (0.1) may be written as:

\frac{∂X}{∂t} = −grad_{H_{0}}A(X). (0.7)

Consider now an approximate solution given by moving (parametrized) geodesic spheres X_{ξ(t),R(t)}. The volume preservation condition implies R(t) is approximately constant, so the projection of ∂_{R}X on T_{X}H_{0} is small, and may be ignored to lowest order. Thus, to lowest order, Equation (0.7) takes the form:

ξ_{t}∂_{ξ}X ∼ −grad_{H_{0}}A(X).

Denoting by ξ_{t}^{T} (u) = ξ_{t} − ⟨ξ_{t}, u⟩u the ‘tangential ’ component of ξ_{t}, we have:

⟨ξ_{t}∂_{ξ}X, ∂_{ξ}X⟩_{H_{0}} ∼ R^{n−1} ∫_{S} ξ_{t}^{T} (u)du.

With ⟨grad_{H_{0}}A(X), ∂_{ξ}X⟩_{H_{0}} = ∂_{ξ}A(ξ, R), this implies:

ω_{n−1}ave_{S}[ξ_{t}^{T}]R^{n−1} ∼ −∂_{ξ}A(ξ, R).

Noting that ξ_{t} = n ave_{S}[ξ_{t}^{T}], this combines with (0.6) to yield:

ξ_{t} ∼ \frac{2n}{3(n + 2)}R^{2}∇^{M}Scal(ξ) + ⋯,

as claimed in Part (iii) of the main theorem.

This formal argument also suggests that our main result enjoys a certain universality, and is probably valid for a large class of geometric evolution laws that preserve the enclosed volume and reduce perimeter, and may be realized as the gradient of the perimeter functional in an appropriate Hilbert manifold H. For the flow considered in this paper, the asymptotic behavior of ‘quasi-spherical’ solutions is controlled by the finite-dimensional gradient flow:

ξ_{t} = c_{n}∇^{M}Scal (ξ).

The scalar curvature increases along solutions; ξ(t) ‘climbs’ towards peaks of maximal scalar curvature. The ω-limit sets of orbits consist of critical points of Scal, with stable equilibria corresponding to local maxima. If all critical points of Scal are nondegenerate (or if M is real-analytic), solutions converge to a critical point as t → ∞, and most orbits (in a topological sense) converge to stable equilibria. This behavior is reflected in the claim of Part (iv) in the main theorem.

Acknowledgements. N.D.A. wants to thank Giorgio Fusco for several discussions regarding this work in its early stages. N.D.A. also acknowledges the hospitality and support of two institutions he visited during the preparation of this paper: The Mathematics Department of Brigham Young University, and the Max Planck Institute for Mathematics in the Sciences in Leipzig.

1. The manifold of small bubbles and the barycentric system

1.1 Existence of the analytic barycenter

Let M be a compact smooth oriented n-dimensional Riemannian manifold without boundary, S ⊂ ℝ^{n} the unit sphere. In this paper we consider the motion by normalized mean curvature flow in the manifold E^{2+α} of ‘small, strictly convex, C^{2+α} embeddings’ X, defined by the conditions:

(i) X : S → M is an embedding of class C^{2+α};

(ii) The image Σ = im (X) lies in a geodesically convex neighborhood of any point in int (X), the ‘small’ open subset of M bounded by Σ;

(iii) Σ = im (X) is a strictly convex hypersurface in M; in particular (given (ii)), int (X) is a geodesically convex subset of M.

E^{2+α} is an open subset in the Banach manifold of C^{2+α} embeddings S → M.

Any ξ ∈ int (X) determines a C^{2+α} function φ_{ξ} : S → ℝ^{+} and a C^{2+α} diffeomorphism F (·, ξ) : S → S_{ξ} (the unit sphere in the tangent space T_{ξ}M) such that Σ = im(X) admits the parametrization:

X_{ξ}(u) = exp_{ξ}[φ_{ξ}(u)F (u, ξ)].

Conversely (by compactness of M), there exists a constant δ_{0} > 0 depending only on M so that, for any ξ ∈ M, F (·, ξ) ∈ Diff^{2+α}(S; S_{ξ}) and φ_{ξ} ∈ C^{2+α}(S; ℝ^{+}) such that ∥φ_{ξ}∥_{C^{0}} < δ_{0}, the embedding X_{ξ} defined by this formula satisfies (i) and (ii). Moreover, there exists ϵ_{0} ∈ (0, 1/10) depending only on M so that if we write φ_{ξ} = R_{ξ}(1 + ψ_{ξ}) with 0 < R_{ξ} < δ_{0}, ∥ψ_{ξ}∥_{C^{2+α}} < ϵ_{0} and ∫_{S} ψ_{ξ} = 0, then (iii) also holds and X_{ξ} ∈ E^{2+α}

The set of C^{2+α} embeddings S → M which can be written in this form is an open subset of E^{2+α}, which we will denote by E_{δ_{0},ϵ_{0}}^{2+α}. From now on we let E = E^{2+α} = E_{δ_{0},ϵ_{0}}^{2+α}.

Consider a motion Σ(t) of hypersurfaces by normalized mean curvature flow, parametrized by X(t) ∈ E^{2+α}. To follow the motion globally on M, we need a notion of ‘barycenter’ of Σ(t). While a geometric notion of barycenter of sets in Riemannian manifolds exists (see [10]), we will need a different one; our ‘analytic barycenter’ will be defined only for small convex hypersurfaces which are sufficiently C^{2+α} close to a geodesic sphere.

We denote by E_{0} the manifold of standard parametrizations of small geodesic spheres:

E_{0} = {X : S → M; X(u) = exp_{ξ}[Ru^{i}e_{i}], ξ ∈ M, R ∈ [0, δ_{0}/2], e ∈ F_{ξ}M},

where F_{ξ}M denotes the space of oriented orthonormal frames of T_{ξ}M. (We do not exclude embeddings which degenerate to a point.) E_{0} is a submanifold of E^{2+α}, diffeomorphic to the orthonormal frame bundle of M cross an interval, 𝔽M × [0, δ_{0}/2]. We will sometimes identify X ∈ E_{0} with the ordered triple (ξ, R, e), and identify the frame e with the isometry e(·, ξ) : S → S_{ξ} it defines via e(u, ξ) = u^{i}e_{i} (written e(u) if ξ is understood).

If X̄ = (ξ̄, R̄, ē) ∈ E_{0}, a basis of neighborhoods of X̄ in E^{2+α} is given by sets of the form:

N_{δ,ϵ}^{2+α}(X̄) = {X ∈ E^{2+α}; ξ̄ ∈ int (X),

∥ψξ̄∥_{C^{2+α}} < ϵ, |R̄ − Rξ̄| < δ, ∥F (·, ξ̄) − e(·, ξ̄)∥_{C^{2+α}} < ϵ}.

For X in a neighborhood N_{δ_{M} ,ϵ_{M}}^{2+α} of E_{0} in E^{2+α}, we wish to define a unique ‘analytic barycenter’. Let U^{2+α} ⊂ E^{2+α} × M be the open subset defined by:

U^{2+α} = {(X, ξ); ξ ∈ int (X)}.

We define a smooth map P : U^{2+α} → ℝ^{n} by:

P^{i}(X, ξ) = ∫_{S} φ_{ξ}(u)u^{i}du = ∫_{S} d_{ξ}(X(u))u^{i}du, i = 1, …, n,

where d_{ξ}(x) = dist_{M}(ξ, x).

Lemma 1.1. There exist δ_{M} ∈ (0, δ_{0}/2), ϵ_{M} ∈ (0, ϵ_{0}/2) such that, defining:

N_{δ_{M} ,ϵ_{M}}^{2+α} = {X ∈ E^{α}; ∃X̄ = (ξ̄, R̄, e) ∈ E_{0} with X ∈ N_{δ_{M} ,ϵ_{M}}^{2+α} (X̄)} ,

one has a smooth map B : N_{δ_{M} ,ϵ_{M}}^{2+α} → M with the properties:

(i) P(X, B(X)) = 0, X ∈ N_{δ_{M} ,ϵ_{M}}^{2+α}.

(ii) B(X) is the only solution of P(X, ·) = 0 in the ball with center B(X), radius ϵ_{M}.

Proof.

Step (i). In the first step we work in E^{2+α′}, for a fixed 0 < α′ < α. Fix X̄ ∈ E_{0} parametrizing SR̄(ξ̄). Clearly P(X̄, ξ̄) = 0. By the Implicit Function Theorem (in the Banach manifold E^{2+α′}), to solve the problem locally near (X̄, ξ̄), it is enough to show that the partial differential (D_{ξ}P)(X̄, ξ̄) : Tξ̄M → ℝ^{n} is an isomorphism.

Writing X̄(u) = expξ̄[R̄e(u, ξ̄)] as above, this follows from the calculation:

(D_{ξ}P^{i})(X̄, ξ̄)v = ∫_{S} ⟨(∇d_{X̄(u)}) (ξ̄), v⟩ u^{i}du

= − ∫_{S}⟨e(u, ξ̄), v⟩u^{i}du

= − ∫_{S} u^{j}v^{j}u^{i}du = −\frac{vol (S)}{n}v^{i},

for each v ∈ Tξ̄M, where v = v^{j}e_{j}.

Thus we obtain a neighborhood of (X̄, ξ̄) in U^{2+α′} (which we may assume to have the form N_{δ̄,ϵ̄}^{2+α′}(X̄) × B_{ϵ̄/2}(ξ̄), with ϵ̄ < ϵ_{0}/2 < 0.1, δ < δ_{0}/2 < 0.1), and a smooth map

BX̄ : N_{δ̄,ϵ̄}^{2+α′}(X̄) → B_{ϵ̄/2}(ξ̄)

such that P(X, BX̄) = 0. Furthermore, by reducing ϵ̄ if needed we may arrange that, for each X ∈ N_{δ̄,ϵ̄}^{2+α′}(X̄), BX̄(X) is the only solution of P(X, ·) = 0 in Bϵ̄(ξ̄).

Step (ii). To define B globally, we use a standard covering argument. By compactness of E_{0} we find X̄_{i} = (ξ̄_{i}, R̄_{i}, ē(·, ξ̄_{i})), i = 1, …, N, and δ̄ ∈ (0, δ_{0}/2)ϵ̄_{i} ∈ (0, ϵ_{0}/2) so that:

E_{0} = ⋃_{i=1}^{N} N_{δ̄_{i}/L,ϵ̄_{i}/L}^{2+α′}(X̄_{i}) ∩ E_{0}

and maps B_{i} = BX̄_{i} : N_{δ̄_{i},ϵ̄_{i}}^{2+α′}(X̄_{i}) → B_{ϵ̄_{i}/2}(ξ̄_{i}) as in Step (i). Here L > 0 is defined by L^{−1} = min{1/10, (10C_{0})^{−1}}, with C_{0} > 0 a constant depending only on M, defined below, in Step (iii) of this proof. We need to check compatibility: B_{i}(X) = B_{j}(X) if X ∈ N_{δ̄_{i},ϵ̄_{i}}^{2+α′}(X̄_{i}) ∩ N_{δ̄_{j},ϵ̄_{j}}^{2+α′}(X̄_{j}). It is easy to show that:

max\{d(ξ_{i}, ξ_{j}), |R̄_{i} − R̄_{j}|\} ≤ ϵ̄_{i}R̄_{i} + ϵ̄_{j}R̄_{j} < \frac{1}{10}(ϵ̄_{i} + ϵ̄_{j}),

since max_{i}{R̄_{i}} < i_{M}/2 < 1/10. In particular (choosing the indices so that ϵ̄_{i} < ϵ̄_{j}):

B_{i}(X) ∈ B_{\frac{ϵ̄_{i}}{2}} (ξ̄_{i}) ⊂ B_{\frac{ϵ̄_{i}}{2} + \frac{1}{10} (ϵ̄_{i}+ϵ̄_{j})}(ξ̄_{j}) ⊂ Bϵ̄_{j}(ξ̄_{j}).

By the uniqueness in Step (i), we must have B_{i}(X) = B_{j}(X). Thus, letting:

W^{2+α′} = ⋃_{i=1}^{N} N_{δ̄_{i},ϵ̄_{i}}^{2+α′}(X̄_{i}) ⊂ E^{2+α′},

we have a smooth map:

B : W^{2+α′} → M, B(X) ∈ int (X),

such that P(X, B(X)) = 0. Furthermore, letting ϵ_{min} = \frac{1}{4} min\{ϵ̄_{i}; 1 ≤ i ≤ N}, B(X) is the unique solution to P(X, ξ) = 0 in the ball B_{ϵ_{min}}(B(X)).

Step (iii). We still need to show that, for some ϵ_{M} > 0 depending only on M and {ϵ̄_{1}, …, ϵ̄_{N}}, W^{2+α′} contains a subset N_{δ_{M} ,ϵ_{M}}^{2+α} as in the statement of the lemma. That is, assuming X ∈ E^{2+α} satisfies, for some (ξ̄, R̄, ē) ∈ E_{0}:

ξ̄ ∈ int (X), |Rξ̄ − R̄| < δ_{M},
∥φξ̄ − R̄∥_{C^{2+α}} < ϵ_{M} R̄, ∥F (·, ξ̄) − ē(·, ξ̄)∥_{C^{2+α}} < ϵ_{M},

we must show that for some i ∈ {1, …, N}:

ξ̄_{i} ∈ int (X), |Rξ̄_{i} − R̄_{i}| < δ̄_{i},
∥φξ̄_{i} − R̄_{i}∥_{C^{2+α′}} < ϵ̄_{i}R̄_{i}, ∥F (·, ξ̄) − ē(·, ξ̄_{i})∥_{C^{2+α′}} < ϵ̄_{i}.

For some i (say i = 1) we have:

|d(x, ξ̄_{1}) − R̄_{1}| ≤ \frac{ϵ̄_{1}}{L} R̄_{1} ≤ \frac{ϵ̄_{1}}{10}R̄_{1} ∀x ∈ SR̄(ξ̄),

∥e(·, ξ̄) − ē(·, ξ̄_{1})∥_{C^{2+α′}} ≤ \frac{ϵ̄_{1}}{L} ≤ \frac{ϵ̄_{1}}{10}.

From this it is not hard to show that:

d(ξ̄, ξ̄_{1}) ≤ \frac{ϵ̄_{1}}{L} R̄_{1} ≤ \frac{ϵ̄_{1}}{10}R̄_{1} and |R̄ − R̄_{1}| ≤ \frac{2ϵ̄_{1}}{L} R̄_{1} ≤ \frac{ϵ̄_{1}}{5} R̄_{1},

which easily implies |Rξ̄− R̄_{1}| < δ̄_{1}, if δ_{M} is chosen small enough. Since d(X(u), ξ̄) ≥ R̄(1 − ϵ_{M}) for all u ∈ S, we have ξ̄_{1} ∈ int(X) provided R̄(1 − ϵ_{M}) > (ϵ̄_{1}/10)R̄_{1}, which holds if ϵ_{M} < 1 − ϵ̄_{1}/(10 − 2ϵ̄_{1}). We also have the estimate:

∥F (·, ξ̄) − ē(·, ξ̄_{1})∥_{C^{2+α′}}
≤ ∥F (·, ξ̄) − e(·, ξ̄)∥_{C^{2+α′}} + ∥e(·, ξ̄) − ē(·, ξ̄_{1})∥_{C^{2+α′}}
≤ ϵ_{M} + ϵ̄_{1}/10 < ϵ̄_{1}.

To show ∥φξ̄ − R̄_{1}∥_{C^{2+α′}} is small, we use:

∥φξ̄_{1} − R̄_{1}∥_{C^{2+α′}} < |R̄_{1} − R̄| + ∥R̄ − φξ̄∥_{C^{2+α′}} + ∥φξ̄ − φξ̄_{1}∥_{C^{2+α′}}

and the following fact:

C_{0} := sup {∥F (·, ξ)∥_{C^{2+α′}}; X ∈ E^{2+α} and

∃X̄ = (ξ̄, R̄, ē) s.t. X ∈ N_{1/10,1/10}^{2+α}(X̄), d(ξ, ξ̄) < \frac{1}{4}R̄\} < ∞,

and C_{0} depends only on M. This follows from compactness of the inclusion C^{2+α} ⊂ C^{2+α′}, by a short argument which will be omitted. Given this bound, one has:

∥φξ̄ − φξ̄_{1}∥_{C^{2+α′}} = ∥∫_{0}^{t} \frac{d}{ds}dist_{M}(X(·), ξ(s))ds∥_{C^{2+α′}}

≤ ∥∫_{0}^{t}⟨∇_{M}d_{X(u)}(ξ(s)), ξ′(s)⟩ds∥_{C^{2+α′}}

≤ sup_{s∈[0,t]} ∥∇_{M}d_{X(u)}(ξ(s))∥_{C^{2+α′}} d(ξ̄, ξ̄_{1})

= sup_{s∈[0,t]} ∥F (·, ξ(s))∥_{C^{2+α′}} d(ξ̄, ξ̄_{1})

≤ C_{0} \frac{ϵ̄_{1}}{L} R̄_{1} ≤ \frac{ϵ̄_{1}}{10}R̄_{1},

since d(ξ̄, ξ̄_{1}) ≤ \frac{ϵ̄_{1}}{L} R̄_{1} ≤ \frac{1}{4} R̄. Thus we have (bearing in mind that R̄ ≤ R̄_{1} + (ϵ̄_{1}/5)R̄_{1}):

∥φξ̄_{1} − R̄_{1}∥_{C^{2+α′}} < \frac{ϵ_{1}}{5} R̄_{1} + ϵ_{M} R̄ + \frac{ϵ̄_{1}}{10}R̄_{1} < ϵ̄_{1}R̄_{1}

for ϵ_{M} sufficiently small, depending only on {ϵ̄_{1}, …, ϵ̄_{N}}. This concludes the proof.

Since α is fixed throughout the paper, from now on we will often omit the superscript in the notation for E^{2+α} and N_{δ_{M} ,ϵ_{M}}^{2+α}.

Definition. For X ∈ N_{δ_{M} ,ϵ_{M}}, we refer to ξ_{X} = B(X) ∈ int (X) as the analytic barycenter of X. The global motion we construct will exist in the open subset N_{δ_{M} ,ϵ_{M}} ⊂ E_{δ_{0},ϵ_{0}}, the manifold of small, almostspherical embeddings, or parametrized ‘bubbles’.

It is important to note that the analytic barycenter B(X) depends on the parametrization X, and not just on the image Σ. In particular, if we wish to find an equation of motion for the barycenter, we must fix an evolution equation for the parametrization X(t); the geometric evolution law (0.1), however, does not fix such an equation (only up to a tangential component). We take up these issues in the next subsection.

1.2 Equations of motion for the barycentric system

From this point on we set δ_{1} = δ_{M}, ϵ_{1} = ϵ_{M} and introduce the notations:

C_{δ_{1},ϵ_{1}}^{2+α} = {φ ∈ C^{2+α}(S); 0 < ave_{S}[φ] < δ_{1}, ∥φ − ave_{S}[φ]∥_{C^{2+α}} < ϵ_{1}};

C_{0}^{k+γ} = {φ ∈ C^{k+γ}(S); ave_{S}[φu^{i}] = 0, i = 1, …, n} (k ∈ ℕ, γ ∈ (0, 1));

B_{δ_{1},ϵ_{1}}^{2+α} = C_{δ_{1},ϵ_{1}}^{2+α} ∩ C_{0}^{2+α}.

For parametrized motions X̄(t) in N_{δ_{1},ϵ_{1}}, the barycenter map makes it possible to choose a parametrization X(t) of Σ(t) = im (X̄(t)) in the space (a submanifold of N):

N_{std} = N_{std}(δ_{1}, ϵ_{1}) = {X ∈ N; X = exp_{ξ}[φe(·, ξ)], φ ∈ B_{δ_{1},ϵ_{1}}^{2+α}}.

X ∈ N_{std} is ‘standard’ in two ways: ξ is the barycenter of X and F_{ξ} : S → S_{ξ} is given by an isometry. Any X ∈ N can be reparametrized into N_{std} (with ξ = B(X)). In this subsection we derive equations of motion on a ‘simpler’ manifold M_{0}, the solutions of which correspond precisely to solutions of normalized mean curvature flow parametrized by maps in N_{std}.

Definition 1.2.

M = M(δ_{1}, ϵ_{1}) = 𝔽M × C_{δ_{1},ϵ_{1}}^{2+α}, M_{0} = M_{0}(δ_{1}, ϵ_{1}) = 𝔽M × B_{δ_{1},ϵ_{1}}^{2+α}.

The standard connection on the frame bundle 𝔽M allows one to identify the tangent space to M_{0} at (ξ, e, φ) ∈ M_{0} with the vector space:

T_{(ξ,e,φ)}M_{0} = {(w, z, χ);

w ∈ T_{ξ}M, z ∈ C^{2+α}(S, T_{ξ}M) e-skew-symmetric , χ ∈ C_{0}^{2+α}}

(where e-skew-symmetric means: ⟨z(u), e(u′)⟩ = −⟨z(u′), e(u)⟩, for all u, u′ ∈ S). There is a natural smooth map Φ : M → N, mapping M_{0} onto N_{std}:

Φ(ξ, e, φ) = X, X(u) = exp_{ξ}[φ(u)e(u)].

In the following lemma we compute the differential of Φ. This of course maps T_{(ξ,e,φ)}M_{0} to the tangent space T_{X}N, the space of C^{2+α} vector fields ℤ on S along X (ℤ(u) ∈ T_{X(u)}M); but we’ll also need to consider the space of C^{α} vector fields along X:

T̃_{X}N := {ℤ : S → T M;

ℤ(u) ∈ T_{X(u)}M, u ↦ exp_{ξ}^{−1}[ℤ(u)] ∈ C^{α}(S, T_{ξ}M) ∀ξ ∈ int (X)},

as well as T̃_{(ξ,e,φ)}M_{0}, defined as T_{(ξ,e,φ)}M_{0} but with z and χ in C^{α}.

In the statement of the next lemma, for v ∈ S_{ξ} 𝕐_{v}(s), s ≥ 0, denotes the Jacobi operator with initial conditions 𝕐_{v}(0) = 𝕀_{T_{ξ}M}, 𝕐̇_{v}(0) = 0 (covariant derivative) along the geodesic γ_{v}(s) = exp_{ξ}[sv]; so 𝕐_{v}(s)w ∈ T_{γ_{v}(s)}M is the value at s of the Jacobi field along γ_{v} with initial value w, initial covariant derivative 0. 𝕁_{v}(s) is the Jacobi operator along the same geodesic with initial conditions 𝕁_{v}(0) = 0, 𝕁̇_{v}(0) = 𝕀_{T_{ξ}M}. Both 𝕐_{v}(s) and 𝕁_{v}(s) preserve the direction spanned by the tangent vector γ′_{v}(s) (and its orthogonal complement) along the geodesic γ_{v}. We have 𝕐_{v}(s)v = γ′_{v}(s), 𝕁_{v}(s)v = 0.

Lemma 1.3. The differential of Φ is given by:

dΦ(ξ, e, φ)[w, z, χ] = 𝕐_{e}(φ)w + χγ′_{e}(φ) + 𝕁_{e}(φ)z.

Here we identify the orthonormal frame e at ξ with the isometry e : S → S_{ξ} it defines, and for simplicity omit u ∈ S from the notation.

Proof. Since Φ(ξ, e, φ) = exp_{ξ}[φe],we have:

dΦ(ξ, e, φ)[w, z, χ] = d_{ξ}(exp_{ξ}[sv])_{|s=φ,v=e}w

+ d_{s}(exp_{ξ}[sv])_{|s=φ,v=e}χ + d_{v}(exp_{ξ}[sv])_{|s=φ,v=e}z.

Since d_{ξ}(exp_{ξ}[sv])w = 𝕐_{v}(s)w, d_{s}(exp_{ξ}[sv])= γ′_{v}(s) and d_{v}(exp_{ξ}[sv])z = 𝕁_{v}(s)z, the lemma follows.

The operators 𝕐_{u}(s) are invertible for s sufficiently small (say, s ∈ (0, δ_{0}/2)). It is often convenient to work with vectors in T_{ξ}M (or maps S → T_{ξ}M); so in general for vectors ℤ(u) ∈ T_{X(u)}M (where X ∈ N_{std}, X = exp_{ξ}[φe]), we set:

z(u) = 𝕐_{e(u)}(φ)^{−1}ℤ(u) ∈ T_{ξ}M.

For a given frame e : S → S_{ξ} and z ∈ T_{ξ}M (possibly depending on u ∈ S), we introduce the orthogonal decomposition:

z = ⟨z, e⟩e + z^{⊥}, z^{⊥} : S → T_{ξ}M.

As a final bit of notation, we denote by ν(u) ∈ T_{ξ}M the vector defined by the property:

⟨w, ν(u)⟩_{=}⟨𝕐_{e(u)}(φ)w, N(u)⟩ ∀w : S → T_{ξ}M,

where N(u) is the normal vector to Σ defined from X by:

N(u) = γ′_{e(u)}(φ) − φ_{k}(h^{kl}J_{l}) ∘ X,

with J_{i} the Jacobi fields defined in Section 2, (2.0). Since ⟨γ′_{e(u)}(φ), N⟩ = 1, it follows that ⟨e, ν⟩ ≡ 1 on S.

We now observe a geometric constraint on the velocity vector of a curve in N_{std}.

Lemma 1.4. Let X(t), t ∈ [0, T) be a curve in N_{std}, X(t) = Φ(ξ(t), e(t), φ(t)). Assume ξ(t) is a C^{1} curve on M, e(t) is parallel along ξ(t) and φ_{t} ∈ C([0, T), C_{0}^{α}(S)) satisfies ave_{S}[φ_{t}u^{i}] = 0, i = 1, …, n. Then the velocity vector ℤ(t) = X_{t} ∈ T̃_{X(t)}N satisfies:

z^{⊥} = (ave_{S}[z])^{⊥}.

(Note both sides of this equation depend on t and on u.)

Proof. From Lemma 1.3, we have:

ℤ(t) = X_{t} = dΦ(ξ(t), e(t), φ(t))(ξ_{t}, ∇_{ξ_{t}}e, φ_{t})
= 𝕐_{e}(φ)ξ_{t} + φ_{t}γ′_{e}(φ),

since ∇_{ξ_{t}}e ≡ 0. Pulling back to T_{ξ}M via 𝕐_{e}(φ)^{−1}, this gives:

z(t) = ξ_{t} + φ_{t}e.

Thus z^{⊥} = ξ_{t}^{⊥}. Since ave_{S}[φ_{t}e] = ave_{S}[φ_{t}u^{i}]e_{i} = 0 we obtain (ave_{S}[z])^{⊥} = (ave_{S}[ξ_{t}])^{⊥} = ξ_{t}^{⊥}, proving the claim.

This lemma motivates the definition (for X = Φ(ξ, e, φ) ∈ N_{std}):

T̃_{X}N_{std} = {ℤ ∈ T̃_{X}N; z^{⊥} = (ave_{S}z)^{⊥}, z = 𝕐_{e}^{−1}(φ)ℤ}.

Lemmas 1.3 and 1.4 show dΦ maps the subspace of T̃_{(ξ,e,φ)}M_{0} defined by z = 0 into T̃_{X}N_{std}.

We face the following problem: if we attempt to solve the parametrized normalized mean curvature flow in N_{std} by setting X_{t} = ℤ(X) = (H^{Σ} − H)N̂ ∈ T̃_{X}N, in general we should not expect that ℤ(X) ∈ T̃_{X}N_{std}. Thus we will not find X(t) in N_{std} solving X_{t} = (H^{Σ} − H)N̂.

Fortunately for ‘radial vector fields’ ℤ(X) ∈ T̃_{X}N it is possible to correct this by adding a ‘tangential component’. Precisely, given X ∈ N_{std}, X = exp_{ξ}[φe], define the linear map:

S : T_{ξ}M → T_{ξ}M, Sw := n(w − ave_{S}[w^{T}]), w^{T} := w − ⟨w, ν⟩e.

The notation w^{T} is justified by the fact that 𝕐_{e}(φ)w^{T} is tangential to Σ: with 𝕎 = 𝕐_{e}(φ)w,

⟨𝕐_{e}(φ)w^{T} , N⟩ = ⟨𝕎, N⟩ − ⟨w, ν⟩⟨γ′_{e}(φ), N⟩ = 0,

since ⟨w, ν⟩ = ⟨𝕎, N⟩ and ⟨γ′_{e}(φ), N⟩ = 1.

The map S is invertible, provided δ_{1}, ϵ_{1} are small enough: since w^{T} = w^{⊥} − ⟨w, ν − e⟩e and ave_{S}[w^{⊥}] = \frac{n−1}{n}w for w ∈ T_{ξ}M, we have:

Sw = w + ave_{S}[n⟨w, ν − e⟩e].

Since ⟨w, ν − e⟩ = ⟨𝕐_{e}(φ)w, N − γ′_{e}(φ)⟩ and:

∥𝕐_{e}(φ)w∥∥γ′_{e}(φ) − N∥ < (1/2n)∥w∥

for all w ∈ T_{ξ}M (for sufficiently small ϵ_{1}, δ_{1}), it follows that S = 𝕀 + E, with E ∈ L(T_{ξ}M), ∥E∥ < 1/2.

We use S to define for X ∈ N_{std} a linear map P_{X} on vector fields ℤ ∈ T̃_{X}N of the form:

ℤ = aγ′_{e}(φ), a ∈ C^{α}(S),

by setting:

P_{X}(ℤ) = ℤ + 𝕋,
𝕋 = 𝕐_{e}(φ)w − ⟨𝕐_{e}(φ)w, N⟩γ′_{e}(φ),
w = nS^{−1}(ave_{S}[z]), z = ae ∈ C^{α}(S, T_{ξ}M).

Lemma 1.5. Let X ∈ N_{std}, X = exp_{ξ}[φe]. Let ℤ = aγ′_{e}(φ) ∈ T̃_{X}N. Then:

(i) ℤ̃ := P_{X}(ℤ) = ℤ + 𝕋 ∈ T̃_{X}N_{std}.

(ii) Conversely, assume ℤ̃ = ℤ + 𝕋 ∈ T̃_{X}N_{std} (with 𝕋(u) ∈ T_{X(u)}Σ)

has the form:

ℤ̃ = 𝕐_{e}(φ)w + χγ′_{e}(φ),

for some w ∈ T_{ξ}M and χ ∈ C_{0}^{α}. Then w = nS^{−1}(ave_{S}[z]) (with z = ae), χ = ⟨z, e⟩ − ⟨w, ν⟩ and ℤ̃ = P_{X}(ℤ).

Proof.

(i) Let z̃ = z + w − ⟨w, ν⟩e = z + w^{T}. Then z^{⊥} = w^{⊥} and:

ave_{S}[z̃] = ave_{S}[z] + ave_{S}[w^{T}] = ave_{S}[z] − \frac{1}{n}Sw + w = w,

since (1/n)Sw = ave_{S}[z]. Thus (ave_{S}[z̃])^{⊥} = z^{⊥}, showing ℤ̃ ∈ T̃_{X}N_{std}. (ii) We have:

w + χe = z̃ = z + t,

where t := 𝕐_{e}(φ)^{−1}𝕋, ave_{S}[χe] = ave_{S}[χu^{i}]e_{i} = 0 and z = ae,

and must show:

t = w − ⟨w, ν⟩e, χ = ⟨z, e⟩ − ⟨w, ν⟩ and
n(w − ave_{S}[w^{T}]) = nave_{S}[z].

The observation that ⟨t, ν⟩ = ⟨𝕋, N⟩ = 0 implies ⟨w, ν⟩ + χ = a = ⟨z, e⟩ and the first equality: t = w + χe − ae = w − ⟨w, ν⟩e = w^{T}. But then:

w − ave_{S}[w^{T}] = ave_{S}[w − t] = ave_{S}[z − χe] = ave_{S}[z],

as desired.

We use this lemma to define a motion on M_{0} which corresponds to parametrized solutions of normalized mean curvature flow in N_{std}.

Definition 1.6. The barycentric system on M_{0} is defined by the equations of motion:

(1.1)  ξ_{t} = n ave_{S}[(H^{Σ} − H)∥N∥e] − n ave_{S}[⟨w, ν − e⟩e]
∇_{ξ_{t}}e = 0
φ_{t} = (H^{Σ} − H)∥N∥ − n⟨ave_{S}[(H^{Σ} − H)∥N∥e], e⟩
− ⟨w, ν − e⟩ + n⟨ave_{S}[⟨w, ν − e⟩e], e⟩,

where:

w = nS^{−1}(ave_{S}[(H^{Σ} − H)∥N∥e]), with S := 𝕀 + n ave_{S}[⟨., ν − e⟩e] ∈ L(T_{ξ}M).

(Here H^{Σ} − H, N, ν, are computed at the embedding Φ(ξ, e, φ).)

Lemma 1.7.

(i) Let (ξ(t), e(t), φ(t)) ∈ M_{0} be a solution of the barycentric system (1.1) in [0, T). Then X(t) = Φ(ξ(t), e(t), φ(t)) ∈ N_{std} is a solution of ⟨X_{t}, N̂⟩ = H^{Σ} − H in [0, T).

(ii) Conversely, let X(t) ∈ N be a solution of the geometric equation ⟨X_{t}, N̂⟩ = H^{Σ} − H in [0, T), with X(0) ∈ N_{std}. Then by reparametrizing X(t) we may obtain X̄(t) ∈ N_{std}, solution of:

X̄_{t} = ℤ(X̄) + 𝕋_{1}, ℤ(X̄) := (H^{Σ} − H)∥N∥γ′_{e}(φ),

where ℤ(X̄) + 𝕋_{1} = PX̄(ℤ(X̄)) ∈ T̃X̄N_{std}. In particular, writing X̄(t) = Φ(ξ(t), e(t), φ̄(t)) with ξ = B(X̄) and e(t) parallel along ξ(t), we obtain a solution of the barycentric system (1.1) in M_{0}.

Remarks.

(i) ν − e ≡ 0 for geodesic spheres, so the terms containing ν − e will be treated as ‘error terms’.

(ii) The factors of n in the equation for φ_{t} in (1.1) can be understood as follows: if f satisfies f_{t} = f −n⟨ave_{S}[fe], e⟩ = f −nave_{S}[fu^{i}]u^{i}, we have:

(ave_{S}[fu^{j}])_{t} = ave_{S}[f_{t}u^{j}]
= ave_{S}[fu^{j}] − nave_{S}[fu^{i}]ave_{S}[u^{i}u^{j}] = 0,

since ave_{S}[u^{i}u^{j}] = δ_{ij}/n. Thus the condition ave_{S}[fu^{j}] = 0 is preserved.

Proof. Throughout the proof we let v_{N}(X) = (H^{Σ}−H)∥N∥, ℍ(X) = (H^{Σ} − H)N̂.

(i) Let z = v_{N}e, z̃ = z + w − ⟨w, ν⟩e. Since ⟨z̃, e⟩ = v_{N} − ⟨w, ν − e⟩, the barycentric system can be written as:

ξ_{t} = n ave_{S}[⟨z̃, e⟩e], φ_{t} = ⟨z̃, e⟩ − n⟨ave_{S}[⟨z̃, e⟩e], e⟩.

As shown in Lemma 1.5(i), z̃ satisfies z^{⊥} = (ave_{S}[z̃])^{⊥}. This implies:

ave_{S}[z^{⊥}] = ave_{S}[(ave_{S}[z̃])^{⊥}]
= \frac{n − 1}{n}ave_{S}[z̃] and ave_{S}[⟨z̃, e⟩e] = (1/n)ave_{S}[z̃].

Thus, we have:

ξ_{t} + φ_{t}e = n ave_{S}[⟨z̃, e⟩e] + ⟨z̃, e⟩e − n⟨ave_{S}[⟨z̃, e⟩e], e⟩e
= n(ave_{S}[⟨z̃, e⟩e])^{⊥} + ⟨z̃, e⟩e = (ave_{S}[z̃])^{⊥} + ⟨z̃, e⟩e
= z^{⊥} + ⟨z̃, e⟩e = z̃.

Thus X(t) satisfies the equation:

X_{t} = 𝕐_{e}(φ)(ξ_{t} + φ_{t}e) = 𝕐_{e}(φ)z̃
= v_{N}γ′_{e}(φ) + 𝕋 ( with 𝕋 = 𝕐_{e}(φ)(w − ⟨w, ν⟩e))
= (H^{Σ} − H)N̂ + 𝕋_{1},

where 𝕋_{1} = 𝕋 + (H^{Σ} − H)∥N∥γ′_{e}(φ) − (H^{Σ} − H)N̂ satisfies ⟨𝕋_{1}, N⟩ ≡ 0.

Remark. Note that the condition z^{⊥} = (ave_{S}[z̃])^{⊥} is used essentially in the proof.

(ii) X(t) satisfies, for some tangential vector 𝕋(t, u):

X_{t} = ℍ(X) + 𝕋.

Leting ξ(t) = B(X(t)) and e(t) be the parallel transport of the frame e(0) along ξ(t), we may reparametrize Σ in the form:

X̄(t, ū) = exp_{ξ(t)}[φ(t, ū)e(t, ū)],

via a diffeomorphism G(t, ·) ∈Diff^{2+α}(S), so u = G(t, ū) and X̄(t, ū) = X(t, G(t, ū)). For the velocity vector:

X̄_{t}(t, ū) = X_{t}(t, G(t, ū)) + X_{u}(t, G(t, ū))G_{t}(t, ū)
= ℍ(X̄) + 𝕋(t, G(t, ū)) + 𝕋̄(t, ū)
= v_{N}(X̄)γ′_{e}(φ) + 𝕋_{2}(t, ū)
= ℤ(X̄) + 𝕋_{2}(t, ū),

where:

𝕋_{2}(t, ū) = 𝕋(t, G(t, ū)) + 𝕋̄(t, ū) + ℍ(X̄)(t, ū)
− v_{N}(X̄)(t, ū)γ′_{e}(φ) ∈ T_{X̄(t,ū)}Σ

and we used the fact that ℍ is ‘geometric’:

ℍ(X)(t, G(t, ū)) = ℍ(X̄)(t, ū).

Since we also have:

X̄_{t} = 𝕐_{e}(φ)w + χγ′_{e}(φ) with w = ξ_{t} ∈ T_{ξ}M and χ = φ_{t} ∈ C_{0}^{2+α}(S),

we may apply Lemma 1.5(ii) to conclude ℤ(X̄) + 𝕋_{2} = PX̄(ℤ(X̄)) and:

ξ_{t} = nS^{−1}(ave_{S}[z]) with z = v_{N}(X̄)e, φ_{t} = v_{N}(X̄) − ⟨ξ_{t}, ν⟩.

From n ave_{S}[z] = Sξ_{t} = ξ_{t} + n ave_{S}[⟨w, ν − e⟩e] (where w = ξ_{t}), we obtain:

ξ_{t} = n ave_{S}[v(X̄)e] − n ave_{S}[⟨w, ν − e⟩e]
φ_{t} = v_{N}(X̄) − ⟨ξ_{t}, e⟩ − ⟨w, ν − e⟩,

which is the barycentric system.

Summary. For f ∈ C^{2+α}(S), denote by ^{∘} the L^{2} projection on the subspace C_{0}^{2+α} defined by ave_{S}[fu^{i}] = 0:

f^{∘} = f − n⟨ave_{S}[fe], e⟩ = f − ave_{S}[nfu^{i}]u^{i}.

Then the barycenter system has the structure:

ξ_{t} = n ave_{S}[(v_{N} − E)e] (1.2)

∇_{ξ_{t}}e = 0
φ_{t} = (v_{N} − E)^{∘};

here E = E(v_{N}) is uniquely defined so that:

z̃:= ξ_{t} + φ_{t}e = satisfies ⟨z̃, ν⟩ = v_{N},

or equivalently for X(t) = Φ(ξ(t), e(t), φ(t)) ∈ N_{std}:

X_{t} = 𝕐_{e}z̃ = v_{N}γ′_{e}(φ) + 𝕋 = (H^{Σ} − H)N̂ + 𝕋_{1}.

Since z̃ is a ‘tangential correction’ of v_{N}e: z̃ = v_{N}e + w − ⟨w, ν⟩e = v_{N}e + w^{⊥} − Ee, we see that ⟨z̃, e⟩ = v_{N} − E, and the system can be written in the alternative form:

ξ_{t} = ave_{S}[z̃]
φ_{t} = ⟨z̃, e⟩^{∘};

note ave_{S}[z̃] = n ave_{S}[⟨z̃, e⟩e]. From the point of view of the system on M_{0}, the reason for the correction is that we want to prescribe ⟨z̃, ν⟩, not ⟨z̃, e⟩. In fact we have:

⟨z̃, ν⟩ = v_{N} ⇔ ⟨z̃, e⟩ = v_{N} − E.

In the proof of global existence, we use a slightly different form of the barycentric system. Any 0 ≤ φ ∈ C_{0}^{2+α}(S) can be written in the form φ = R(1 + ψ), where R = ave_{S}[φ] ≥ 0 and ψ = (1/R)(φ − R) (ψ ≡ 0 if φ ≡ 0) is in the subspace K^{2+α} of C_{0}^{2+α}, where we denote:

K^{2+α} : = {ψ ∈ C_{0}^{2+α}(S); ave_{S}[ψ] = 0},
K_{ϵ_{1}}^{2+α} = {ψ ∈ K^{2+α}; ∥ψ∥_{C^{2+α}} < ϵ_{1}}.

Thus we may write:

M_{0} = M_{0}(δ_{1}, ϵ_{1}) = (0, δ_{1}) × 𝔽M × K_{ϵ_{1}}^{2+α},

and derive differential equations for R and ψ via:

φ_{t} = (1 + ψ)R_{t} + Rψ_{t}, R_{t} = ave_{S}[φ_{t}], Rψ_{t} = φ_{t} − (1 + ψ)ave_{S}[φ_{t}].

Denoting by f ↦ (f)_{K} the L^{2} projection C^{2+α} → K^{2+α}, the R, ψ equations may be written in the form:

R_{t} = ave_{S}[v_{N} − E]
Rψ_{t} = (v_{N} − E)_{K} − ψ ave_{S}[v_{N} − E]

Thus the equations of motion take the form:

R_{t} = ave_{S}[v_{N} − E] (1.3)

ξ_{t} = ave_{S}[n(v_{N} − E)e]
∇_{ξ_{t}}e = 0
Rψ_{t} = (v_{N} − E)_{K} − ψ ave_{S}[v_{N} − E].

2. Asymptotics in Riemannian normal coordinates

2.1 Second fundamental form and mean curvature

In this section we apply some classical expansions in Riemannian normal coordinates to obtain the leading terms in the barycentric system on M_{0}(δ_{0}, ϵ_{0}). The main results are the asymptotics of the normal velocity v = H^{Σ} − H (Lemma 2.1), an a priori estimate for R(t) (Lemma 2.2), the asymptotics of the tangential correction E (Lemma 2.4), and, finally, of the barycentric system (Lemma 2.5). Only the statements of these lemmas are needed in Section 3; the expansions themselves are obtained in a completely standard way.

Ultimately we would like to identify the terms of order up to R^{2}, and up to first order jointly in (ψ, ∇^{S}ψ, D_{S}^{2}ψ) explicitly. Intermediate expansions will be obtained to varying orders; for instance, since v is of order R^{−1}, in Lemma 2.1 we actually state the asymptotics of Rv to order 3 in R.

We begin by considering the second fundamental form and mean curvature of a hypersurface Σ = im(X) in a Riemannian n-manifold M, parametrized by a ‘radial embedding’:

X : S → M, X(u) = exp_{ξ}[φ(u)u_{ξ}],

where ξ ∈ M is fixed and u ↦ u_{ξ} is a fixed isometry from S to the unit tangent sphere S_{ξ} ⊂ T_{ξ}M. We assume Σ ⊂ C_{ξ}^{∗} := C_{ξ} − {ξ}, where C_{ξ} is a totally convex open neighborhood of ξ in M. (In other sections of the paper u_{ξ} is denoted e(u, ξ) or e(u); but in this section-except for Subsection 2.4, when we estimate E- we will not keep track of the frame e explicitly.)

It is easy to describe a local basis of tangent vectors and an outward normal vector for Σ. Let (e_{i}) be a local orthonormal frame on S, (ē_{i}) the corresponding (under the isometry fixed above) local frame of S_{ξ}. We assume (ē_{i}) is extended to an open sector U_{ξ} ⊂ C_{ξ}^{∗} by parallel translation along radial geodesics from ξ. Denoting by (ρ, ω) polar normal coordinates in C_{ξ}^{∗} (ρ > 0, ω ∈ S_{ξ}), and by γ_{ω}(ρ) the unit speed geodesic from ξ with initial tangent vector ω, we define n vector fields in U_{ξ}:

e_{0}(ρ, ω) = d exp_{ξ}(ρω)ω = γ′_{ω}(ρ);
J_{i}(ρ, ω) = d exp_{ξ}(ρω)[ρē_{i}],

the Jacobi field along γ_{ω}(ρ) with initial conditions J_{i}(0, ω) = 0, J′_{i}(0, ω) = ē_{i}(ω) ∈ T_{ω}S_{ξ}. By the Gauss lemma < J_{i}, e_{0} >≡ 0 in U_{ξ}. A basis of tangent vector fields to Σ is given by:

E_{i} = dXe_{i} = d exp_{ξ}(φu_{ξ})[φ_{i}u_{ξ}] + d exp_{ξ}(φu_{ξ})[φē_{i}]
= φ_{i}(e_{0} ∘ X) + J_{i} ∘ X,

where we have set φ_{i} = (d^{S}φ)e_{i} (differential on S). The E_{i} are ‘vector fields on M along X’, in the sense that E_{i}(u) ∈ T_{X(u)}M, i = 1, … n − 1. Setting h_{ij} = ⟨J_{i}, J_{j}⟩ : U_{ξ} → ℝ, we obtain an outward normal vector to Σ:

(2.0)  N = e_{0} ∘ X − φ_{k}(h^{kl}J_{l}) ∘ X,
∥N∥^{2} = 1 + (h^{ij} ∘ X)φ_{i}φ_{j} : U_{ξ} → ℝ^{+}.

N is also a vector field along X, and it is easy to check that it is orthogonal to the E_{i}.

The second fundamental form of Σ with respect to N is defined by:

A^{N}(E_{i}, E_{j}) = ⟨∇_{E_{i}}N, E_{j}⟩.

Using the above definitions for N and E_{i}, one easily obtains the expression:

A^{N}(E_{i}, E_{j}) = H(J_{i}, J_{j}) + h^{kl}(H(J_{i}, J_{l})φ_{j} + H(J_{j}, J_{l})φ_{i})φ_{k}
− J_{i}(φ̃_{j}) + h^{kl}⟨∇_{J_{i}}J_{j}, J_{l}⟩,

where:

H(X, Y) = ⟨∇_{X}e_{0}, Y ⟩ is the Hessian of the distance function d_{ξ} (∇d_{ξ} = e_{0});

φ̃_{j} is the function on U_{ξ} defined by:

φ̃_{j}(ρ, u_{ξ}) = φ_{j}(u), or φ̃_{j}(exp_{ξ}[ρu_{ξ}]) = φ_{j}(u).

A little more precisely, the functions H(J_{i}, J_{j}), h^{kl} and ⟨∇_{J_{i}}J_{j}, J_{l}⟩ on U_{ξ} define by composition with the embedding X the functions A_{ij}^{N} on S:

(2.1)  A_{ij}^{N} = H(J_{i}, J_{j}) ∘ X + φ_{k}φ_{j}[h^{kl}H(J_{i}, J_{k})] ∘ X
+ φ_{k}φ_{i}[h^{kl}H(J_{i}, J_{l})] ∘ X + φ_{k}[h^{kl}⟨∇_{J_{i}}J_{j}, J_{l}⟩] ∘ X
− ⟨∇_{e_{i}}^{S}e_{j}, φ_{k}e_{k}⟩_{S} − (H^{S}φ)(e_{i}, e_{j}),

where H^{S} denotes the Hessian on S:

(H^{S}φ)(e_{i}, e_{j}) = (d^{S}φ_{j})e_{i} − ⟨∇_{e_{i}}^{S}e_{j}, φ_{k}e_{k}⟩_{S}.

We also used the fact that:

(d^{S}φ_{j})e_{i} = dφ̃_{j} ∘ d exp_{ξ}(ρu_{ξ})[ρē_{i}] = J_{i}(φ̃_{j}).

We also need an expression for the induced metric on Σ:

(2.2)  g_{ij} = ⟨E_{i}, E_{j}⟩ = φ_{i}φ_{j} + h_{ij}

= φ_{i}φ_{j} + φ^{2} [δ_{ij} − \frac{1}{3}R_{ij}φ^{2} − \frac{1}{12}R′_{ij}φ^{3}] + O(|φ|^{6}),

using the expression for h_{ij} given below.

The functions on U_{ξ} ⊂ M defined above have well-known Taylor expansions in normal polar coodinates; in order to state them, we introduce the notation:

R_{ij}(ω) = ⟨Rm^{M}(ξ)(ω, ē_{i})ω, ē_{j}⟩, R′_{ij}(ω) = ⟨(∇_{ω}Rm^{M})(ξ)(ω, ē_{i})ω, ē_{j}⟩.

The following Taylor expansions in ρ, at a fixed ω ∈ S_{ξ} and ρ = 0, are easily obtained from the Jacobi equation:

J_{i}(ρ, ω) = ρē_{i} − \frac{1}{6}ρ^{3}Rm^{M}(ξ)(ω, ē_{i})ω

− \frac{1}{24}ρ^{4}(∇_{ω}Rm^{M}(ξ))(ω, ē_{i})ω + O(ρ^{5});

H(J_{i}, J_{j})(ρ, ω) = δ_{ij}ρ − \frac{2}{3}R_{ij}(ω)ρ^{3} − \frac{5}{12}R′_{ij}(ω)ρ^{4} + O(ρ^{5}).

Rm^{M} denotes the (3, 1) Riemann curvature tensor of M. (Here we used:

H(J_{i}, J_{j}) = ⟨∇_{J_{i}}e_{0}, J_{j}⟩ = ⟨J′_{i}, J_{j}⟩ + ⟨[J_{i}, e_{0}], J_{j}⟩ = ⟨J′_{i}, J_{j}⟩,

since for the geodesic variation f(ρ, τ) := exp_{ξ}(ρϕ_{τ}^{i} (ω)), with ϕ_{τ}^{i} the local flow of ē_{i} on S_{ξ}:

[e_{0}, J_{i}] = [d exp_{ξ}(ρω)ω, d exp_{ξ}(ρω)ē_{i}]

= [\frac{∂f}{∂ρ} , \frac{∂f}{∂τ} ]_{|τ=0} = df [\frac{∂}{∂ρ}, \frac{∂}{∂τ} ]_{|τ=0} = 0.)

h_{ij}(ρ, ω) = ρ^{2} [δ_{ij} − \frac{1}{3}R_{ij}(ω)ρ^{2} − \frac{1}{12}R′_{ij}(ω)ρ^{3}] + O(ρ^{6});
h^{ij}(ρ, ω) = ρ^{−2} [δ_{ij} + \frac{1}{3}R_{ij}(ω)ρ^{2} + \frac{1}{12}R′_{ij}(ω)ρ^{3}] + O(ρ^{2});
⟨∇_{J_{i}}J_{j}, J_{l}⟩(ρ, ω) = Γ̄_{ijl}(ω)ρ^{2} + T_{4ijl}(ω)ρ^{4} + T_{5ijl}(ω)ρ^{5} + O(ρ^{6}),

where in the last line we set:

Γ̄_{ijl}(ω) = ⟨∇S_{ξ} ; ē_{i} ē_{j}, ē_{l}⟩_{S_{ξ}}(ω), so ⟨∇_{e_{i}}^{S}e_{j}, e_{l}⟩_{S}(u) = Γ̄_{ijl}(u_{ξ}).

These are the only expansions in normal coordinates we’ll need; all other expansions obtained below follow from these and algebraic computation.

Introducing the decomposition φ = R(1 + ψ) with ∫_{S} ψ = 0, we define smooth functions on SM × [0, δ_{0}) × [0, ϵ_{0}) × ℝ^{n} × Sym^{2}(ℝ^{n}):

A_{ij}(ξ, ω, r, z, p, q), g_{ij}(ξ, ω, r, z, p), N (ξ, ω, r, z, p),

so that the second fundamental form and induced metric of Σ = im(X(ξ, R, ψ)) and the length squared of N are given by:

A^{N}(E_{i}, E_{j}) = A_{ij}(ξ, u_{ξ}, R, ψ, ∇^{S}ψ, H^{S}ψ)
g_{ij} = g_{ij}(ξ, u_{ξ}, R, ψ, ∇^{S}ψ), ∥N∥^{2} = N (ξ, u_{ξ}, R, ψ, ∇^{S}ψ).

In terms of the Riemannian functions defined above:

A_{ij} = H(J_{i}, J_{j})(r(1 + z), ω) + r^{2}p_{k}p_{j}[h^{kl}H(J_{i}, J_{k})](r(1 + z), ω)
+ r^{2}p_{k}p_{i}[h^{kl}H(J_{j}, J_{k})](r(1 + z), ω)
+ rp_{k}⟨h^{kl}∇_{J_{i}}J_{j}, J_{l}⟩(r(1 + z), ω) − rp_{k}Γ̄_{ijk}(ω) − rq(e_{i}, e_{j}),
g_{ij} = r^{2}p_{i}p_{j} + h_{ij}(r(1 + z), ω),
N = 1 + r^{2}p_{i}p_{j}h^{ij}(r(1 + z), ω).

The functions A_{ij}, g_{ij}, g^{ij} and N have Taylor expansions (at fixed (ξ, ω) and (r, z, p, q) = (0, 0, 0, 0)) given as follows.

A_{ij}(r, z, p, q) = δ_{ij}r − \frac{2}{3}R_{ij}r^{3} − \frac{5}{12}R′_{ij}r^{4} + δ_{ij}rz − 2R_{ij}r^{3}z − \frac{5}{3}R′_{ij}r^{4}z

+ (T_{4ijk} + \frac{1}{3}R_{kl}Γ_{ijl}) r^{3}p_{k} + (T_{5ijk} + \frac{1}{6}R′_{kl}Γ_{ijl}) r^{4}p_{k}

− rq + B_{ij}(u, r, z, p), where B_{ij} satisfies the estimate:

|B_{ij}(u, r, z, p)| ≤ C[r^{5} + |z|^{2} + |p|^{2}],

provided r ∈ (0, δ_{0}) and |z| + |p| < ϵ_{0}, for a constant C depending only on M.

g_{ij}(r, z, p) = r^{2} (δ_{ij} − \frac{1}{3}R_{ij}r^{2} − \frac{1}{12}R′_{ij}r^{3})

+ [2δ_{ij} − \frac{4}{3}R_{ij}r^{2} − \frac{5}{12}R′_{ij}r^{3}] r^{2}z + b_{ij},

where we have the estimate:

|b_{ij}(u, r, z, p)| ≤ Cr^{2}(r^{4} + |z|^{2} + |p|^{2}),

for r ∈ (0, δ_{0}) and |z| + |p| < ϵ_{0}, with C depending only on M. For the inverse metric tensor g^{ij}, we have:

g^{ij}(r, z, p)

= r^{−2} [δ_{ij} + \frac{1}{3}R_{ij}r^{2} + \frac{1}{12}R′_{ij}r^{3} − 2δ_{ij}z + \frac{4}{3}R_{ij}r^{2}z + \frac{5}{12}R′_{ij}r^{3}z] + b^{ij},

|b^{ij}(r, z, p)| ≤ C(r^{4} + |z|^{2} + |p|^{2}),

for C, r, z, p as before.

(2.2′) N (r, z, p) = 1 + p_{i}p_{j} (δ_{ij} + \frac{1}{3}R_{ij}r^{2} + \frac{1}{6}R′_{ij}r^{3} + B′_{N}(r, z))
= 1 + B_{N}(r, z, p),

where |B′_{N}| ≤ C(r^{4} + |z|), |B_{N}| ≤ C|p|^{2} for C, r, z, p as before. (In particular, we see that for expansions up to first order in (z, p), the superscript N in A^{N} may be supressed.)

The mean curvature of Σ with respect to the unit normal vector N̂ is the trace:

H = \frac{1}{∥N∥}g^{ij}A^{N}(E_{i}, E_{j}).

As before, we use this expression to define a smooth function H : SM × [0, δ_{0}) × [0, ϵ_{0}) × ℝ^{n} × Sym^{2}(ℝ^{n}) → ℝ so that:

(RH)_{|X=X_{(R,ξ,ψ)}} = H(ξ, u_{ξ}, R, ψ, ∇^{S}ψ, H^{S}ψ).

H is linear in q and admits the Taylor expansion at (ξ, ω, 0, 0, 0, 0) (to third order in r and first order jointly in (z, p, q):

H(ξ, ω, r, z, p, q) = n − 1 − [(n − 1)z + tr (q)] (2.3)

− \frac{1}{3}(Ric )r^{2} + (Ric )r^{2}z + ⟨T′_{4}, p⟩r^{2} − \frac{1}{3}⟨Ric , q⟩r^{2}

− \frac{1}{3}(Ric′)r^{3} − \frac{1}{3}(Ric′)r^{3}z + ⟨T′_{5}, p⟩r^{3} − \frac{1}{12}⟨Ric′, q⟩r^{3}

+ B(ξ, ω, r, z, p) + ⟨C(ξ, ω, r, z, p), q⟩

:= h_{0} + h_{2}r^{2} + h_{3}r^{3} + B + ⟨C, q⟩,

where q_{ij} = q(e_{i}, e_{j}) and we have set:

tr R_{ij} = Ric (ξ)(ω, ω) = Ric
tr R′_{ij} = (∇_{ω}Ric )(ξ)(ω, ω) = Ric′,
⟨T′_{4}, p⟩ = (T_{4iik} + \frac{1}{3}R_{kl}Γ_{iil})p_{k},
⟨T′_{5}, p⟩ = (T_{5iik} + \frac{1}{6}R′_{kl}Γ_{iil})p_{k}
⟨Ric , q⟩ = ⟨Rm^{M}(ξ)(ω, ē_{i})ω, ē_{j}⟩q(e_{i}, e_{j}),
⟨Ric′, q⟩ = ⟨(∇_{ω}Rm^{M})(ξ)(ω, ē_{i})ω, ē_{j}⟩q(e_{i}, e_{j}).

Here Ric (ξ)(ω, ω) denotes the Ricci curvature of M at ξ in the direction ω.

The remainder terms satisfy the estimates (for r ∈ (0, δ_{0}), |z| + |p| < ϵ_{0}):

|B(ξ, ω, r, z, p)| ≤ C(r^{4} + |z|^{2} + |p|^{2}),
|C^{ij}(ξ, ω, r, z, p)| ≤ C(r^{4} + |z| + |p|).
(2.3′)  |d_{z}B| + |d_{p}B| ≤ C(r^{4} + |z| + |p|),
|d_{z}C^{ij}| + |d_{p}C^{ij}| ≤ C(r^{4} + 1).

We record the lowest order contribution (in R) to the linear term in ψ in H(ξ, u_{ξ}, R, ψ, ∇^{S}ψ, H^{S}ψ), the linear operator in C^{2+α}:

Aψ = Δ^{S}ψ + (n − 1)ψ,

where Δ^{S} is the Laplace-Beltrami operator in S.

Remark. In the case M = ℝ^{n}, we have the exact expression for H:

H = [(1 + z)^{2} + |p|^{2}]^{−1/2}[(1 + z)^{2}δ_{ij} + p_{i}p_{j}]^{−1}
· [(1 + z)^{2}δ_{ij} + 2p_{i}p_{j} − (1 + z)q_{ij}]
= n − 1 − [(n − 1)z + tr(q)] + O(|p|^{2} + |z|^{2} + |p∥q| + |z∥q|),

where the remainder term is independent of r.

2.2 The Jacobian of X and the average mean curvature

The Jacobian of the embedding X = X(ξ, R, ψ) : S → M is given in terms of the induced metric by:

Jac(X) = [det(g_{ij})]^{1/2}.

As before, there is a smooth function J : SM ×[0, δ_{0})×[0, ϵ_{0})×ℝ^{n} → ℝ^{+} so that:

J (ξ, u_{ξ}, R, ψ, ∇^{S}ψ) = Jac(X_{(R,ξ,ψ)}).

We are interested in the Taylor expansion of J at (r, z, p) = (0, 0, 0), for fixed (ξ, ω) ∈ SM, to third order in r and first order jointly in (z, p). From (2.2), it is easy to see that we may write:

det(g_{ij}) = r^{2(n−1)}[det(I + h̃) + O(|p|^{2})],

where:

h̃_{ij} = −\frac{1}{3}R_{ij}r^{2} − \frac{1}{12}R′_{ij}r^{3} + [2δ_{ij} − \frac{4}{3}R_{ij}r^{2} − \frac{5}{12}R′_{ij}r^{3}] z + b̃_{ij},

|b̃_{ij}| ≤ C(r^{4} + |z|^{2}).

From the well-known formula:

\frac{d}{dz} det(I + h̃)_{|z=0} = det(I + h̃)_{|z=0} tr (\frac{d}{dz} h̃)_{|z=0} ,

one easily computes:

(2.4)  \frac{J}{r^{n−1}} = 1 + (n − 1)z − [\frac{1}{6}(Ric ) + \frac{2}{3}(Ric )z] r^{2}

− [\frac{1}{24}(Ric′) + \frac{5}{24}(Ric′)z] r^{3} + B_{1}(ξ, ω, r, z, p)

: = J_{0} + J_{2}r^{2} + J_{3}r^{3} + B_{1},

where the estimate:

|B_{1}| ≤ C(r^{4} + |z|^{2} + |p|^{2})

holds whenever r ∈ (0, δ_{0}), |z|+|p| ≤ ϵ_{0} (with C depending only on M). We now use (2.4) to compute the average mean curvature:

H^{Σ} = \frac{1}{vol (Σ)} ∫_{Σ} Hdσ = \frac{ave_{S}[HJac(X)]}{ave_{S}[Jac(X)]},

or equivalently:

(RH^{Σ})_{|X_{(R,ξ,ψ)}} = \frac{ave_{S}[(HJ)(ξ, u_{ξ}, R, ψ, ∇^{S}ψ, H^{S}ψ)]}{ave_{S}[J (ξ, u_{ξ}, R, ψ, ∇^{S}ψ, H^{S}ψ)]}

= \frac{ave_{S}[HJ]/R^{n−1}}{ave_{S}[J]/R^{n−1}} .

From now on when evaluating averages on S (denoted by bar) we always assume (ξ, u_{ξ}, R, ψ, ∇^{S}ψ, H^{S}ψ) as the argument of J , J H, etc; and we also assume ψ ∈ K^{2+α}(S). From (2.4) we compute:

\frac{1}{R^{n−1}} ave_{S}[J] = 1 − \frac{1}{6n}Scal(ξ)R^{2} − \frac{2}{3}ave_{S}[Ric (u_{ξ})ψ(u)]R^{2}

− \frac{5}{24}ave_{S}[Ric′(u_{ξ})ψ(u)]R^{3} + B̄_{1}

:= J̄_{0} + J̄_{2}(ξ)R^{2} + J̄_{3}(ξ)R^{3} + B̄_{1},

where Scal(ξ) denotes the scalar curvature at ξ and we used the easily verified fact: ave_{S}[Ric′(u_{ξ})] = 0.

Similarly, for H we have:

ave_{S}[H] = h̄_{0} + h̄_{2}R^{2} + h̄_{3}R^{3} + ave_{S}[B + ⟨C, q⟩],

where:

h̄_{0} = n − 1
h̄_{2} = −\frac{1}{3n}Scal(ξ)

+ ave_{S}[(Ric (u_{ξ})ψ(u)] + ave_{S}[⟨T′_{4}, ∇^{S}ψ⟩] − \frac{1}{3}ave_{S}[⟨R_{ij}, H_{ij}^{S}ψ⟩]

h̄_{3} = −\frac{1}{3}ave_{S}[(Ric′(u_{ξ})ψ] + ave_{S}[⟨T′_{5}, ∇^{S}ψ⟩] − \frac{1}{12}ave_{S}[⟨R′_{ij}, H_{ij}^{S}ψ⟩].

normalized mean curvature flow 279

We are interested in the difference:

ave_{S} [\frac{HJ}{R^{n−1}} ] − ave_{S} [\frac{J}{R^{n−1}} ] ave_{S}[H]

= (h̄_{0}J_{2} − h̄_{0}J̄_{2} + h̄_{2}J_{0} − h̄_{2}J̄_{0})R^{2}

+ (h̄_{0}J_{3} − h̄_{0}J̄_{3} + h̄_{3}J_{0} − h̄_{3}J̄_{0})R^{3} + B̄_{2} := C + B̄_{2}.

A straightforward computation yields for the ‘correction’ C:

(2.5) C = ave_{S}[(Ric )Δψ]\frac{R^{2}}{6} + ave_{S}[(Ric′)(−3(n − 1)ψ + Δψ)]\frac{R^{3}}{24}
:= C_{1}[ψ]R^{2} + C_{2}[ψ]R^{3},

while B̄_{2} satisfies the estimate:

|B̄_{2}| ≤ C(R^{4} + ∥ψ∥_{C^{1}}∥ψ∥_{C^{2}}).

For the average mean curvature we obtain the expression: RH^{Σ} = ave_{S}[H] + C + B_{2}, (2.6) where B_{2} = B̄_{2}R^{n−1}/ave_{S}[J] satisfies the same estimate as B̄_{2}.

Evaluated at (R, ψ, ∇^{S}ψ, H^{S}ψ), the expression (2.3) for H may be written as:

(2.6′) H = n − 1 − \frac{1}{3}(Ric )R^{2} − \frac{1}{3}(Ric′)R^{3} − Aψ

+ L_{1}[ψ]R^{2} + L_{2}[ψ]R^{3} + B + ⟨C, q⟩,

where:

L_{1}[ψ] := (Ric )ψ + ⟨∇ψ, T′_{4}⟩ − \frac{1}{3}⟨R_{ij}, H_{ij}ψ⟩
L_{2}[ψ] := −\frac{1}{3}(Ric′)ψ + ⟨∇ψ, T′_{5}⟩ − \frac{1}{12}⟨R′_{ij}, H_{ij}ψ⟩.

Combining the above (and using ψ ∈ K^{2+α}), we obtain for the average mean curvature the expansion:

RH^{Σ} = n − 1 − \frac{1}{3n}Scal(ξ)R^{2} + L̄_{1}[ψ]R^{2} + L̄_{2}[ψ]R^{3} + B_{3}, (2.7)

where L̄_{1}, L̄_{2} denote averages over S of L_{1}, L_{2} and B_{3} satisfies the estimate:

|B_{3}| ≤ C(R^{4} + ∥ψ∥_{C^{1}}^{2}) + C(R^{4} + ∥ψ∥_{C^{1}})∥D^{2}ψ∥_{C^{0}}, (2.7′)

provided 0 < R < δ_{0} and ∥ψ∥_{C^{1}} < ϵ_{0}.

Combining (2.3) and (2.7), we obtain an expansion for the ‘speed’ H^{Σ} − H, which we state as a lemma:

Lemma 2.1. There exist constants δ_{0}, ϵ_{0} depending only on M with the following property. The expression Rv = R(H^{Σ} − H) evaluated for the embedding X_{(R,ξ,e,ψ)} defines a continuous (in fact, smooth) nonlinear map:

R(H^{Σ} − H) : M_{0}(δ_{0}, ϵ_{0}) → C^{α}(S),

which admits the expression:

Rv = R(H^{Σ} − H) = Aψ + \frac{1}{3}(Ric_{0})R^{2} + \frac{1}{3}(Ric′)R^{3} (2.8)

+ (L̄_{1} − L_{1})[ψ]R^{2} + (L̄_{2} − L_{2})[ψ]R^{3} + C + B_{v},

where C is given in (2.5), L_{1}, L_{2} are given in (2.6′) and Ric_{0} denotes the trace-free Ricci tensor: Ric_{0}(ξ, u_{ξ}) = Ric (ξ, u_{ξ}) − \frac{1}{n}Scal(ξ).

The term B_{v} satisfies the estimates:

∥B_{v}∥_{C^{0}} ≤ C(R^{4} + ∥ψ∥_{C^{1}}^{2}) + C(R^{4} + ∥ψ∥_{C^{1}})∥D^{2}ψ∥_{C^{0}},
(2.8′) [B_{v}]_{C^{α}} ≤ C(R^{4} + ∥ψ∥_{C^{2}}∥ψ∥_{C^{1+α}}) + C(R^{4} + ∥ψ∥_{C^{1}})[D^{2}ψ]_{C^{α}},

provided 0 < R < δ_{0} and ∥ψ∥_{C^{1}} < ϵ_{0}.

Proof. Only the C^{α} estimate remains to be shown. We have:

B_{v} = B_{3} − B − C^{ij}(H_{ij}^{S}ψ),

where B(ψ, ∇^{S}ψ), C^{ij}(ψ, ∇^{S}ψ) are functions on S and B_{3} is a real number satisfying the desired estimate (2.7). For B we have:

|B(ψ, ∇^{S}ψ)(u) − B(ψ, ∇^{S}ψ)(ū)|

≤ sup_{|z|+|p|≤ϵ_{0}}(|d_{z}B| + |d_{p}B|)∥ψ∥_{C^{1+α}}|u − ū|^{α},

so from (2.3′):

[B(ψ, ∇^{S}ψ)]_{C^{α}} ≤ C(R^{4} + ∥ψ∥_{C^{1}})∥ψ∥_{C^{1+α}}.

normalized mean curvature flow 281

The estimate of the remaining term is similar, again using (2.3′): |C^{ij}(ψ, ∇^{S}ψ)(u)H_{ij}^{S}ψ(u) − C^{ij}(ψ, ∇^{S}ψ)H_{ij}^{S}ψ(ū)|

≤ sup_{|z|+|p|≤ϵ_{0}}(|d_{z}C^{ij}| + |d_{p}C^{ij}|)∥ψ∥_{C^{1+α}}∥D^{2}ψ∥_{C^{0}}|u − ū|^{α}

+ (sup_{S} |C^{ij}|)|H^{S}ψ(u) − H^{S}ψ(ū)|

≤ {C(1 + R^{4})∥D^{2}ψ∥_{C^{0}}∥ψ∥_{C^{1+α}} + C(R^{4} + ∥ψ∥_{C^{1}})[D^{2}ψ]_{C^{α}}}|u − ū|^{α}. We conclude:

[C^{ij}H_{ij}^{S}ψ]_{C^{α}} ≤ C∥ψ∥_{C^{1+α}}∥D^{2}ψ∥_{C^{0}} + C(R^{4} + ∥ψ∥_{C^{1}})[D^{2}ψ]_{C^{α}}, as desired.

Remark. We have the bounds:

(2.9)  R∥v∥_{C^{0}} ≤ C(R^{2} + ∥Aψ∥_{C^{0}} + R^{2}∥ψ∥_{C^{2}} + ∥B_{v}∥_{C^{0}}),
R[v]_{C^{α}} ≤ C(R^{2} + [Aψ]_{C^{α}} + R^{2}∥ψ∥_{C^{2+α}} + [B_{v}]_{C^{α}})

2.3 The enclosed volume and an ‘a priori estimate’ for

R.

The volume enclosed by Σ = im(X_{(R,ξ,e,ψ)}) may be computed from the expression:

V (Σ) := vol_{n}(int (X)) = ∫_{0}^{R} ∫_{S} Jac(X)(ξ, τ, ψ)dudτ

= ω_{n−1} ∫_{0}^{R} ave_{S}[\frac{J}{τ^{n−1}} (τ, ψ, ∇^{S}ψ)]τ^{n−1}dτ,

where ω_{n−1} = vol (S). From (2.4), we have:

\frac{V (Σ)}{ω_{n−1}} = ∫_{0}^{R}(1 + J̄_{2}τ^{2} + J̄_{3}τ^{3} + B̄_{1})τ^{n−1}dτ

= R^{n} (\frac{1}{n} + \frac{1}{n + 1}J̄_{2}R^{2} + \frac{1}{n + 2}J̄_{3}R^{3}) + R^{n}B̄_{5},

where B̄_{5} satisfies the estimate:

R^{n}|B̄_{5}| = |∫_{0}^{R} B̄_{1}τ^{n−1}dτ| ≤ C(R^{n+4} + ∥ψ∥_{C^{1}}^{2}R^{n}).

This gives the expansion for V (Σ):

\frac{V (Σ)}{R^{n}ω_{n−1}/n} = 1 − \frac{1}{6n(n + 1)}Scal(ξ)R^{2} − \frac{1}{3}ave_{S}[Ric (u_{ξ})ψ(u)]R^{2}

− \frac{2n + 3}{12(n + 2)}ave_{S}[Ric′(u_{ξ})ψ(u)]R^{3} + B̄_{5}

= 1 + B̄_{6},

where |B̄_{6}| ≤ CR^{2} if R < δ_{0}, ∥ψ∥_{C^{1}} < ϵ_{0}. Thus, assuming we have V (Σ) = V (Σ_{0}) for two such hypersurfaces, we obtain:

R^{n}(1 + B̄_{6}(R)) = R_{0}^{n}(1 + B̄_{6}(R_{0})).

This easily implies the following lemma.

Lemma 2.2. Assume that for hypersurfaces Σ = im(X), Σ_{0} = im(X_{0}) as above we have equality of the enclosed volumes: V (Σ) = V (Σ_{0}). Then:

\frac{R}{R_{0}} = 1 + B_{7}(R, R_{0}),

assuming max{R, R_{0}} < δ_{0}, where |B_{7}| ≤ C(R_{0}^{2} + R^{2}), for C depending only on M. Moreover, there exist functions R̄_{min}(V_{0}, ϵ_{0}) < R̄_{max}(V_{0}), taking values in (0, δ_{0}) and defined for V_{0} ∈ {vol (X); X = X_{(R,ξ,e,ψ)}, 0 < R < δ_{0}, ∥ψ∥_{C^{0}} < ϵ_{0}} so that if V (X) = V_{0}, X = X_{(R,ξ,e,ψ)} ∈ E_{δ_{0},ϵ_{0}}, then:

R̄_{min}(V_{0}, ϵ_{0}) ≤ R ≤ R̄_{max}(V_{0}).

Proof. Only the last claim remains to be shown. Since the volume enclosed by geodesic spheres V (S_{ξ}(ρ)) is a monotone increasing function of ρ (for fixed ξ), for each 0 < v ≤ max{V (S_{ξ}(ρ)); 0 < ρ ≤ δ_{0}} there exists a unique ρ = ρ(v, ξ) ≤ δ_{0} such that V (S_{ξ}(ρ)) = v. In particular, we may define R_{min}(V_{0}, ϵ, ξ) < R_{max}(V_{0}, ϵ, ξ) by:

(1 + ϵ)R_{min} = ρ(V_{0}, ξ), (1 − ϵ)R_{max} = ρ(V_{0}, ξ),

and then minimize R_{min} and maximize R_{max} over ξ and ϵ to obtain:

R̄_{min} = (1 + ϵ_{0})^{−1} min_{ξ∈M} ρ(V_{0}, ξ) > 0, R̄_{max} = max_{ξ∈M} ρ(V_{0}, ξ) < δ_{0}.

2.4 The ‘tangential correction’

In this subsection we estimate the ‘correction term’ in the barycenter system. Recall from Section 2.1 that, given (ξ, e, φ) ∈ M_{0}, we define:

N(u) = γ′_{e(u)}(φ) − h^{kl}φ_{k}J_{l}(φ),

where φ_{k} are components of the gradient ∇^{S}φ with respect to a local orthonormal frame (ē_{i}) on S (defined in a neighborhood of u) and the J_{l} are Jacobi fields along the geodesic γ_{e(u)}, orthogonal to γ′_{e(u)}. We then set:

E = ⟨w, ν − e⟩, w = nS^{−1}(ave_{S}[v_{N}e]), v_{N} = v∥N∥,

where v = H^{Σ} − H and S is the linear map:

S := 𝕀 + ave_{S}[n⟨·, ν − e⟩e] : T_{ξ}M → T_{ξ}M.

Define the functions of u ∈ S:

w_{j} = ⟨𝕐_{e}(φ)w, 𝕐_{e}(φ)e_{j}⟩, y_{ij} = ⟨𝕐_{e}(φ)e_{i}, 𝕐_{e}(φ)e_{j}⟩;

Here (e_{i}) is the o.n. frame on S_{ξ}, image of (ē_{i}) under the isometry e : S → S_{ξ}. With (y^{ij}) the inverse matrix, we have:

w = y^{kl}w_{l}e_{k}, 𝕐_{e}(φ)w = y^{kl}w_{l}𝕐_{e}(φ)e_{k}.

Then, from:

⟨𝕐_{e}Sw, 𝕐_{e}e_{j}⟩ = n ave_{S}[v_{N}u^{i}]⟨𝕐_{e}e_{i}, 𝕐_{e}e_{j}⟩

and:

⟨𝕐_{e}Sw, 𝕐_{e}e_{j}⟩ = ⟨𝕐_{e}w, 𝕐_{e}e_{j}⟩ + ave_{S}[n⟨𝕐_{e}w, N − γ′_{e}⟩u^{i}]⟨𝕐_{e}e_{i}, 𝕐_{e}e_{j}⟩
= w_{j} + ave_{S}[n⟨𝕐_{e}e_{k}, N − γ′_{e}⟩u^{i}y^{kl}w_{l}]y_{ij},

we obtain:

w_{j} + ave_{S}[nE_{k}u^{i}y^{kl}w_{l}]y_{ij} = ave_{S}[nv_{N}u^{i}]y_{ij},

where:

E_{k} := ⟨𝕐_{e}e_{k}, N − γ′_{e}⟩.

Thus, defining w^{m} = y^{jm}w_{j} and ṽ_{m} = ave_{S}[v_{N}u^{m}], we conclude (w^{m}) is independent of u ∈ S, and is a solution of the linear system:

w^{m} + ave_{S}[nE_{k}u^{m}]w^{k} = nṽ_{m}.

We are interested in:

E = ⟨𝕐_{e}w, N − γ′_{e}⟩ = w^{k}E_{k}.

Note that N − γ′_{e}(φ) = −h^{kl}φ_{k}J_{l}(φ). This motivates the definition in the following lemma.

Lemma 2.3. Given a frame (e_{i}) on S_{ξ}M, defined in a neighborhood of ω ∈ S_{ξ}, let E_{i}(ξ, ω, r, z, p) := −⟨𝕐_{ω}(ρ)e_{i}, h^{kl}p_{k}J_{l}(ρ)⟩_{|ρ=r(1+z)}. We have the Taylor expansion:

−E_{i} = p_{i} − \frac{R_{ik}}{3} p_{k}r^{2} − \frac{R′_{ik}}{24} p_{k}r^{3} + β_{i}
:= p̃_{i} + β_{i},

where |β_{i}| ≤ C(r^{4} + |z|^{2} + |p|^{2}) for r < δ_{0}, |z| + |p| < ϵ_{0}.

Proof. With Y_{i} = 𝕐_{ω}e_{i}, we have the expansion:

Y_{i}(ρ, ω) = ρe_{i} − \frac{1}{2}ρ^{2}Rm^{M}(ω, e_{i})ω − \frac{1}{6}ρ^{3}(∇_{ω}Rm^{M})(ω, e_{i})ω + O(ρ^{4}).

Combining this with the previously obtained expansions of h^{kl} and J_{l}, we easily obtain:

h^{kl}⟨Y_{i}(ρ), J_{l}(ρ)⟩ = \frac{δ_{ik}}{r(1 + z)} − \frac{R_{ik}}{3} r(1 + z) − \frac{R′_{ik}}{24} r^{2}(1 + z)^{2} + O(r^{3}).

Since −E_{i} = rp_{k}h^{kl}⟨Y_{i}(ρ), J_{l}(ρ)⟩, this yields the result.

Remark. For the evolution in euclidean space (M = ℝ^{n}), we have:

−E_{i} = \frac{p_{i}}{1 + z}

(in particular, E_{i} is independent of r).

The lemma implies:

|E_{i}| < C|p|(1 + r^{3}) + O(r^{4} + |z|^{2} + |p|^{2}).

In particular, for a given (ξ, e, R, ψ) and evaluating E_{i} at (ξ, e(u), R, ψ, ∇^{S}ψ), we have n|E_{i}| < 1/2 for ϵ_{0}, δ_{0} small enough, and the system for (w^{m}) is uniquely solvable:

w^{k} = (δ_{mk} − ave_{S}[nE_{k}u^{m}])nṽ_{m} + ṽ_{m}B_{mk},

where B_{mk} is independent of u and |B_{mk}| < C(R^{4} + ∥ψ∥_{C^{1}}^{2}).

For the expression E, evaluated at (ξ, e, R, ψ), we obtain:

E = w^{k}E_{k} = −n(δ_{mk} − ave_{S}[nE_{k}u^{m}])ṽ_{m}(ψ̃_{k} + β_{k})
= −nṽ_{k}ψ̃_{k} + B̃_{E},

where:

ψ̃_{i} = ψ_{k} (δ_{ik} − \frac{R_{ik}}{3} R^{2} − \frac{R′_{ik}}{24} R^{3})

and:

B̃_{E} = −nṽ_{k}β_{k} − n ave_{S}[n(ψ̃_{k} + β_{k})u^{m}]ṽ_{m}(ψ̃_{k} + β_{k}) ∈ C^{α}(S)

satisfies:

∥B̃_{E}∥_{C^{0}} ≤ C∥v∥_{C^{0}}(∥ψ∥_{C^{1}}^{2} + R^{4})
∥B̃_{E}∥_{C^{α}} ≤ C∥v∥_{C^{0}}(∥ψ∥_{C^{1+α}}^{2} + R^{4}).

From the expression (2.12) for ave_{S}[vu^{k}] obtained in the next subsection, we obtain for ṽ_{k} = ave_{S}[v_{N}u^{k}]:

ṽ_{k} = \frac{2}{3(n + 2)}e_{k}(Scal)R^{2} + Rave_{S}[L_{1}u^{k}] + R^{2}ave_{S}[L_{2}u^{k}]

+ \frac{1}{R}ave_{S}[B_{4}u^{k}] + ave_{S}[vB_{N}u^{k}].

Thus we have our final expression for E, which we state as a lemma.

Lemma 2.4. E = −S_{k}ψ_{k}R^{2} + B_{E}, where S_{k} = \frac{2n}{3(n+2)}e_{k}(Scal)(ξ) and B_{E}, defined as:

B_{E} = \frac{2nR^{2}}{3(n + 2)}e_{k}(Scal) (\frac{R_{ik}}{3} R^{2}ψ_{i} + \frac{R′_{ik}}{24} R^{3}ψ_{i})

− nψ̃_{k}(ave[L_{1}u^{k}]R + ave[L_{2}u^{k}]R^{2})

− (nψ̃_{k})\frac{1}{R}ave_{S}[B_{4}u^{k}] − nψ̃_{k}ave_{S}[vB̃_{N}u^{k}] + B̃_{E}

satisfies:

∥B_{E}∥_{C^{0}} ≤ C(R^{4}∥ψ∥_{C^{1}} + ∥ψ∥_{C^{1}}∥ψ∥_{C^{2}}R + ∥v∥_{C^{0}}(∥ψ∥_{C^{1}}^{2} + R^{4})
+ R^{−1}∥ψ∥_{C^{1}}∥B_{v}∥_{C^{0}}),
∥B_{E}∥_{C^{α}} ≤ C(R^{4} + ∥ψ∥_{C^{1+α}}∥ψ∥_{C^{2}}R + ∥v∥_{C^{0}}(∥ψ∥_{C^{1+α}}^{2} + R^{4})
+ R^{−1}∥ψ∥_{C^{1+α}}∥B_{v}∥_{C^{0}}).

2.5 Asymptotics for the barycenter system

Using ∥N∥ = 1 + B_{N}, we recall the barycenter system (1.3):

(2.10) ξ_{t} = n ave_{S}[vu^{i}]e_{i} − ave_{S}[nEu^{i}]e_{i} + n ave_{S}[vB_{N}u^{i}]e_{i} ∇_{ξ_{t}}e = 0

R_{t} = ave_{S}[v] − ave_{S}[E] + ave_{S}[vB_{N}]
Rψ_{t} = (v − E + vB_{N})_{K} − ψave_{S}[v − E + vB_{N}],

where f ↦ (f)_{K} is the L^{2} projection C^{2+α} → K^{2+α}. Our goal in this subsection is to identify the main terms in the asymptotics of (2.10), with estimates for the error terms.

From (2.9) it follows easily that:

(2.11)  R_{t} = CR^{−1} + ave_{S}[S_{k}ψ_{k}]R + B_{R}
= C_{1}[ψ]R + ave_{S}[S_{k}ψ_{k}]R + C_{2}[ψ]R^{2} + B_{R}

where S_{k} := \frac{2n}{3(n+2)}e_{k}(Scal)(ξ) and B_{R}, given by:

B_{R} = R^{−1}B̄_{v} − ave_{S}[B_{E}] + ave_{S}[vB_{N}]

satisfies the estimate:

(2.11′)  |B_{R}| ≤ CR^{−1}∥B_{v}∥_{C^{0}} + C∥v∥_{C^{0}}(∥ψ∥_{C^{1}}^{2} + R^{4})
+ C∥ψ∥_{C^{1}}∥ψ∥_{C^{2}}R + CR^{4}∥ψ∥_{C^{1}}.

Furthermore, we have the estimate:

|R_{t}| < C∥ψ∥_{C^{2}}R + |B_{R}|.

Turning to ξ_{t}, we consider the average (using (2.8′)):

ave_{S}[vu^{i}] = \frac{R}{3}ave_{S}[Ric_{0}u^{i}] + \frac{R^{2}}{3} ave_{S}[Ric′u^{i}] + \frac{1}{R}ave_{S}[(Aψ)u^{i}] (2.12)

− Rave_{S}[L_{1}[ψ]u^{i}] − R^{2}ave_{S}[L_{2}[ψ]u^{i}] + \frac{1}{R}ave_{S}[B_{4}u^{i}].

It is easy to see that the first term vanishes (since it is the average of an odd function of u), and so does the third (since ψ ∈ K^{2+α}). For the second term we use:

ave_{S}[Ric′u^{l}] = \frac{1}{ω_{n−1}} ∫_{S}⟨∇_{u_{ξ}}(Riem^{M})(u_{ξ}, ē_{i})u_{ξ}, ē_{i}⟩u^{l}du

= \frac{1}{ω_{n−1}} R_{jimi;n}(ξ) ∫_{S} u^{n}u^{j}u^{m}u^{l}du

= \frac{2}{n + 2}ē_{l}(Scal)(ξ),

by the calculation in Lemma 1.2 in [14].

We conclude:

(2.13) ξ_{t} = \frac{2n}{3(n + 2)}∇^{M}Scal(ξ)R^{2} − nRave_{S}[L_{1}[ψ]u^{i}]e_{i}

− nR^{2}ave_{S}[L_{2}[ψ]u^{i}]e_{i} + R^{2}ave_{S}[nS_{k}ψ_{k}u^{i}]e_{i} + B_{ξ},

where B_{ξ} = nave_{S}[vB_{N}u^{i}]e_{i} − ave_{S}[nB_{E}u^{i}]e_{i} + nR^{−1}ave_{S}[B_{v}u^{i}]e_{i} satisfies the bound:

(2.13′) |B_{ξ}| ≤ C∥v∥_{C^{0}}(∥ψ∥_{C^{1}}^{2} + R^{4}) + CR^{−1}∥B_{v}∥_{C^{0}}

+ C∥ψ∥_{C^{1}}∥ψ∥_{C^{2}}R + CR^{4}∥ψ∥_{C^{1}}.

With (2.11) and (2.12) we obtain the equation for ψ_{t}:

(2.14)  Rψ_{t} = \frac{1}{R}Aψ + \frac{1}{3}Ric_{0}(ξ, u)R + \frac{1}{3}Ric′(ξ, u)R^{2} − S_{k}u^{k}R^{2}
− (L_{1}[ψ])_{K}R − (L_{2}[ψ])_{K}R^{2} + (S_{k}ψ_{k})_{K}R^{2} + B_{ψ},

with B_{ψ} given by:

B_{ψ} = R^{−1}(B_{v})_{K} − (B_{E})_{K} + (vB_{N})_{K} − ψave_{S}[v − E + vB_{N}].

From this definition, (2.2′), (2.8′) and Lemma 2.4 one sees easily that:

∥B_{ψ}∥_{C^{0}} ≤ C∥v∥_{C^{0}}(∥ψ∥_{C^{1}}^{2} + R^{4}) + CR^{−1}∥B_{v}∥_{C^{0}}
+ CR∥ψ∥_{C^{1}}∥ψ∥_{C^{2}} + CR^{4}∥ψ∥_{C^{1}};
(2.14′)  ∥B_{ψ}∥_{C^{α}} ≤ C∥v∥_{C^{α}}(∥ψ∥_{C^{1+α}}^{2} + R^{4}) + CR^{−1}∥B_{v}∥_{C^{α}}
+ CR∥ψ∥_{C^{1+α}}∥ψ∥_{C^{2}} + CR^{4}∥ψ∥_{C^{1+α}},

(still under the assumptions R < δ_{0}, ∥ψ∥_{C^{1+α}} < ϵ_{0}).

Combining (2.11′), (2.13′) and (2.14′) with the estimates (2.8) and (2.8′) for v and B_{v}, we obtain a more useful estimate for the remainder terms, stated in the final lemma of this section.

Lemma 2.5. The barycentric system (2.10) in M_{0}(δ_{0}, ϵ_{0}) may be written in the form (2.11), (2.13), (2.14), plus the parallel transport equation for the frame e. The remainder terms satisfy the estimates:

|B_{R}| + |B_{ξ}| + ∥B_{ψ}∥_{C^{0}} ≤ cR^{3} + cR^{−1}∥ψ∥_{C^{1}}∥ψ∥_{C^{2}} + cR^{3}∥D^{2}ψ∥_{C^{0}}, (2.15)
∥B_{ψ}∥_{C^{α}} ≤ cR^{3} + cR^{−1}∥ψ∥_{C^{1+α}}∥ψ∥_{C^{2+α}} + cR^{3}∥D^{2}ψ∥_{C^{α}}.

3. Local and global existence

3.1 Local existence via maximal regularity

Our goal in this subsection is to prove local existence for the barycenter system on M_{0}:

ξ_{t} = ave_{S}[n(v_{N} − E)e] (3.1)

∇_{ξ_{t}}e = 0
φ_{t} = (v_{N} − E)^{∘},

where v_{N} = (H^{Σ} − H)∥N∥ and f ↦ f^{∘} is the L^{2} projection C^{α} → C_{0}^{α}. In Section 2 we derived the expression for H^{N} = ∥N∥H:

H^{N} = −P (φ, ∇^{S}φ)[φ] + B(φ, ∇^{S}φ),

where:

P (φ, ∇^{S}φ)[χ] = g^{ij}(H^{S}χ)(e_{i}, e_{j}),

g_{ij} = φ_{i}φ_{j} + h_{ij} is the induced metric on Σ and H^{S} is the Hessian on S. Defining:

P^{Σ}[χ] := \frac{∥N∥}{vol (Σ)} ∫_{Σ} ∥N∥^{−1}P [χ]dσ, B^{Σ} := \frac{∥N∥}{vol (Σ)} ∫_{Σ} ∥N∥^{−1}Bdσ,

we may write:

v_{N} = P [φ] − P^{Σ}[φ] − B + B^{Σ}.

To apply the results in [3] on local existence, we introduce appropriate function spaces. We denote by h^{k+γ}(S) (for an integer k ≥ 0 and γ ∈ (0, 1)) the closure of smooth functions in C^{k+γ}(S), with the standard C^{k+γ} norm. h_{0}^{k+γ}(S) denotes the corresponding subspace of C_{0}^{k+γ}(S). We also introduce local coordinates on the manifold 𝔽M, locally modelled on ℝ^{n} × so(n). For fixed 0 < α < β < β_{0} < 1, define the Banach spaces:

E_{1} = ℝ^{n} × so(n) × h_{0}^{2+α}(S), E_{0} = ℝ^{n} × so(n) × h_{0}^{α}(S)
E_{θ} = ℝ^{n} × so(n) × h_{0}^{1+β}(S), E_{σ} = ℝ^{n} × so(n) × h_{0}^{1+β_{0}}(S).

In the notation of [3], we seek a solution in the space:

W^{T} (ℝ^{n}) = C([0, T], E_{σ}) ∩ C_{σ}([0, T], E_{1}) ∩ C_{σ}^{1}((0, T], E_{0}).

The initial data (ξ(0), e(0), φ(0)) will be taken in the open subset of E_{σ}:

O_{δ_{2},ϵ_{2}}^{1+β_{0}}(U_{0}) = {(x, A, φ) ∈E_{σ}; x ∈U_{0}, e(x) = e^{A}x ∈ 𝔽_{g}(U_{0}), φ ∈ B_{δ_{2},ϵ_{2}}^{1+β_{0}}} .

Here U_{0} ⊂ R^{n} is the image of a normal coordinate neighborhood of ξ(0) in M, and we identify U_{0} and this neighborhood in the notation. 𝔽_{g}U_{0} is the orthonormal frame bundle of U_{0}, with respect to the metric g pulled back from M via the coordinate chart. We also use the fact that any 0 ≤ φ ∈ C^{0}(S) may be written uniquely in the form φ = R(1 + ψ) with ave_{S}[ψ] = 0.

Prior to proving the local existence lemma, we need an observation regarding the term E.

Lemma 3.1. The linear assignment v_{N} ↦ E defines a smooth map:

𝔼 : O_{δ_{1},ϵ_{1}}^{1+β}(U) → L(C^{α}, h^{1+β}).

Proof. Given (ξ, e, φ) ∈ O_{δ_{1},ϵ_{1}}^{1+β}(U), the map 𝔼_{ξ,e,φ} is the composition 𝔼 = Y ∘ S of:

v_{N} ↦ S[v_{N}] ∈ T_{ξ}M, (i)

where S[w] = nS_{ξ,e,φ}^{−1}(ave_{S}[we]) defines a smooth map O_{δ_{1},ϵ_{1}}^{1+β}(U) → L(C^{α}, T M), preserving ξ ∈ M;

Y_{(ξ,e,φ)} : η ↦ ⟨𝕐_{e}(φ)η, N − γ′_{e}(φ)⟩, (ii) which defines a smooth map O_{ϵ_{1},δ_{1}}^{1+β}(U) → L(T_{ξ}M, h^{1+β}(S)). These observations prove the lemma.

Denoting P_{(ξ,e,φ)}[χ] = (P − P^{Σ})[χ], B(ξ, e, φ) = B − B^{Σ}, we may write:

v_{N} − E = (v_{N} − E)_{1}[φ] + (v_{N} − E)_{2},

where

(v_{N} − E)_{1}[χ] = (P_{(ξ,e,φ)} − 𝔼_{(ξ,e,φ)} ∘ P_{(ξ,e,φ)})[χ],
(v_{N} − E)_{2}(ξ, e, φ) = −B(ξ, e, φ) + 𝔼_{(ξ,e,φ)}(B(ξ, e, φ)).

In local coordinates (x^{a}, e_{i}^{a}, φ) ∈ U × ℝ^{n^{2}} × h_{0}^{2+α}(S), e_{i} = e_{i}^{a} \frac{∂}{∂x^{a}}, the barycentric system is written:

x_{t}^{a} = ave_{S}[n(v_{N} − E)u^{i}]e_{i}^{a}
(e_{i}^{a})_{t} = −Γ_{bc}^{a}(x)ave_{S}[n(v_{N} − E)u^{j}]e_{j}^{b}e_{i}^{c}
φ_{t} = (v_{N} − E)^{∘},

where Γ_{bc}^{a}(x) are the Christoffel symbols of the Riemannian connection. We may write this as the quasilinear system in U × ℝ^{n^{2}} × h_{0}^{2+α}(S):

(x^{a}, e_{i}^{a}, φ)_{t} = ℙ_{(x,e_{i},φ)}[φ] + 𝔹(x, e_{i}^{a}, φ), (3.2)

with:

ℙ_{(x,e,φ)}[χ] := (ave_{S}[n(v_{N} − E)_{1}[χ]u^{i}]e_{i}^{a},

− Γ_{bc}^{a}(x)ave_{S}[n(v_{N} − E)_{1}[χ]u^{j}]e_{j}^{b}e_{i}^{c}, (v_{N} − E)_{1}^{∘}[χ])

and:

𝔹(x, e_{i}^{a}, φ) = (ave_{S}[n(v_{N} − E)_{2}u^{i}]e_{i}^{a},

− Γ_{bc}^{a}ave_{S}[n(v_{N} − E)_{2}u^{j}]e_{j}^{b}e_{i}^{c}, (v_{N} − E)_{2}^{∘}).

(Since parallel translation preserves orthonormality, it is harmless to consider ℝ^{n^{2}} instead of so(n).)

To state the local and global existence results for (3.1), we introduce the space:

W_{δ_{1},ϵ_{1}}^{T} = C^{0}([0, T], M_{0}(δ_{1}, ϵ_{1})) ∩ C^{1}((0, T], M^{α}(δ_{1}, ϵ_{1}))

and the open subset:

W_{δ_{1},ϵ_{1}}^{T}(U) = C^{0}([0, T], O_{δ_{1},ϵ_{1}}^{2+α}(U)) ∩ C^{1}((0, T], O_{δ_{1},ϵ_{1}}^{α}(U)).

Lemma 3.2. Let 0 < α < β_{0} < 1.

(i) There exist constants δ_{2} ∈ (0, δ_{1}/2), ϵ_{2} ∈ (0, ϵ_{1}/2) depending only on M and T = T (δ_{2}, ϵ_{2}) > 0 so that for any (ξ(0), e(0), φ(0)) ∈ M_{0}^{1+β_{0}}(δ_{2}, ϵ_{2}) such that φ(0) ∈ C^{∞}(S), there exists a unique solution (ξ(t), e(t), φ(t)) of (3.1) in W_{δ_{1},ϵ_{1}}^{T}(U_{0}) (where U_{0} is a normal coordinate neighborhood of ξ(0)).

(ii) φ(t, ·) ∈ C^{∞}(S) and (ξ(t), e(t)) is a smooth curve in 𝔽_{g}U_{0} for t ∈ [0, T].

(iii) The hypersurfaces Σ(t), t ∈ [0, T] parametrized by X(t) ∈

N_{std}(δ_{1}, ϵ_{1}):

X(t, u) = exp_{ξ_{0}}[φ(t, u)e(t, u)]

are smooth for t > 0 and satisfy:

X_{t} = (H^{Σ} − H)N̂ + 𝕋,

where 𝕋(t, u) is tangent to Σ(t) at X(t, u). Thus Σ(t) is a motion by normalized mean curvature.

Proof.

(i) We apply Theorem 2.11 in [3] to the quasilinear system (3.2). There are three conditions to verify.

Claim (a): ℙ defines a smooth map O_{δ_{2},ϵ_{2}}^{1+β}(U_{0}) → L(E_{1}, E_{0}). In Section 2 we showed that:

g_{ij}(φ, ∇_{S}φ) = φ^{2} (δ_{ij} + \frac{φ_{i}φ_{j}}{φ^{2}} + h̃_{ij}(φ)) , ∥N∥ = (1 + h^{ij}φ_{i}φ_{j})^{1/2},

where h̃_{ij}, the smooth function of τ > 0 defined by:

h_{ij} = ⟨J_{i}(τ), J_{j}(τ)⟩ = τ^{2}(δ_{ij} + h̃_{ij})

satisfies |h̃_{ij}(φ)| ≤ c|φ|^{2} for |φ| < δ_{2}, for some c > 0 depending only on M. From the definitions of P and P^{Σ}, it is clear that for δ_{1} > 0 sufficiently small, P defines a smooth map P : O_{δ_{2},ϵ_{2}}^{1+β}(U_{0}) → L(h_{0}^{2+α}, h_{0}^{α}). From Lemma 3.1, 𝔼 defines a smooth map O_{δ_{2},ϵ_{2}}^{1+β}(U) → L(h_{0}^{α}, h^{1+β}(S)). This shows (v_{N} − E)_{1} = P − 𝔼 ∘ P defines a smooth map from the same set to L(h_{0}^{2+α}, h^{α}). It is then clear that ℙ defines a smooth map into L(h_{0}^{2+α}, E_{0}), which may be regarded as a map into L(E_{1}, E_{0}).

Claim (b): 𝔹 defines a smooth map O_{δ_{2},ϵ_{2}}^{1+β} → E_{0}.

Since Hd_{ξ_{0}}(τ)(J_{i}(τ), J_{j}(τ)) (where Hd_{ξ_{0}} denotes the Hessian of the distance function to ξ_{0}) is a smooth function of τ satisfying:

|Hd_{ξ_{0}}(φ)(J_{i}(φ), J_{j}(φ))| ≤ c|φ| for |φ| < δ_{0},

with c depending only on M, the expression for B(φ, ∇_{S}φ) (see Section 2):

B = g^{ij}[Hd_{ξ_{0}}(J_{i}, J_{j}) + h^{kl}φ_{k}(φ_{j}Hd_{ξ_{0}}(J_{i}, J_{l}) + φ_{i}Hd_{ξ_{0}}(J_{j}, J_{l}))

+ h^{kl}φ_{k}⟨∇_{J_{i}}J_{j}, J_{l}⟩]

and the expressions above for g_{ij} immediately imply that B = B − B^{Σ} defines a smooth map B : O_{δ_{2},ϵ_{2}}^{1+β} → h_{α}, for δ_{2} ∈ (0, δ_{0}) sufficiently small. Given that 𝔼 is smooth into L(h_{0}^{α}, h^{1+β}) (Lemma 3.1), this shows (v_{N} − E)_{2} is smooth in h^{α}, and hence 𝔹 is smooth from the same set into E_{0}.

Claim (c): ℙ_{(ξ,e,φ)} ∈ M_{σ}(E_{1}, E_{0}) for (ξ, e, φ) ∈ O_{δ_{2},ϵ_{2}}^{1+β_{0}}(U).

It follows from theorem (2.14) in [3] (which generalizes [4]) that P_{(ξ,e,φ)} ∈ M_{σ}(h^{2+α}, h^{α}), provided we can verify that P_{(ξ,e,φ)} generates an analytic semigroup in (F_{1}, F_{0}), where we set F_{1} = h^{2+α_{0}}, F_{0} = h^{α_{0}} (for some α_{0} ∈ (0, α)):

P(φ, ∇_{S}φ) ∈ Hol(h^{2+α_{0}}, h^{α_{0}}), for all φ ∈ O_{δ_{1}}^{1+β_{0}}.

This follows from the fact that, for such φ, P (φ, ∇_{S}φ)[.] is a second order, uniformly elliptic operator in S, and P^{Σ} defines a perturbation with zero relative norm: since P^{Σ}[χ] ∈ ℝ with |P^{Σ}[χ]| ≤ ∥χ∥_{C^{2}}, we have for each ϵ > 0:

∥P^{Σ}[χ]∥_{h^{α_{0}}} = |P^{Σ}[χ]| < ϵ∥χ∥_{h^{2+α_{0}}} + C_{ϵ}∥χ∥_{h^{α_{0}}} .

Since, in the notation of Lemma 3.1 above:

∥E ∘ P[χ]∥_{h^{1+β}} ≤ c∥(Y ∘ S)[Pχ]∥_{h^{1+β}}
≤ c|S[Pχ]|_{T M} ≤ c∥Pχ∥_{C^{0}} ≤ ∥χ∥_{C^{2}},

the same argument shows:

(v_{N} − E)_{1} = P − 𝔼 ∘ P ∈ M_{σ}(h^{2+α}, h^{α}).

Likewise, since:

|ave_{S}[n(v_{N} − E)_{1}[χ]u^{i}]e_{i}^{a}| + |Γ_{bc}^{a}(x)ave_{S}[n(v_{N} − E)[χ]u^{j}]e_{j}^{b}e_{i}^{c}|
≤ c(x, e, φ)∥χ∥_{C^{2}},

ℙ_{(x,e,φ)} may be regarded as a ‘lower order perturbation’ of (0, 0, (v_{N} − E)_{1}) ∈ M_{σ}(E_{1}, E_{0}), which suffices for the claim.

(ii) (Higher regularity.) This follows from a standard ‘bootstrapping’ argument. From Part (i), φ(t) ∈ h^{2+α} for each t > 0. Fix s > 0 and apply Theorem 2.11 in [3] with initial data Ψ(s) and spaces E = E_{0}, E_{σ}, E_{θ}, E_{1} defined as E = ℝ × so(n) × h, where h = h_{0}^{1+α} for E_{0}, h = h_{0}^{2+α} for E_{σ}, h = h_{0}^{2+α_{0}} for E_{θ}, h = h_{0}^{3+α} for E_{1}. It follows as above that P(φ, ∇_{S}φ) ∈ Hol(F_{1}; F_{0}), where F_{1} = h^{3+α_{0}}, F_{0} ∈ h^{1+α_{0}}. We obtain a local solution Ψ_{1}:

Ψ_{1} ∈ C^{0}([s, T_{1}]; M_{0}^{2+α}) ∩ C^{0}((s, T_{1}]; M_{0}^{3+α}) ∩ C^{1}((s, T_{1}]; M_{0}^{1+α}),

which must coincide with Ψ on some interval (s, s_{1}). Since s ∈ (0, T) is arbitrary, this shows Ψ ∈ C^{0}((0, T); M_{0}^{3+α}) ∩ C^{1}((0, T); M_{0}^{1+α}). Iterating this argument yields the conclusion of (ii).

(iii) This follows directly from Lemma 1.7.

The same argument yields local existence in a slightly different space.

Lemma 3.3. There exist constants δ_{2} ∈ (0, δ_{1}/2), ϵ_{2} ∈ (0, ϵ_{1}/2) depending only on M, and T = T (δ_{2}, ϵ_{2}) > 0 so that if (ξ(0), e(0), φ(0) ∈ M_{0}(δ_{2}, ϵ_{2}) with φ(0) smooth, there exists a unique solution of (3.1) in W_{δ_{1},ϵ_{1}}^{T}(U_{0}), where U_{0} is a normal coordinate neighborhood of ξ(0). Statements (ii) and (iii) of Lemma 3.2 still hold.

Proof. Identical to Lemma 3.2. Since no ‘smoothing effect’ is needed, Theorem 2.7 in [3] could also be used.

3.2 Global existence for the scaled system

Prior to establishing global existence, we introduce two small scale parameters ϵ ∈ (0, ϵ_{1}), δ ∈ (0, δ_{1}) and write a general X ∈ N_{std} in the form:

X = Φ(δR, ξ, e, ϵψ), X(u) = exp_{ξ}[δR(1 + ϵψ(u))e(u)],

where we always assume 0 < R < 1, ψ ∈ K^{2+α}, ∥ψ∥_{C^{2+α}} < 1. (Thus δ measures the ‘size’ of the bubble and ϵ the ‘deviation from sphericity relative to size’.) With these scale parameters, the equations of motion take the form:

δR_{t} = ave_{S}[v_{N} − E] (3.3)_{δ,ϵ}

ξ_{t} = ave_{S}[n(v_{N} − E)e]
∇_{ξ_{t}}e = 0
δϵRψ_{t} = (v_{N} − E)_{K} − δψ ave_{S}[v_{N} − E],

with v_{N} and E computed at (δR, ξ, e, ϵψ) ∈ M_{0}.

To prove global existence, we use the expansions obtained in Section 2 (Lemma 2.5) to write the system in the form:

δR_{t} = C_{1}[ψ]δϵR + C_{2}[ψ]δ^{2}ϵR^{2} + ave_{S}[S_{k}ψ_{k}]δϵR + B_{R}(ξ, e, δR, ϵψ) (3.4)
ξ_{t} = c_{n}∇^{M}Scal(ξ)δ^{2}R^{2} − ave_{S}[nL_{1}[ψ]e]δϵR − ave_{S}[nL_{2}[ψ]e]δ^{2}ϵR^{2}
+ ave_{S}[nS_{k}ψ_{k}e]δ^{2}ϵR^{2} + B_{ξ}(ξ, e, δR, ϵψ)
∇_{ξ_{t}}e = 0
δϵRψ_{t} = δ^{−1}R^{−1}ϵAψ + \frac{1}{3}(Ric_{0})δR + \frac{1}{3}(Ric′)δ^{2}R^{2} + S_{k}u^{k}δ^{2}R^{2}
− (L_{1}[ψ])_{K}ϵδR − (L_{2}[ψ])_{K}ϵδ^{2}R^{2} + B_{ψ}(ξ, e, δR, ϵψ).

For the remainder terms, we have the estimates:

|B_{R}| + |B_{ξ}| + ∥B_{ψ}∥_{C^{0}} ≤ cδ^{3}R^{3} + cδ^{−1}R^{−1}∥ψ∥_{C^{1}}∥ψ∥_{C^{2}}ϵ^{2}

+ cR^{3}δ^{3}ϵ∥D^{2}ψ∥_{C^{0}},

∥B_{ψ}∥_{C^{α}} ≤ cδ^{3}R^{3} + cδ^{−1}R^{−1}∥ψ∥_{C^{1+α}}∥ψ∥_{C^{2+α}}ϵ^{2} + cR^{3}δ^{3}ϵ∥D^{2}ψ∥_{C^{α}}.

Remark. The traceless Ricci curvature Ric_{0} vanishes identically on surfaces, so the δR term is not present in the equation for ψ_{t} when n = 2.

We re-state Lemma 3.3 as follows:

Lemma 3.4. There exist δ_{2} > 0, ϵ_{2} > 0 depending only on M and T = T (δ_{2}, ϵ_{2}) > 0, with the following property. For each Ψ(0) = (δR(0), ξ(0), e(0), ϵψ(0)) in M_{0}(δ_{2}, ϵ_{2}) with ϵ < ϵ_{2}, δ < δ_{2}, there exists a unique solution Ψ(t) = (δR(t), ξ(t), e(t), ϵψ(t)) ∈ M_{0}(δ_{2}, ϵ_{2}) to (3.3)_{δ,ϵ} in the space W_{δ_{1},ϵ_{1}}^{T}(U_{0}), where U_{0} is a normal coordinate neighborhood of ξ_{0}. In particular, we have ∥ψ(t)∥_{C^{1+α}}(S) < 1 in [0, T].

From Lemma 2.2, the following estimates hold for R(t), t ∈ [0, T].

(3.4′)  δR_{min} ≤ δR(t) ≤ δR_{max},

where R_{max} and R_{min}, given by:

R_{max} = R̄_{max}(vol (X(0))) ≤ R̄_{max}(vol (X_{(δ_{2},ξ(0),e(0),ϵ_{2})})
R_{min} = R̄_{min}(vol (X(0)), ϵ_{2}) ≥ R̄_{min}(vol (X_{(δ_{2},ξ(0),e(0),ϵ_{2})})

may be taken to depend only on M.

For t ∈ [0, T_{δ,ϵ}), we have the estimates for remainder terms (using the bound ∥ψ∥_{C^{1+α}} < 1):

|B_{R}| + |B_{ξ}| + ∥B_{ψ}∥_{C^{0}} ≤ cδ^{3}R_{max}^{3}(1 + ϵ∥D^{2}ψ∥_{C^{0}})
+ cδ^{−1}R_{min}^{−1}(ϵ^{2} + ϵ^{2}∥D^{2}ψ∥_{C^{0}}),
∥B_{ψ}∥_{C^{α}} ≤ R_{max}^{3}δ^{3}(1 + ϵ∥D^{2}ψ∥_{C^{α}})
+ cδ^{−1}R_{min}^{−1}(ϵ^{2} + ϵ^{2}∥D^{2}ψ∥_{C^{α}}).

We will need some facts from semigroup theory. Consider the operator Aψ = Δ^{S}ψ + (n − 1)ψ on the Hilbert space:

X = {ψ ∈ L^{2}(S); ave_{S}[ψ] = 0 = ave_{S}[ψu^{i}], i = 1, …, n}.

A is self-adjoint on X with eigenvalues λ_{n} < λ_{max} < 0. A ∈ M_{1}(h_{0}^{2+α}, h_{0}^{α}), i.e., A has the maximal regularity property in the Banach couple (h_{0}^{2+α}, h_{0}^{α}) (see [3]). Denote by e^{sA}, s > 0, the analytic semigroup generated by A.

Lemma 3.5. There exist constants μ > 0, M_{0} > 0 such that the semigroup generated by A in h_{0}^{2+α} satisfies the estimates:

∥e^{sA}ϕ∥_{C^{2+α}} ≤ M_{0}e^{−μs}∥ϕ∥_{C^{2+α}}, ϕ ∈ h_{0}^{2+α}; (1)

∥e^{sA}ϕ∥_{C^{2+α}} ≤ \frac{M_{0}}{s} e^{−μs}∥ϕ∥_{C^{α}}, ϕ ∈ h_{0}^{α}; (2)

sup_{0<s≤s̄} ∥∫_{0}^{s} e^{(s−σ)A}ϕ(σ)dσ∥_{C^{2+α}(S)} ≤ M_{0} sup_{0<s≤s̄}∥ϕ(s)∥_{C^{α}(S)}, (3)

if ϕ : (0, s̄] → h_{0}^{α} is continuous (where M_{0} is independent of s̄).

Proof. The first two estimates are well-known consequences of the fact that A generates an analytic semigroup on h_{0}^{2+α}. Estimate (3) is equivalent to the statement A ∈ M_{1}(h_{0}^{2+α}, h_{0}^{α}). The fact that M_{0} may be taken independent of s̄ uses the negative upper bound on the spectrum of A.

We may now state and prove the global existence lemma. For Ψ ∈ M_{0}(δ_{2}, ϵ_{2}), define T_{δ,ϵ}^{∗}(Ψ) as the supremum (possibly infinite) of all T > 0 such that the solution of (3.3)_{δ,ϵ} found in Lemma 3.4, with initial condition Ψ, is in W_{δ_{2},ϵ_{2}}^{T}.

Lemma 3.6. There exist δ_{3} ∈ (0, δ_{2}), ϵ_{3} ∈ (0, ϵ_{2}) so that if 0 < δ < δ_{3}, 0 < ϵ < ϵ_{3} and δ^{2} ≪ ϵ, we have T_{δ,ϵ}^{∗}(Ψ) = ∞, for any Ψ ∈ M_{0}(δ_{3}, ϵ_{3}). In addition, ∥ϵψ(t)∥_{C^{2+α}} < ϵ_{2}, for all t ≥ 0.

Proof. By contradiction, suppose T_{δ,ϵ}^{∗} is finite. Let ξ_{∞} ∈ M be a limit point of ξ(t) as t ↗ T_{δ,ϵ}^{∗}; choose a normal coordinate neighborhood U_{∞} = B_{ξ_{∞}}(d_{∞}) of ξ_{∞} in M (identified with the open set U_{∞} in ℝ^{n}). Now take τ_{δ,ϵ} ∈ (0, T_{δ,ϵ}^{∗}) so that ξ(τ_{δ,ϵ}) ∈ U_{∞}. Our goal is to show that Ψ(·) extends continuously to [τ_{δ,ϵ}, T_{δ,ϵ}^{∗}], taking values in O_{δ_{2},ϵ_{2}}^{2+α}(U_{∞}). Then local existence and continuity (Lemma 3.4) contradict the maximality of T_{δ,ϵ}^{∗} (note that in particular we would have ψ(T_{δ,ϵ}^{∗}) ∈ h_{0}^{2+α}). Thus, we must show:

(i) R(t) extends to a C^{1} function [τ_{δ,ϵ}, T_{δ,ϵ}^{∗}] → (0, δ_{2}).

(ii) (ξ(t), e(t)) extends continuously to a C^{1} curve [τ_{δ,ϵ}, T_{δ,ϵ}^{∗}]→ 𝔽_{g}U_{∞}; (by standard results in differential geometry, it is enough to show this for ξ(t)).

(iii) ψ(t) extends to a C^{0} map [τ_{δ,ϵ}, T_{δ,ϵ}^{∗}] → O_{δ_{2},ϵ_{2}}^{2+α}.

We begin by re-scaling time, setting

\frac{ds}{dt} = δ^{−2}R^{−2}(t).

Since R(t) is uniformly bounded from above and from below, if we let S_{δ,ϵ}^{∗} < ∞ correspond to T_{δ,ϵ}^{∗} (and σ_{δ,ϵ} correspond to τ_{δ,ϵ}), we are assuming S_{δ,ϵ}^{∗} < ∞, and must show (i)-(iii) for Ψ(t(s)) and [σ_{δ,ϵ}, S_{δ,ϵ}^{∗}]. We abuse notation by writing ψ(s) = ψ(t(s)),etc. Omitting the parallel transport equation for the frame from now on, we write the system in the s variable as:

R_{s} = ρ(R(s), ξ(s), e(s), ψ(s))
ξ_{s} = α(R(s), ξ(s), e(s), ψ(s))
ψ_{s} = Aψ + ϕ(R(s), ξ(s), e(s), ψ(s)),

where:

ρ = C_{1}[ψ]δ^{2}ϵ^{2}R^{3} + C_{2}[ψ]δ^{3}ϵR^{4} + ave_{S}[S_{k}ψ_{k}]δ^{2}ϵR^{3} + δR^{2}B_{R}
α = c_{n}∇^{M}Scal(ξ)δ^{4}R^{4} − ave_{S}[nL_{1}[ψ]e]δ^{3}ϵR^{3} − ave_{S}[nL_{2}[ψ]e]δ^{4}ϵR^{4}

+ ave_{S}[nS_{k}ψ_{k}]δ^{4}ϵR^{4} + δ^{2}R^{2}B_{ξ}

ϕ = \frac{1}{3}(Ric_{0})δ^{2}R^{2}ϵ^{−1} + \frac{1}{3}(Ric′)δ^{3}ϵ^{−1}R^{3} − S^{k}u^{k}δ^{3}ϵ^{−1}R^{3}

− (L_{1}[ψ])_{K}δ^{2}R^{2} − (L_{2}[ψ])_{K}δ^{3}R^{3} + (S_{k}ψ_{k})_{K}δ^{3}R^{3} + δϵ^{−1}RB_{ψ}.

The main step in showing (i)-(iii) consists of estimating:

Z_{δ,ϵ} := sup_{s∈[0,S_{δ,ϵ}^{∗})} ∥ψ(s)∥_{C^{2+α}}.

(This supremum may in principle be infinite.) Choosing local coordinates x on U_{∞}, with ξ_{∞} corresponding to x = 0, we have the representation formulas for the solution:

R(s) = R(0) + ∫_{0}^{s} ρ(σ)dσ, s ∈ [0, S_{δ,ϵ}^{∗});
x(s) = x(σ_{δ,ϵ}) + ∫_{σ_{δ,ϵ}}^{s} α(σ)dσ, as long as x(s) ∈ U_{∞};
ψ(s) = e^{sA}ψ(0) + ∫_{0}^{s} e^{(s−σ)A}ϕ(σ)dσ, s ∈ [0, S_{δ,ϵ}^{∗}).

Claim 3.7. sup_{s∈[0,S_{δ,ϵ}^{∗})} ∥ϕ∥_{C^{α}} ≤ C(δ, ϵ)(1+Z_{δ,ϵ}), where C(δ, ϵ) → 0 for δ → 0_{+}, ϵ, → 0_{+}.

Assuming the claim, from Lemma 3.5(1) and (3) we obtain, for all 0 ≤ s < S_{δ,ϵ}^{∗}:

∥ψ(s)∥_{C^{2+α}} ≤ M_{0}e^{−μs}∥ψ(0)∥_{C^{2+α}} + M_{0} sup_{s∈0,S_{δ,ϵ}^{∗})} ∥ϕ(s)∥_{C^{α}}
≤ M_{0}∥ψ(0)∥_{C^{2+α}} + M_{0}C(δ, ϵ)(1 + Z_{δ,ϵ}).

Thus:

Z_{δ,ϵ} ≤ M_{0}C(δ, ϵ)(Z_{δ,ϵ} + 1) + M_{0}∥ψ(0)∥_{C^{2+α}}.

If δ, ϵ are chosen small enough that C(δ, ϵ) < 1 and C(δ, ϵ) < ϵ_{2}/4M_{0} < 1/2M_{0}, we have:

Z_{δ,ϵ} ≤ 2M_{0}∥ψ(0)∥_{C^{2+α}} + \frac{ϵ_{2}}{2} . (3.5)

Thus Z_{δ,ϵ} < ∞. This and the bound on ϕ (from the claim) easily imply (iii). In particular, one sees immediately from the expressions for α, ρ, B_{R} and B_{ξ} (and still assuming ϵ^{2} ≪ δ) that:

(3.5′)  sup_{s∈(σ_{δ,ϵ},S_{δ,ϵ}^{∗})}(|ρ| + |α|_{ℝ^{n}}) < c_{δ,ϵ},

where c_{δ,ϵ} depends only on the manifold M, the constants M_{0}, R_{min} and R_{max}, and on ∥ψ(0)∥_{C^{2+α}}. Thus we may choose σ̃_{δ,ϵ} ∈ (σ_{δ,ϵ}, S_{δ,ϵ}) so that x(σ̃_{δ,ϵ}) < d_{∞}/3 and

|x(s) − x(σ̃_{δ,ϵ})| = |∫ s ; σ̃_{δ,ϵ} α(σ)dσ| < d_{∞}/3,

as long as x(s) ∈ U_{∞}. In particular x(s) ∈ U_{∞} for all s ∈ [σ̃_{δ,ϵ}, S_{δ,ϵ}^{∗}). This bound on |x(s)|, the bound R_{min} < R(s) < R_{max} < δ_{2} and the estimates (3.5), (3.5′) easily imply (i) and (ii).

In addition, assume ψ(0) satisfies:

2M_{0}∥ψ(0)∥_{C^{2+α}} < ϵ_{2}/2.

Then (reverting to the original time variable t) ∥ψ(T_{δ,ϵ}^{∗})∥_{C^{2+α}} < ϵ_{2}. By the local existence Lemma 3.4, this contradicts the maximality of T_{δ,ϵ}^{∗} (and also implies the last claim in the lemma).

Proof of Claim 3.7.

The claim follows directly from the estimate:

∥ϕ∥_{C^{α}} ≤ cδ^{2}ϵ^{−1}R_{max}^{2} + cδ^{3}ϵ^{−1}R_{max}^{3} + c∥ψ∥_{C^{2+α}}δ^{2}R_{max}^{2} (3.6)
+ c∥ψ∥_{C^{2+α}}δ^{3}R_{max}^{3}

+ cδ^{2}ϵ^{−1}R_{max}[R_{max}^{3}δ^{3}(1+ϵ∥ψ∥_{C^{2+α}})+δ^{−1}R_{min}^{−1}ϵ^{2}(1+∥ψ∥_{C^{2+α}})]

≤ cδ^{2}ϵ^{−1}R_{max}^{2} + cδ^{2}R_{max}^{2}∥ψ∥_{C^{2+α}}
+ cδ^{4}ϵ^{−1}R_{max}^{4} + cδ^{4}R_{max}^{4}∥ψ∥_{C^{2+α}} + cδϵ(1 + ∥ψ∥_{C^{2+α}})
≤ cδ^{2}ϵ^{−1}R_{max}^{2} + cδϵ + c(δ^{2}R_{max}^{2} + δϵ)∥ψ∥_{C^{2+α}}
≤ c(δ, ϵ)(1 + ∥ψ∥_{C^{2+α}}) ≤ c(δ, ϵ)(1 + Z(δ, ϵ)),

assuming (on the last step) δ^{2} ≪ ϵ.

Remark. One can easily trace the origin of the condition δ^{2} ≪ ϵ to the equation for ψ_{t}: it is needed so that the (euclidean) linear operator term \frac{1}{δ^{2}R^{2}} Aψ dominates the largest Riemannian term \frac{1}{3ϵ}Ric_{0}.

Motion of the analytic barycenter.

Under the assumption δ^{2} ≪ ϵ, the largest terms in the equation for ξ_{t} are:

(a) the ψ-independent Riemannian term c_{n}∇^{M}Scal(ξ)δ^{2}R^{2};

(b) the Riemannian term linear in ψ: −ave_{S}[nL_{1}[ψ]e]δϵR;

(c) the Euclidean term R^{−1}∥ψ∥_{C^{1}}∥ψ∥_{C^{2}}δ^{−1}ϵ^{2}.

We may choose the relative scales so that the term (a) dominates the euclidean term (c) by imposing δ^{−1}ϵ^{2} ≪ δ^{2}. This also implies (a) dominates (b). Thus we have the lemma (Part (iii) of the main theorem):

Lemma 3.8. Assume the scale parameters ϵ and δ satisfy:

δ^{2} ≪ ϵ ≪ δ^{3/2}.

Then the leading terms in the equations of motion for ξ and ψ are:

ξ_{t} = c_{n}∇^{M}Scal(ξ)R^{2}δ^{2} + O(ϵ^{2}δ^{−1}),
ψ_{t} = δ^{−2}R^{−2}Aψ + O(ϵ^{−1}).

4. Asymptotic behavior

In this section we discuss the asymptotic behavior of a global solution of (0.1), and prove Part (iv) of the main theorem, asserting the convergence of solutions to parametrized hypersurfaces of constant mean curvature.

It is simpler to argue subconvergence of solutions, based on the fact that (0.4) is a gradient system on M_{0}, satisfying (0.2):

\frac{dA(Σ(t))}{dt} = − ∫_{Σ}(H − H^{Σ})^{2}dσ ≤ 0.

By parabolic regularity the solution curves of (0.4) are relatively compact in C^{2+α}(S) (cf. the last statement in Lemma 3.6), and so by the Invariance Principle ([7]) the ω−limit set of a complete orbit is contained in the maximal invariant set of:

Ω = {(R, ξ, e, ψ) ∈ M_{0}; H ≡ H^{Σ}, Σ = im(X_{(R,ξ,e,ψ)})} . (4.1)

Thus, for any initial embedding X_{0} for which a global solution exists, the ω−limit set ω(X_{0}) is contained in the set of hypersurfaces (topological spheres) in M with constant mean curvature.

In the proof of convergence we will need a lemma contained in R. Ye’s paper [14]. We need to point out a slight difference between Ye’s parametrization of quasi-spherical hypersurfaces and ours. Fix p ∈ M and a totally convex neighborhood U_{p} ⊂ M. For τ ∈ U_{p}, χ ∈ C^{2}(S) and 0 < r < δ_{0}, consider the embedding:

Y_{(r,τ,χ)} : S → M, Y_{(r,τ,χ)}(u) = exp_{τ} [r(1 − χ)ē(u, τ)],
Σ_{(r,τ,χ)} = im(Y_{(r,τ,χ)}),

where the frame ē(·, τ) is defined by radial parallel translation of a fixed frame at p. (In contrast with the notation in the present paper, it is not assumed that ave_{S}[χ] = 0.) To obtain a local foliation by constant mean curvature spheres, R. Ye considers the function:

H(r, τ, χ) = r × (mean curvature of Σ_{(r,τ,χ)}),

defined on an open subset of (0, δ_{0}) × U_{p} × K^{⊥}, taking values in C^{α}(S); here K = C_{0}^{2+α} ⊂ C^{2+α}(S) denotes the kernel of A = Δ + (n − 1), and K^{⊥} its L^{2} orthogonal complement.

Lemma 4.1 ([14]).

(i) If p ∈ M is a nondegenerate critical point of scalar curvature, there exists δ̄ > 0 and smooth functions (τ, χ) : [0, δ̄) → U_{p} × K^{⊥} with τ(0) = p, such that H(r, τ(r), r^{2}χ(r)) ≡ n − 1 for 0 ≤ r < δ̄. The family F = (Σ_{r})_{r∈(0,δ̄)}, Σ_{r} = Σ_{(r,τ(r),r^{2}χ(r))} defines a foliation of a deleted neighborhood of p by hypersurfaces of constant mean curvature (n − 1)/r.

(ii) Conversely, there is a neighborhood V_{p} ⊂ U_{p} of p and δ̂ ∈ (0, δ̄) so that if Σ = im(X), X = X_{(R,ξ,ψ)} ∈ N_{std}(δ_{1}, ϵ_{1}), ξ ∈ V_{p}, is a hypersurface of constant mean curvature (n − 1)/r with r ∈ (0, δ̂), then Σ = Σ_{r} is a leaf of the local foliation at p.

Proof. R. Ye [14] uses the implicit function theorem to argue the existence of functions τ(r), χ(r) such that H(r, τ(r), r^{2}χ(r)) ≡ n − 1. Conversely (as observed in the ‘intermediate remark’ on p. 389 and in the proof of Theorem 2.1 in [14]), there exist δ̄ > 0, V_{p} ⊂ U_{p} so that if, for some fixed r_{0} ∈ (0, δ̄) and some ξ ∈ V_{p} we have Σ = Σ_{(r_{0},τ,r_{0}^{2}χ)} with constant mean curvature (n − 1)/r_{0} contained in U_{p}, then χ = χ(r_{0}) and ξ = τ(r_{0}), and Σ is a leaf of the local foliation at p.

Switching to our parametrization, we observe that if

Σ = im(X_{(R,ξ,e,ψ)}) = im(Y_{(r_{0},ξ,r_{0}^{2}χ)}),

we have: R(1 + ψ) = r_{0}(1 + r_{0}^{2}χ). To conclude (ii), we just have to observe that r_{0} ∈ (0, δ̄) (defined as (n − 1) times the reciprocal of the mean curvature of Σ) satisfies r_{0} → 0 as R → 0 (uniformly for ξ and χ varying in compact sets). This follows immediately from the expression R = r_{0}(1 + r_{0}^{2}ave_{S}[χ]).

We now conclude the proof of Part (iv) of the main theorem

Lemma 4.2. There exist constants δ_{4} > 0, ϵ_{4} > 0 depending only on M so that any limit point:

lim Ψ(t_{i}) = Ψ_{∗} = (δR_{∗}, ξ_{∗}, e_{∗}, ϵψ_{∗}), t_{i} → ∞,

of (3.3)_{δ,ϵ} with ϵ < ϵ_{4}, δ < δ_{4}, ϵ^{2} ≪ δ^{3} corresponds to X_{∗} ∈ N_{std}(δ_{4}, ϵ_{4}) parametrizing a hypersurface Σ_{∗} of constant mean curvature, which is contained in some V_{p} as in Lemma 4.1 (with p a critical point of Scal) and is a leaf of the local c.m.c. foliation at p.

In particular, since the set of leaves of local c.m.c. foliations enclosing a given volume is finite (with at most one in each V_{p}, p a critical point of Scal), we must have convergence Ψ(t) → Ψ_{∗}, where Ψ_{∗} corresponds to X_{∗} parametrizing a c.m.c hypersurface Σ_{∗} enclosing the same volume as X(0) — as claimed in Part (iv) of the main theorem.

Proof. We first argue that ξ_{∗} ∈ ⋃_{p} V_{p} if δ_{4}, ϵ_{4} are small enough and ϵ_{4}^{2} ≪ δ_{4}^{3}. Otherwise we can find sequences δ_{k} → 0, ϵ_{k} → 0 with ϵ_{k}^{2} ≪ δ_{k}^{3}, and solutions Ψ^{k}(t) of (3.3)_{(δ_{k},ϵ_{k})} admitting a limit point Ψ_{∗}^{k} = (δ_{k}R_{∗}^{k}, ξ_{∗}^{k}, e_{∗}^{k}, ϵ_{k}ψ_{∗}^{k}) as t → ∞, where ξ_{∗} ∉ ⋃_{p∈Crit(Scal)} V_{p} for any k ≥ 1.

The corresponding embedding X_{∗}^{k} parametrizes a hypersurface of constant mean curvature- a stationary solution of (0.1). From (3.3) we see that ave_{S}[n(v_{N} − E)e] = 0 (evaluated at Ψ_{∗} ∈ M_{0}.) From (3.4)

(after division by δ_{k}^{2}), we obtain:

c_{n}∇^{M}Scal(ξ_{∗}^{k}) = ave_{S}[nL_{1}[ψ_{∗}^{k}]e_{∗}^{k}]\frac{ϵ_{k}}{δ_{k}} R_{∗}^{k} + ave_{S}[nL_{2}[ψ_{∗}^{k}]e_{∗}^{k}]ϵ_{k}(R_{∗}^{k})^{2}
+ ave_{S}[nS_{∗j}ψ_{∗j}^{k}e_{∗}^{k}]ϵ_{k}(R_{∗}^{k})^{2}
+ (δ_{k})^{−2}B_{ψ}(ξ_{∗}^{k}, e_{∗}^{k}, δ_{k}R_{∗}^{k}, ϵ_{k}ψ_{∗}^{k}).

Since ϵ_{k}^{2} ≪ δ_{k}^{3} (in particular also ϵ_{k} ≪ δ_{k}) and ∥ψ_{∗}^{k}∥_{C^{2+α}}, |R_{∗}^{k}| are bounded independently of k, we see that ξ_{∗}^{k} ∉ ⋃_{p∈Crit(Scal)} would lead to a contradiction for k large enough.

In particular, for δ_{4}, ϵ_{4} sufficiently small, given the estimate (3.5) we obtain ξ ∈ V_{p} and Σ_{∗} ⊂ V_{p} for some p ∈ Crit(Scal). It then follows from Lemma 4.1.(ii) that Σ_{∗} is a leaf of the local c.m.c. foliation at p.

Remark. We expect a stronger result linking the dynamics of (0.4) to the gradient flow of scalar curvature on M, namely that there exists an n-dimensional attracting invariant submanifold of M_{0} on which (3.3) is conjugate to the gradient flow of Scal via a time-preserving homeomorphism (for δ, ϵ small enough).

References

[1] N.D. Alikakos & G. Fusco, The equations of Ostwald ripening for dilute systems, J. Statistical Phys. 95 (1999) 851–866, MR 2000e:74082, Zbl 0952.76089.

[2] H. Amann, Linear and quasilinear parabolic problems, Vol. 1: abstract linear theory, Monographs in Mathematics, 89, Birkhäuser, Boston, 1995, MR 96q:34088, Zbl 0819.35001.

[3] S. Angenent, Nonlinear Analytic Semiflows, Proc. Roy. Soc. Edinburgh, Sect. A 115 (1990) 91–107, MR 91c:34085, Zbl 0723.34047.

[4] G. Da Prato & P.Grisvard, Equations d’évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl. (4) 120 (1979) 329–396, MR 81d:34052, Zbl 0471.35036.

[5] J. Escher & G. Simonett, The volume-preserving mean curvature flow near spheres, Proc. Amer. Math. Soc. 126(9) (1998) 2789–2796, MR 98m:53054, Zbl 0909.53043.

[6] M. Gage, On an area-preserving evolution equation for plane curves, in ‘Nonlinear problems in geometry’, Mobile, AL 1985, Contemp. math., 51, Amer. Math. Soc., Providence, 1986, MR 87q:53003, Zbl 0608.53002.

[7] J.K. Hale, Dynamical Systems and Stability, J. Math. Anal. Appl. 26 (1969) 39–59, MR 39 #5896, Zbl 0179.13303.

[8] G. Huisken, The volume-preserving mean curvature flow, J. Reine Angew. Mat. 382 (1987) 35–48, MR 89d:53015, Zbl 0621.53007.

[9] G. Huisken & S.-T. Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Inventiones Math. 124 (1996) 281–311. MR 96m:53037, Zbl 0858.53071.

[10] H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. math 30(5) (1977) 509–541, MR 56 #1350, Zbl 0354.57005.

[11] A. Lunardi, Analytic Semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, Vol.16, Birkhäuser, Basel, 1995, MR 96e:47039, Zbl 0816.35001.

[12] U. Mayer & G. Simonett, Self-intersections for the surface diffusion and the volume-preserving mean curvature flow, Differential Integral Equations 13 (2000) 1189–1199, MR 2001i:35157, Zbl 1013.53045.

[13] N. Owen & P. Sternberg, Gradient flow and front propagations with boundary contact energy, Proc. Roy. Soc. London Ser. A 437 (1992) 715–728, MR 93e:82053.

[14] R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147(2) (1991) 381–396, MR 92f:53030, Zbl 0722.53022.

Department of Mathematics

University of Athens

Panepistemiopolis-GR 15784 Athens, Greece

Department of Mathematics University of Tennessee Knoxville, TN 37996-1300