j. differential geometry 57 (2001) 301-338

MEAN CURVATURE FLOW OF SURFACES IN EINSTEIN FOUR-MANIFOLDS

MU-TAO WANG

Abstract

Let Σ be a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold (M, ω). We consider the evolution of Σ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms ω′ and ω′′ that determine different orientations and Σ is symplectic with respect to both ω′ and ω′′, we prove the mean curvature flow of Σ exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

1. Introduction

Let (M, g) be a Riemannian manifold and let α be a calibrating kform on M i.e., dα = 0 and comass(α) = 1. In this article, we shall assume additionally α is parallel. This in particular implies M is of special holonomy.

A k-dimensional submanifold is said to be calibrated by α if the restriction of α gives the volume form of the submanifold. A simple application of Stokes’ theorem shows a calibrated submanifold minimizes the volume functional in its homology class. To produce a calibrated submanifold, it is thus natural to consider the gradient flow of the volume functional. By the first variation formula of volume, this is equivalent to evolving a submanifold Σ_{0} in the direction of its mean curvature vector. To make it precise, the mean curvature flow is the solution of the following system of parabolic equations.

\frac{dF}{dt} (x, t) = H(x, t)

Received October 19, 2000, and, in revised form, January 30, 2001

where F : Σ × [0, T) → M is a one parameter family of immersions F_{t}(·) = F (·, t) of Σ into M. H(x, t) is the mean curvature vector of F_{t}(Σ) at F_{t}(x). We say F is the mean curvature flow of the immersed submanifold F_{0}(Σ). For a fixed t, the submanifold F_{t}(Σ) is denoted by Σ_{t}. If we assume M = ℝ^{n}, then in terms of coordinate x^{1}, ⋯, x^{k} on Σ, the mean curvature flow is the following system of parabolic equations

F = F^{A}(x^{1}, ⋯, x^{k}, t), A = 1, …, n

\frac{∂F^{A}}{∂t} = ∑_{i,j,B} g^{ij} P_{B}^{A} \frac{∂^{2}F^{B}}{∂x^{i}∂x^{j}} , A = 1, …, n

where g^{ij} is the inverse matrix to g_{ij} = \frac{∂F^{A}}{∂x^{i}} \frac{∂F^{A}}{∂x^{j}} and P_{B}^{A} = δ_{B}^{A} − g^{kl} \frac{∂F^{A}}{∂x^{k}} \frac{∂F^{B}}{∂x^{l}} is the projection to the normal part.

The mean curvature flow of hypersurfaces has been studied extensively in the last decade. In this case, the mean curvature H is essentially a scalar function and the positivity of H is preserved along the flow. Very little is known in higher codimension except for the curve flows.

This article considers the next simplest higher codimension mean curvature flow, namely a surface flow in a four dimensional manifold. We impose a positivity condition on the initial submanifold. An oriented submanifold Σ is said to be almost calibrated by α if ∗α > 0 where ∗ is the Hodge star operator on Σ.

The following question arises naturally. Can an almost calibrated submanifold be deformed to a calibrated one along the mean curvature flow? We study this question in the case when M is a four-dimensional Einstein manifold and Σ_{0} is almost calibrated by a parallel calibrating form. When M is a Kähler-Einstein surface and the calibrating form is the Kähler form, an almost calibrated surface is a symplectic curve with the induced symplectic structure. A calibrated submanifold in this case is a holomorphic curve.

We use blow up analysis to characterize the singularities of mean curvature flow of symplectic surfaces. It turns out they are all so-called type II singularities.

Theorem A. Let M be a four-dimensional Kähler-Einstein manifold, then a symplectic surface remains symplectic along the mean curvature flow and the flow does not develop any type I singularities.

When M is locally a product and the initial surface is almost calibrated by two calibrating forms, we prove the following long time existence theorem.

Theorem B. Let M be an oriented four-dimensional Einstein manifold with two parallel calibrating forms ω′, ω′′ such that ω′ is self-dual and ω′′ is anti-self-dual. If Σ is a compact oriented surface immersed in M such that ∗ω′, ∗ω′′ > 0 on Σ, then the mean curvature flow of Σ exists smoothly for all time.

We remark that the assumption implies M is locally a product of two surfaces. As for convergence at infinity, we prove the following theorem in the nonnegative curvature case.

Theorem C. Under the same assumption as in Theorem B. When M has nonnegative curvature, there exists a constant 1 > ϵ > 0 such that if Σ is a compact oriented surface immersed in M with ∗ω′, ∗ω′′ > 1 − ϵ on Σ, the mean curvature flow of Σ converges smoothly to a totally geodesic surface at infinity.

This is proved by an uniform estimate of the norm of the second fundamental form.

When M = S^{2} × S^{2}, the combination of Theorem B and C yields:

Theorem D. Let M = (S^{2}, ω_{1}) × (S^{2}, ω_{2}). If Σ is a compact oriented surface embedded in M such that ∗ω_{1} > | ∗ ω_{2}|. Then the mean curvature flow of Σ exists for all time and converges smoothly to an S^{2} × {p}.

This theorem in particular applies to the graph of maps between two Riemann surfaces. Namely, let f : (Σ_{1}, ω_{1}) → (Σ_{2}, ω_{2}) be a map between Riemann surfaces of the same constant curvature and ω_{i} is the volume form of Σ_{i}. We consider the product M = Σ_{1} × Σ_{2} and let ω′ = ω_{1} + ω_{2} and ω′′ = ω_{1} − ω_{2}. If the Jacobian of f is less than one, then we have ∗ω′ > 0 and ∗ω′′ > 0 on the graph of f. Therefore this formulation gives a natural way to deform the map f to a constant map.

Corollary D. Any smooth map between two-spheres with Jacobian less than one deforms to a constant map through the mean curvature flow of the graph.

The article is organized as follows. In Section 2, the parabolic equation satisfied by a general parallel form along the mean curvature flow is derived. Section 3 discusses general calibrating two-forms in a four dimensional space. Section 4 computes the equation satisfied by a Kähler form along the mean curvature flow. Section 5 studies the singularities of mean curvature flow of symplectic surfaces and proves Theorem A. Section 6 concerns long time existence and Theorem B is proved there. Convergence at infinity is discussed in Section 7. Theorem C is proved at the end of this section. Section 8 discusses applications in the positive curvature case and proves Theorem D.

This project began in the fall of 1998 in an attempt to answer Professor S.-T. Yau’s question, “How can a symplectic submanifold be deformed to a holomorphic one.” Theorem A, in particular the result “symplectic remains symplectic” and the exclusion of type I singularity, was obtained in the summer of 1999. It has been presented in the geometry seminars at Stanford, U. C. Berkeley, U. C. Santa Cruz and U of Minnesota between February 2000 and May 2000. I would like to thank Professor R. Schoen and Professor S.-T. Yau for their constant encouragement and invaluable advice. I also have benefitted greatly from the many discussions I had with Professor G. Huisken, Professor L. Simon and Professor B. White.

2. Evolution equations of parallel forms

Let F : Σ^{2} → M^{4} be an isometric immersion of an orientable surface into a four-dimensional Riemannian manifold. We fixed an orientation on Σ. The restriction of the tangent bundle of M to Σ splits as the direct sum of the tangent bundle of Σ and the normal bundle:

T M|_{Σ} = T Σ ⊕ NΣ.

The Levi-Civita connection on M induces a connection on T Σ. We denote the connection on M by \overline{∇} and the induced connection on T Σ by ∇. Therefore,

∇_{X}Y = (\overline{∇}_{X}Y)^{T}

for any tangent vector fields X, Y. Here (·)^{T} denotes the projection from T M onto T Σ and (·)^{N} shall denote the projection onto NΣ.

The second fundamental form A : T Σ × T Σ ↦ NΣ is defined by A(X, Y) = (\overline{∇}_{X}Y)^{N}. We also define B : T Σ×NΣ ↦ T Σ by B(X, N) = (\overline{∇}_{X}N)^{T}. The relation between A and B is

⟨A(X, Y), N⟩ = −⟨Y, B(X, N)⟩.

Note that we have identified X ∈ T Σ with F_{∗}(X) ∈ T M.

Fix a point p ∈ Σ. Let {x^{i}} be a normal coordinate system for Σ at p and {y^{A}} a normal coordinate system for M at F (p). We denote \frac{∂}{∂x^{i}} by ∂_{i} and identify it with \frac{∂F}{∂x_{i}}. The induced metric on Σ is given by g_{kl} = ⟨∂_{k}, ∂_{l}⟩. The mean curvature vector along Σ is the trace of A, i.e., H = g^{kl}A(∂_{k}, ∂_{l}).

Let \overline{ω} be a parallel two form on M and ω = F^{∗}\overline{ω} be the pull-back of ω on Σ. We first compute the rough Laplacian of ω on Σ:

Δω = g^{kl}∇_{∂_{k}}∇_{∂_{l}}ω.

Lemma 2.1.

\begin{matrix}(Δω)(X, Y) = & \overline{ω}((\overline{∇}_{X}H)^{N}, Y) − \overline{ω}((\overline{∇}_{Y} H)^{N}, X) \\ − g^{kl}\overline{ω}((K(∂_{k}, X)∂_{l})^{N}, Y) \\ + g^{kl}\overline{ω}((K(∂_{k}, Y)∂_{l})^{N}, X) \\ + g^{kl}\overline{ω}(B(∂_{k}, A(∂_{l}, X)), Y) \\ − g^{kl}\overline{ω}(B(∂_{k}, A(∂_{l}, Y)), X) \\ + 2g^{kl}\overline{ω}(A(∂_{k}, X), A(∂_{l}, Y)) \end{matrix} (2.1)

where K(X, Y)Z = −\overline{∇}_{X} \overline{∇}_{Y} Z+\overline{∇}_{Y} \overline{∇}_{X}Z−\overline{∇}_{[X,Y]}Z is the curvature operator of M. Notice that ⟨K(X, Y)X, Y ⟩ > 0 if M has positive sectional curvature.

Proof. Since both sides are tensors, we calculate at the point p using a normal coordinate system. Therefore g_{kl} = δ_{kl} and all connection terms vanish. Now

(Δω)(∂_{i}, ∂_{j}) = ∂_{k}[∂_{k}(ω(∂_{i}, ∂_{j})) − ω(∇_{∂_{k}}∂_{i}, ∂_{j}) − ω(∂_{i}, ∇_{∂_{k}}∂_{j})].

The term in the bracket is

∂_{k}(ω(∂_{i}, ∂_{j}))−ω(∇_{∂_{k}}∂_{i}, ∂_{j}) − ω(∂_{i}, ∇_{∂_{k}}∂_{j})
= (\overline{∇}_{∂_{k}} \overline{ω})(∂_{i}, ∂_{j}) + \overline{ω}(\overline{∇}_{∂_{k}}∂_{i}, ∂_{j}) + \overline{ω}(∂_{i}, \overline{∇}_{∂_{k}}∂_{j})
− \overline{ω}(∇_{∂_{k}}∂_{i}, ∂_{j}) − \overline{ω}(∂_{i}, ∇_{∂_{k}}∂_{j})
= \overline{ω}(A(∂_{k}, ∂_{i}), ∂_{j}) + \overline{ω}(∂_{i}, A(∂_{k}, ∂_{j}))

where we have used the fact that \overline{ω} is parallel and \overline{∇}_{∂_{k}}∂_{i} − ∇_{∂_{k}}∂_{i} = A(∂_{k}, ∂_{i}).

Therefore

Δω(∂_{i}, ∂_{j}) = ∂_{k}[\overline{ω}(A(∂_{k}, ∂_{i}), ∂_{j}) + \overline{ω}(∂_{i}, A(∂_{k}, ∂_{j}))]. (2.2)

Use Leibnitz rule and the parallelity of \overline{ω} again.

∂_{k}(\overline{ω}(A(∂_{k}, ∂_{i}), ∂_{j})
= \overline{ω}(\overline{∇}_{∂_{k}}A(∂_{k}, ∂_{i}), ∂_{j}) + \overline{ω}(A(∂_{k}, ∂_{i}), \overline{∇}_{∂_{k}}∂_{j})
= \overline{ω}((\overline{∇}_{∂_{k}}A(∂_{k}, ∂_{i}))^{T} + (\overline{∇}_{∂_{k}}A(∂_{k}, ∂_{i}))^{N}, ∂_{j})
+ \overline{ω}(A(∂_{k}, ∂_{i}), ∇_{∂_{k}}∂_{j} + A(∂_{k}, ∂_{j}))
= \overline{ω}(B(∂_{k}, A(∂_{k}, ∂_{i})), ∂_{j}) + \overline{ω}(A(∂_{k}, ∂_{i}), A(∂_{k}, ∂_{j}))
+ \overline{ω}((\overline{∇}_{∂_{k}}A(∂_{k}, ∂_{i}))^{N}, ∂_{j})

where we have used ∇_{∂_{k}}∂_{j} = 0 at the point p in normal coordinates.

\overline{ω}((\overline{∇}_{∂_{k}}A(∂_{k}, ∂_{i}))^{N}, ∂_{j}) = \overline{ω}((\overline{∇}_{∂_{k}} \overline{∇}_{∂_{i}}∂_{k})^{N}, ∂_{j}) − \overline{ω}((\overline{∇}_{∂_{k}}∇_{∂_{i}}∂_{k})^{N}, ∂_{j}) = \overline{ω}((−K(∂_{k}, ∂_{i})∂_{k}, +\overline{∇}_{∂_{i}} \overline{∇}_{∂_{k}}∂_{k})^{N}, ∂_{j}) − \overline{ω}(B(∂_{k}, ∇_{∂_{i}}∂_{k}), ∂_{j}) = −\overline{ω}((K(∂_{k}, ∂_{i})∂_{k})^{N}, ∂_{j}) + \overline{ω}((\overline{∇}_{∂_{i}}H)^{N} + (\overline{∇}_{∂_{i}}∇_{∂_{k}}∂_{k})^{N}, ∂_{j})

= −\overline{ω}((K(∂_{k}, ∂_{i})∂_{k})^{N}, ∂_{j}) + \overline{ω}((\overline{∇}_{∂_{i}}H)^{N}, ∂_{j})
+ \overline{ω}(B(∂_{i}, ∇_{∂_{k}}∂_{k})), ∂_{j}).

The last term vanishes in normal coordinates.

Thus we have proved

\begin{aligned}∂_{k}(\overline{ω}(A(∂_{k}, ∂_{i}), ∂_{j})) & = \overline{ω}(B(∂_{k}, A(∂_{k}, ∂_{i})), ∂_{j}) + \overline{ω}(A(∂_{k}, ∂_{i}), A(∂_{k}, ∂_{j})) \\ & − \overline{ω}((K(∂_{k}, ∂_{i})∂_{k})^{N}, ∂_{j}) + \overline{ω}((\overline{∇}_{∂_{i}}H)^{N}, ∂_{j}). \end{aligned}

Plug this equation back into Equation (2.2) and anti-symmetrize i, j and the lemma is proved. q.e.d.

Let’s represent the fixed orientation on Σ by a two-form dμ. Let F : Σ × [0, T) → M be the mean curvature flow of Σ. The immersion F_{t} induces a pull-back metric g_{t} on Σ. The volume form of g_{t} is denoted by dμ_{t} = \sqrt{det g_{t}} dμ.

Now we consider the evolution equation of ω_{t} = F_{t}^{∗}(\overline{ω}). This is a family of time-dependent two forms on the fixed surface Σ. Let the one-form α_{t} be defined by α_{t}(X) = \overline{ω}(H_{t}, X).

Lemma 2.2. Along the mean curvature flow

\frac{d}{dt}ω_{t} = dα_{t}.

For any vector field X, Y ∈ T Σ,

\frac{d}{dt}ω_{t}(X, Y) = \overline{ω}((\overline{∇}_{X}H)^{N}, Y) + \overline{ω}(X, (\overline{∇}_{Y} H)^{N})
+ \overline{ω}(B(X, H), Y) + \overline{ω}(X, B(Y, H)).

Proof.

\frac{d}{dt}ω_{t}(∂_{i}, ∂_{j}) = \overline{ω}(\overline{∇}_{H}∂_{i}, ∂_{j}) + \overline{ω}(∂_{i}, \overline{∇}_{H}∂_{j}) = \overline{ω}(\overline{∇}_{∂_{i}}H, ∂_{j}) + \overline{ω}(∂_{i}, \overline{∇}_{∂_{j}}H).

By definition \overline{∇}_{∂_{i}}H = (\overline{∇}_{∂_{i}}H)^{N} + B(∂_{i}, H). On the other hand,

dα_{t}(∂_{i}, ∂_{j}) = ∂_{i}(\overline{ω}(H, ∂_{j})) − ∂_{j}(\overline{ω}(H, ∂_{i})) = \overline{ω}(\overline{∇}_{∂_{i}}H, ∂_{j}) − \overline{ω}(\overline{∇}_{∂_{j}}H, ∂_{i}).

q.e.d.

The volume form dμ_{t} determines a Hodge operator ∗_{t}. Therefore ∗_{t}ω_{t} becomes a time-dependent function on Σ.

Proposition 2.1. Let \overline{ω} be a parallel two-form on M. F_{t} : Σ ↦ M be the t slice of a mean curvature flow and ω_{t} = F_{t}^{∗}(\overline{ω}) be the pull-back form on Σ. Then η_{t} = ∗_{t}ω_{t} satisfies the following parabolic equation:

\frac{d}{dt}η_{t} = (Δ_{t}η_{t}) + |A|^{2} η_{t}
− 2\overline{ω}(A(e_{k}, e_{1}), A(e_{k}, e_{2})) + \overline{ω}((K(e_{k}, e_{1})e_{k})^{N}, e_{2})
− \overline{ω}((K(e_{k}, e_{2})e_{k})^{N}, e_{1})

where |A| is the norm of the second fundamental form,

|A|^{2} = g^{ij}g^{kl}⟨A(∂_{i}, ∂_{k}), A(∂_{j}, ∂_{l})⟩

and {e_{1}, e_{2}} any orthonormal basis with respect to g_{t}.

Proof. Combining the previous two lemmas, we get

\begin{aligned}\frac{d}{dt}ω_{t}(X, Y) = & (Δ_{t}ω_{t})(X, Y) + \overline{ω}(B(X, H), Y) \\ & + \overline{ω}(X, B(Y, H)) + g^{kl}\overline{ω}((K(∂_{k}, X)∂_{l})^{N}, Y) \\ & − g^{kl}\overline{ω}((K(∂_{k}, Y)∂_{l})^{N}, X) \\ & − g^{kl}\overline{ω}(B(∂_{k}, A(∂_{l}, X)), Y) \\ & + g^{kl}\overline{ω}(B(∂_{k}, A(∂_{l}, Y)), X) \\ & − 2g^{kl}\overline{ω}(A(∂_{k}, X), A(∂_{l}, Y)). \end{aligned} (2.3)

Now ∗_{t}ω_{t} = \frac{ω(∂_{1},∂_{2})}{\sqrt{det g_{t}}} where \{∂_{1}, ∂_{2}\} is a fixed coordinate system on Σ and det g_{t} is the determinant of (g_{t})_{ij} = ⟨(F_{t})_{∗}∂_{i}, (F_{t})_{∗}∂_{j}⟩.

It is easy to compute

\frac{d}{dt}\sqrt{det g_{t}} = −|H|^{2}\sqrt{det g_{t}}

where |H| is the norm of the mean curvature vector.

Thus

\frac{d}{dt} ∗_{t} ω_{t} = \frac{1}{\sqrt{det g_{t}}} \frac{d}{dt}ω_{t}(∂_{1}, ∂_{2}) + |H|^{2} ∗_{t} ω_{t}.

Now we use Equation (2.3) with X = ∂_{1} and Y = ∂_{2}.

The first term is \frac{1}{\sqrt{det g_{t}}} (Δ_{t}ω_{t})(∂_{1}, ∂_{2}) = ∗_{t}Δ_{t}ω_{t} = Δ_{t} ∗_{t} ω_{t} because the Hodge ∗_{t} operator is parallel.

For other terms we can take any orthonormal basis {e_{1}, e_{2}} with respect to the metric g_{t} to calculate.

It is not hard to see

\overline{ω}(B(e_{1}, H), e_{2})−\overline{ω}(B(e_{2}, H), e_{1})
= ∗_{t}ω_{t}(⟨B(e_{1}, H), e_{1}⟩ + ⟨B(e_{2}, H), e_{2}⟩)
= − ∗_{t} ω_{t}(⟨A(e_{1}, e_{1}), H⟩ + ⟨A(e_{2}, e_{2}), H⟩)
= − ∗_{t} ω_{t}|H|^{2}.

Likewise,

\overline{ω}(B(e_{k},A(e_{k}, e_{1})), e_{2}) − \overline{ω}(B(e_{k}, A(e_{k}, e_{2})), e_{1})
= ∗_{t}ω_{t}(⟨B(e_{k}, A(e_{k}, e_{1})), e_{1}⟩ + ⟨B(e_{k}, A(e_{k}, e_{2})), e_{2}⟩)
= − ∗_{t} ω_{t}(⟨A(e_{k}, e_{1}), A(e_{k}, e_{1})⟩ + ⟨A(e_{k}, e_{2}), A(e_{k}, e_{2})⟩).

q.e.d.

3. Calibrating two-forms in four-dimensional spaces

Let V ⋍ ℝ^{4} be an inner product space and α ∈ ∧^{2}V^{∗} a two form. We shall use the inner product to identify V and V^{∗} and this induces inner product on all ∧^{k}V^{∗}. First let’s recall the definition of comass of α,

comass(α) = max_{x∈G(2,V)} α(x)

where G(2, V) is the Grassmanian of all two-planes in V. G(2, V) can be described by

G(2, V) = {x ∈ ∧^{2}V, x ∧ x = 0 and |x|^{2} = 1}.

Now we fix an orientation ν ∈ ∧^{4}V^{∗} and normalize so that |ν| = 1. Given any orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}} for V such that ν(e_{1}, e_{2}, e_{3}, e_{4}) = 1, the following two-forms give an orthonormal basis for ∧^{2}V^{∗}.

\begin{matrix}α_{1} = \frac{1}{\sqrt{2}}(e_{1}^{∗} ∧ e_{2}^{∗} + e_{3}^{∗} ∧ e_{4}^{∗}) & β_{1} = \frac{1}{\sqrt{2}}(e_{1}^{∗} ∧ e_{2}^{∗} − e_{3}^{∗} ∧ e_{4}^{∗}) \\ α_{2} = \frac{1}{\sqrt{2}}(e_{1}^{∗} ∧ e_{3}^{∗} − e_{2}^{∗} ∧ e_{4}^{∗}) & β_{2} = \frac{1}{\sqrt{2}}(e_{1}^{∗} ∧ e_{3}^{∗} + e_{2}^{∗} ∧ e_{4}^{∗}) \\ α_{3} = \frac{1}{\sqrt{2}}(e_{1}^{∗} ∧ e_{4}^{∗} + e_{2}^{∗} ∧ e_{3}^{∗}) & β_{3} = \frac{1}{\sqrt{2}}(e_{1}^{∗} ∧ e_{4}^{∗} − e_{2}^{∗} ∧ e_{3}^{∗}). \end{matrix}

These forms serve as coordinate functions on G(2, V), under the identification

x → (α_{i}(x), β_{i}(x)).

An element x in G(2, V) satisfies ∑_{i}(α_{i}(x))^{2} = ∑_{i}(β_{i}(x))^{2} = \frac{1}{2}. Therefore G(2, V) ⋍ S^{2} (\frac{1}{\sqrt{2}}) × S^{2} (\frac{1}{\sqrt{2}}).

Now for any given α ∈ ∧^{2}V^{∗}. We identify α with an element K in End(V) by α(X, Y) = ⟨K(X), Y ⟩. Since \sqrt{−1} K is Hermitian symmetric and purely imaginary, it has real eigenvalues ±λ_{1}, ±λ_{2}. We can require α ∧ α = λ_{1}λ_{2}ν. λ_{1}λ_{2} is actually the Pfaffian of α and det K = (λ_{1}λ_{2})^{2}. A form is self-dual (anti-self-dual ) if λ_{1}λ_{2} = 1(−1).

Lemma 3.1.

comass(α) = max{|λ_{1}|, |λ_{2}|}.

Proof. We can find an orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}} with ν(e_{1}, e_{2}, e_{3}, e_{4}) = 1 such that α = λ_{1}e_{1}^{∗} ∧ e_{2}^{∗} + λ_{2}e_{3}^{∗} ∧ e_{4}^{∗}. In terms of the self-dual and anti-self-dual bases associated with {e_{1}, e_{2}, e_{3}, e_{4}}.

α = \frac{1}{\sqrt{2}}(λ_{1} + λ_{2})α_{1} + \frac{1}{\sqrt{2}}(λ_{1} − λ_{2})β_{1}.

Therefore

α(x) = \frac{1}{\sqrt{2}}(λ_{1} + λ_{2})α_{1}(x) + \frac{1}{\sqrt{2}}(λ_{1} − λ_{2})β_{1}(x)

≤ \frac{1}{2}(|λ_{1} + λ_{2}| + |λ_{1} − λ_{2}|)

= max{λ_{1}, λ_{2}}.

We notice that if |λ_{1}| ≠ |λ_{2}|, a unique plane is calibrated by α. However if |λ_{1}| = |λ_{2}|, then a two-dimensional family of planes in G(2, V) are calibrated by α. q.e.d.

Lemma 3.2. A self-dual or anti-self-dual calibrating form α can be written as α(·, ·) = ⟨K(·), ·⟩ with K ∈ O(4, V), the orthogonal group.

Proof. If α is calibrating and self-dual or anti-self-dual, then max{λ_{1}, λ_{2}} = 1 and λ_{1}λ_{2} = ±1, therefore λ_{1} = ±1 and it is not hard to see that K is an isometry.

On the other hand, if α is induced by an isometry J, then det J = ±1, therefore λ_{1}λ_{2} = ±1 and α is self-dual or anti-self-dual. q.e.d.

Proposition 3.1. Let (x, μ) be an oriented two-plane in V. Let α be a self-dual calibrating form and β be a anti-self-dual calibrating form. Then there exists an orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}} for V with {e_{1}, e_{2}} a basis for x such that μ(e_{1}, e_{2}) > 0 , ν(e_{1}, e_{2}, e_{3}, e_{4}) > 0, α(e_{A}, e_{B}), A, B = 1 ⋯ 4 is of the form

(\begin{matrix}0 & η_{1} & ζ_{1} & 0 \\ −η_{1} & 0 & 0 & −ζ_{1} \\ −ζ_{1} & 0 & 0 & η_{1} \\ 0 & ζ_{1} & −η_{1} & 0 \end{matrix}) (3.1)

where η_{1} = α(e_{1}, e_{2}), ζ_{1}^{2} + η_{1}^{2} = 1, and β(e_{A}, e_{B}) is of the form.

(\begin{matrix}0 & η_{2} & ζ_{2} & 0 \\ −η_{2} & 0 & 0 & ζ_{2} \\ −ζ_{2} & 0 & 0 & −η_{2} \\ 0 & −ζ_{2} & η_{2} & 0 \end{matrix}) (3.2)

where η_{2} = β(e_{1}, e_{2}) and η_{2}^{2} + ζ_{2}^{2} = 1.

Proof. Let K, L be the elements in End(V) corresponding to α and β. If η_{1} ≠ ±1, we take any orthonormal basis {e_{1}, e_{2}} with μ(e_{1}, e_{2}) > 0. Notice that (Ke_{1})^{T} = η_{1}e_{2}. Let

\begin{matrix}e_{3} = \frac{1}{\sqrt{1 − η_{1}^{2}}} (Ke_{1} − η_{1}e_{2}) \\ e_{4} = \frac{−1}{\sqrt{1 − η_{1}^{2}}} (Ke_{2} + η_{1}e_{1}). \end{matrix}

Therefore α_{A,B} is of the required form. If η_{1} = ±1, then any {e_{1}, e_{2}, e_{3}, e_{4}} compatible with μ and ν works.

It is not hard to check that K and L as elements in End(V) commute and KL is a self-adjoint operator. Therefore we can rotate {e_{1}, e_{2}} to get a new basis so that ⟨KLe_{1}, e_{2}⟩ = 0.

This implies

⟨Le_{1}, e_{4}⟩ = \frac{−1}{\sqrt{1 − η_{1}^{2}}} ⟨Le_{1}, Ke_{2} + η_{1}e_{1}⟩
= \frac{1}{\sqrt{1 − η_{1}^{2}}} ⟨KLe_{1}, e_{2}⟩ = 0.

Likewise ⟨Le_{2}, e_{3}⟩ = 0. That ⟨Le_{1}, e_{3}⟩ = −⟨Le_{2}, e_{4}⟩ follows from the fact that β is anti-self-dual. q.e.d.

Finally, we make a remark about α + β. In the above basis α + β is of the form

(\begin{matrix}0 & η_{1} + η_{2} & ζ_{1} + ζ_{2} & 0 \\ −η_{1} − η_{2} & 0 & 0 & −ζ_{1} + ζ_{2} \\ −ζ_{1} − ζ_{2} & 0 & 0 & η_{1} − η_{2} \\ 0 & ζ_{1} − ζ_{2} & −η_{1} + η_{2} & 0 \end{matrix}) . (3.3)

If the eigenvalues of \sqrt{−1}(α+β) are ±λ_{1} and ±λ_{2}, then it is not hard to compute that λ_{1}^{2}λ_{2}^{2} = ((η_{1}+η_{2})(η_{1}−η_{2})−(ζ_{1}+ζ_{2})(−ζ_{1}+ζ_{2}))^{2} = 0 and λ_{1}^{2} + λ_{2}^{2} = (η_{1} + η_{2})^{2} + (ζ_{1} + ζ_{2})^{2} + (η_{1} − η_{2})^{2} + (ζ_{1} − ζ_{2})^{2} = 4. Therefore \frac{1}{2}(α + β) is a calibrating form and calibrates a unique two-plane.

4. Surfaces in Kähler manifolds

In this section, we assume \overline{ω} is a parallel self-dual calibrating two form and \overline{ω}(X, Y) = ⟨J(X), Y ⟩. J is then a parallel almost complex structure. M is therefore a Kähler manifold with Kähler form \overline{ω}.

We shall compute the equation of η_{t} = ∗_{t}ω_{t} along the mean curvature flow.

The following Lemma is well-known.

Lemma 4.1. Let K(·, ·) be the curvature operator of M and Ric(·, ·) be the Ricci tensor of M. In terms of any orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}}, the Ricci form is

Ric(JX, Y) = \frac{1}{2}K(X, Y, e_{A}, J(e_{A})).

Proof. This is seen by the following calculation:

K(JX, e_{A}, Y, e_{A})

= K(JX, e_{A}, J(Y), J(e_{A}))
= −K(JX, JY, J(e_{A}), e_{A}) − K(JX, J(e_{A}), e_{A}, J(Y))
= K(X, Y, e_{A}, J(e_{A})) − K(JX, J(e_{A}), Y, J(e_{A})).

Now K(JX, e_{A}, Y, e_{A}) = K(JX, J(e_{A}), Y, J(e_{A})) since {J(e_{A})} is also an orthonormal basis. q.e.d.

Let F : Σ → M be an isometric immersion. Σ is equipped with a fixed orientation dμ. By Proposition 3.1, for any point p ∈ Σ it is possible to choose an orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}} for T_{p}M such that dμ(e_{1}, e_{2}) > 0 and

\overline{ω}^{2}(e_{1}, e_{2}, e_{3}, e_{4}) = \overline{ω}(e_{1}, e_{2})\overline{ω}(e_{3}, e_{4}) − \overline{ω}(e_{1}, e_{3})\overline{ω}(e_{2}, e_{4})
+ \overline{ω}(e_{1}, e_{4})\overline{ω}(e_{2}, e_{3}) > 0

and such that \overline{ω}_{A,B} = \overline{ω}(e_{A}, e_{B}), A, B = 1 ⋯ 4 is of the form.

(\begin{matrix}0 & η & \sqrt{1 − η^{2}} & 0 \\ −η & 0 & 0 & −\sqrt{1 − η^{2}} \\ −\sqrt{1 − η^{2}} & 0 & 0 & η \\ 0 & \sqrt{1 − η^{2}} & −η & 0 \end{matrix}) (4.1)

where η = ω(e_{1}, e_{2}) = ∗ω.

We first use this basis to calculate the curvature term in Proposition (2.1).

Proposition 4.1. Let Ric(·, ·) be the Ricci tensor of M. \overline{ω} a parallel Kähler form. Then η = ∗_{t}ω_{t} satisfies the following equation:

\frac{d}{dt}η = Δη + η [(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] + (1 − η^{2}) Ric(Je_{1}, e_{2}) (4.2)

where {e_{1}, e_{2}, e_{3}, e_{4}} is any orthonormal basis for T_{p}M such that {e_{1}, e_{2}} forms an orthonormal basis for T Σ_{t}, dμ(e_{1}, e_{2}) > 0 and \overline{ω}^{2}(e_{1}, e_{2}, e_{3}, e_{4}) > 0. A(e_{i}, e_{j}) = h_{3ij}e_{3} + h_{4ij}e_{4} is the second fundamental form.

Remark 4.1. Notice that the term (h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2} depends only on the orientation of {e_{1}, e_{2}, e_{3}, e_{4}} but not on the particular orthonormal basis we choose.

Proof. First we show

\overline{ω}((K(e_{k}, e_{1})e_{k})^{N}, e_{2}) − \overline{ω}((K(e_{k}, e_{2})e_{k})^{N}, e_{1}) = (1 − η^{2}) Ric(Je_{1}, e_{2}). (4.3)

By definition,

\begin{matrix}\overline{ω}((K(e_{k}, e_{1})e_{k})^{N}, & e_{2}) − \overline{ω}((K(e_{k}, e_{2})e_{k})^{N}, e_{1}) \\ = −⟨(Je_{2})^{N}, K(e_{k}, e_{1})e_{k}⟩ + ⟨(Je_{1})^{N}, K(e_{k}, e_{2})e_{k}⟩. \end{matrix}

Therefore when η = ±1, Equation (4.3) is obvious. Therefore we may assume η ≠ ±1 and apply the basis in Equation (4.1) and get

\begin{matrix}\sqrt{1 − η^{2}}(K(e_{k}, e_{2}, e_{k}, e_{3}) + K(e_{k}, e_{1}, e_{k}, e_{4})) \\ = \sqrt{1 − η^{2}}(K(e_{1}, e_{2}, e_{1}, e_{3}) + K(e_{2}, e_{1}, e_{2}, e_{4})). \end{matrix}

By the previous lemma,

Ric(JX, Y) = \frac{1}{2}J_{AB}K(X, Y, e_{A}, e_{B})
= η(K(X, Y, e_{1}, e_{2}) + K(X, Y, e_{3}, e_{4}))
+ \sqrt{1 − η^{2}}(K(X, Y, e_{1}, e_{3}) − K(X, Y, e_{2}, e_{4})).

Since J is parallel and isometry, the curvature tensor is J invariant, therefore we have

K(X, Y, e_{1}, e_{2}) = K(X, Y, J(e_{1}), J(e_{2})).

Use (4.1) again, this is the same as

\begin{matrix}(1 − η^{2})(K(X, Y, e_{1}, e_{2}) + K(X, Y, e_{3}, e_{4})) \\ = η\sqrt{1 − η^{2}}(K(X, Y, e_{1}, e_{3}) − K(X, Y, e_{2}, e_{4})). \end{matrix}

Therefore

Ric(JX, Y) = \frac{1}{\sqrt{1 − η^{2}}} (K(X, Y, e_{1}, e_{3}) − K(X, Y, e_{2}, e_{4})).

Equation (4.3) now follows by substituting X = e_{1}, Y = e_{2}.

We use the basis in Equation (4.1) to calculate the rest terms in Proposition 2.1.

\begin{matrix}η|A|^{2} − 2\overline{ω}(A(e_{k}, e_{1}), A(e_{k}, e_{2})) \\ = η(|A|^{2} − 2h_{31k}h_{42k} + 2h_{41k}h_{32k}). \end{matrix}

Equation (4.2) follows by completing squares. q.e.d.

Remark 4.2. When M is a Kähler manifold with Kähler form \overline{ω} and almost complex structure J. The second fundamental form of a holomorphic submanifold has the symmetry h_{31k} = h_{42k}, h_{41k} = −h_{32k}.

5. Asymptotics of singularities

In this section, we study the asymptotic behavior of singularities of the mean curvature flow. In particular, we show that no type I singularity will occur in the mean curvature flow of symplectic surface in a four-dimensional Kähler-Einstein manifold. Techniques involved are blow-up analysis and monotonicity formula of backward heat kernel.

The following lemma says singularity forms only when the second fundamental form blows up.

Lemma 5.1. Given any mean curvature flow F : Σ × [0, t_{0}) → M, suppose sup_{t∈[0,t_{0})} sup_{x∈Σ} |A|(x, t) is bounded where |A|(x, t) is the norm of the second fundamental form for F_{t}(Σ) at F_{t}(x). Then F can be extended to Σ × [0, t_{0}) for some t_{0} > t_{0}.

Proof. It can be shown that all higher covariant derivatives of the second fundamental form are uniformly bounded. For the detail see [2] for the hypersurface case. q.e.d.

Since the study of singularities is local, it is more convenient to adopt an unparametrized definition of mean curvature flow introduced in [12]. Let M be an m-dimensional Riemannian manifold of bounded geometry. An immersed smooth submanifold S ⊂ M × ℝ is a smooth flow if the function τ : M × ℝ → ℝ, τ(y, t) = t has no critical points in S. S_{t} = S ∩M ×{t} is called the t-slice of S. At each point (y, t) ∈ S, the normal velocity v(y, t) is the unique vector that satisfies v is normal to S_{t} and v+ \frac{∂}{∂t} is tangent to S. H(y, t) is the mean curvature vector of S_{t} at (y, t). We allow M and S_{t} to have boundary. In fact, all unparametrized flow considered in this article is of the form ∪_{t∈[0,t_{0})}(F_{t}(Σ)∩B)×{t}, where F is a parameterized mean curvature flow of a compact manifold Σ without boundary and B is an neighborhood of y in a complete Riemannian manifold. Therefore ∂S_{t} ⊂ ∂M.

Definition 5.1. A smooth flow S is called a (unparametrized) mean curvature flow if

v(y, t) = H(y, t)

at each point (y, t) ∈ S.

Let (y_{0}, t_{0}) be an interior point in M ×ℝ. When M is the Euclidean space, in [3] Huisken introduces the backward heat kernel to study the asymptotic behavior near singular points. Recall the (n-dimensional) backward heat kernel ρ_{y_{0},t_{0}} at (y_{0}, t_{0}):

ρ_{y_{0},t_{0}}(y, t) = \frac{1}{(4π(t_{0} − t))^{\frac{n}{2}}} exp (\frac{−|y − y_{0}|^{2}}{4(t_{0} − t)} ) . (5.1)

The monotonicity formula of Huisken asserts for t < t_{0}

\frac{d}{dt} ∫ ρ_{y_{0},t_{0}}dμ_{t} ≤ 0.

For general Riemannian manifold M, following [11], we isometrically embed M into ℝ^{N}. The mean curvature flow of Σ in M now reads.

\frac{d}{dt}F = H = H + E

where F is the coordinate function in ℝ^{N}, H is the mean curvature vector of Σ in M, H is the mean curvature vector of Σ in ℝ^{N}, and

E = ∑_{i} \overline{A}(e_{i}, e_{i}).

Here \overline{A} denotes the second fundamental form of M in ℝ^{N} and \{e_{i}\} is an orthonormal basis for T Σ_{t}.

In the general case ∫ ρ_{y_{0},t_{0}}dμ_{t} is no longer decreasing, however the following is still true:

Proposition 5.1. Let S ⊂ M × ℝ be a mean curvature flow such that ∂S_{t} ⊂ ∂M. We fix an isometric embedding M ↪ ℝ^{N} and let ρ_{y_{0},t_{0}} be the (n-dimensional) backward heat kernel at (y_{0}, t_{0}). Then the limit

lim_{t→t_{0}} ∫ ρ_{y_{0},t_{0}}dμ_{t}

exists, where dμ_{t} is the Radon measure associated with S_{t} ⊂ M.

Proof. See Proposition 11 in [11]. q.e.d.

The limit is called the Gaussian density of S at (y_{0}, t_{0}) in [12]. The Gaussian density can be used to detect singularities of mean curvature flow. The following theorem of White in [12] is a parabolic analogue of Allard’s regularity theorem.

Theorem 5.1. There is an ϵ > 0 such that whenever

lim_{t→t_{0}} ∫ ρ_{y_{0},t_{0}}dμ_{t} < 1 + ϵ,

it can concluded that (y_{0}, t_{0}) is a regular point of S.

A regular point is a point where the second fundamental form is locally bounded in Hölder norm.

To study singularity, we consider the parabolic blow-up near a possible singular point. Let F : Σ × [0, t_{0}) → M ↪ ℝ^{N} be a parameterized mean curvature flow. Let B be a ball about y_{0} of radius r in ℝ^{N}. Take S = ∪_{t∈[0,t_{0})}(F_{t}(Σ) ∩ B) × {t}, then S is an unparametrized mean curvature flow in B.

For any λ > 1, the parabolic dilation D_{λ} at (y_{0}, t_{0}) is defined by

\begin{aligned}D_{λ} : ℝ^{N} × [0, t_{0}) & → ℝ^{N} × [−λ^{2}t_{0}, 0) \\ (y, t) & → (λ(y − y_{0}), λ^{2}(t − t_{0})). \end{aligned} (5.2)

For any s, −λ^{2}t_{0} ≤ s < 0, the two slices S_{s}^{λ} and S_{t_{0}+ \frac{s}{λ^{2}}} can be identified and dμ_{s}^{λ} = λ^{n}dμ_{t}.

It is not hard to check that if we denote F_{s}^{λ}(x) = λ(F_{t}(x) − y_{0}) for s = λ^{2}(t − t_{0}), then

ρ_{0,0}(F_{s}^{λ}(x), s) = ρ_{0,0}(λ(F_{t}(x) − y_{0}), λ^{2}(t − t_{0})) = \frac{1}{λ^{n}} ρ_{y_{0},t_{0}}(F_{t}(x), t).

Therefore

∫ ρ_{y_{0},t_{0}}dμ_{t} = ∫ ρ_{0,0}dμ_{s}^{λ}

is invariant under the parabolic dilation.

The singularity of S near (y_{0}, t_{0}) is reflected in the asymptotic behavior of S^{λ} as λ → ∞.

Take any sequence λ_{i} → ∞, it can be proved as in [4] and [11] that a subsequence of S^{λ_{i}} converges to a Brakke flow S^{∞} ⊂ ℝ^{N} × (−∞, 0). S^{∞} is called a tangent flow of S at (y_{0}, t_{0}).

Now we state and prove the main proposition in this section.

Proposition 5.2. If F : Σ × [0, t_{0}) → M ↪ ℝ^{N} is a mean curvature flow of an orientable surface in a (real) four dimensional Kähler manifold M. Assume the second fundamental form of M ↪ ℝ^{N} is bounded. Let ω(·, ·) = ⟨J(·), ·⟩ be a Kähler form on M. If there exist δ, C > 0 such that η_{t} = ∗ω_{t} > δ on F_{t}(Σ) for t ∈ [0, t_{0}) and such that |A|^{2} ≤ \frac{C}{t_{0}−t}, then F can be extended to Σ × [0, t_{0}) for some t_{0} > t_{0}.

Proof. For y_{0} ∈ M, we shall consider the blow up of the mean curvature flow at (y_{0}, t_{0}). Let B be a ball of radius r about y_{0} in ℝ^{N} and ψ be a cut-off function supported in B so that ψ ≡ 1 in the ball of radius \frac{r}{2} about y_{0}. We assume

|\overline{∇}ψ| + |\overline{∇}\overline{∇}ψ| ≤ C

where \overline{∇} is the covariant derivative on ℝ^{N}. Recall the equation for η is

\frac{d}{dt}η = Δη + η [(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] + (1 − η^{2}) Ric(Je_{1}, e_{2}).

The backward heat kernel ρ_{y_{0},t_{0}} satisfies the following parabolic equation along the mean curvature flow. Notice that ∇ and Δ are the covariant derivative and the Laplace operator on Σ_{t} respectively.

\frac{d}{dt}ρ_{y_{0},t_{0}} = −Δρ_{y_{0},t_{0}} − ρ_{y_{0},t_{0}} (\frac{|F^{⊥}|^{2}}{4(t_{0} − t)^{2}} + \frac{F^{⊥} · H}{t_{0} − t} + \frac{F^{⊥} · E}{2(t_{0} − t)}) (5.3)

where F^{⊥} is the component of F ∈ T ℝ^{N} in T ℝ^{N}/T Σ_{t}. This equation for mean curvature flow in a Euclidean space is essentially derived by Huisken [3] and in a general ambient manifold by White [11]. It is derived in the next paragraph for completeness. Recall that

\frac{d}{dt}F (x, t) = H = H + E

where H ∈ T M/T Σ is the mean curvature vector of Σ_{t} in M and H ∈ T ℝ^{N}/T Σ is the mean curvature vector of Σ_{t} in ℝ^{N}.

We may assume y_{0} is the origin and then

ρ_{y_{0},t_{0}}(F (x, t), t) = \frac{1}{(4π(t_{0} − t))^{\frac{n}{2}}} exp (\frac{−|F (x, t)|^{2}}{4(t_{0} − t)} ) .

Abbreviate ρ_{y_{0},t_{0}}(F (x, t), t) by ρ, it is not hard to see

\frac{d}{dt}ρ = ρ [\frac{n}{2(t_{0} − t)} − \frac{|F (x, t)|^{2}}{4(t_{0} − t)^{2}} − \frac{F · H}{2(t_{0} − t)}] . (5.4)

We shall compute ∑_{i}(\overline{∇}_{e_{i}} \overline{∇}ρ) · e_{i} in two different ways, where \overline{∇} denotes the covariant derivative in ℝ^{N} and {e_{i}} is an orthonormal basis for T Σ.

∑_{i}(\overline{∇}_{e_{i}} \overline{∇}ρ) · e_{i} = ∑_{i} \overline{∇}_{e_{i}}(∇ρ + (\overline{∇}ρ)^{T ℝ^{N} /T Σ}) · e_{i} = Δρ − \overline{∇}ρ · H.

\overline{∇}ρ = −\frac{ρ}{2(t_{0}−t)}F, thus

∑_{i}(\overline{∇}_{e_{i}} \overline{∇}ρ) · e_{i} = Δρ + \frac{1}{2(t_{0} − t)}F · H.

On the other hand,

∑_{i}(\overline{∇}_{e_{i}} \overline{∇}ρ) · e_{i} = ∑_{i} \overline{∇}_{e_{i}} (−\frac{ρ}{2(t_{0} − t)}F ) · e_{i}

= −\frac{1}{2(t_{0} − t)} (∇ρ · F + ρ ∑_{i} \overline{∇}_{e_{i}}F · e_{i})

= −\frac{1}{2(t_{0} − t)} (−\frac{ρ}{2(t_{0} − t)}F^{T Σ} · F + nρ)

= ρ (\frac{1}{4(t_{0} − t)^{2}} |F^{T Σ}|^{2} − \frac{n}{2(t_{0} − t)}) .

Compare these two equalities, we get

Δρ = ρ [−\frac{1}{2(t_{0} − t)}F · H + \frac{1}{4(t_{0} − t)^{2}} |F^{T Σ}|^{2} − \frac{n}{2(t_{0} − t)}] . (5.5)

Now add Equations (5.4) and (5.5), we get

\begin{aligned}\frac{d}{dt}ρ + Δρ & = \frac{ρ}{4(t_{0} − t)^{2}} (|F^{T Σ}|^{2} − |F |^{2}) − \frac{ρ}{2(t_{0} − t)}(F · H + F · H) \\ & = −\frac{ρ}{4(t_{0} − t)^{2}} |F^{⊥}|^{2} − \frac{ρ}{2(t_{0} − t)}(F^{⊥} · H + F^{⊥} · H) \end{aligned}

where F^{⊥} = (F)^{T ℝ^{N} /T Σ}. Recall that H = H + E and we get Equation (5.3).

The minus sign in front of the Laplacian in Equation (5.3) indicates the fact that ρ satisfies the backward heat equation. The following inequality is particularly useful when deal with backward heat kernels:

(5.6) g(−Δρ) + (Δg)ρ = −div(∇ρ g) + div(ρ∇g).

The volume form dμ_{t} of Σ_{t} satisfies the equation

\frac{d}{dt}dμ_{t} = −|H|^{2}dμ_{t} = −H · (H + E)dμ_{t}.

mean curvature flow 319

Therefore,

\frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t} = ∫ [\frac{d}{dt}ψ(1 − η)] ρ_{y_{0},t_{0}} dμ_{t}

+ ∫ ψ(1 − η) [\frac{d}{dt}ρ_{y_{0},t_{0}}] dμ_{t}

− ∫ ψ(1 − η)ρ_{y_{0},t_{0}} H · (H + E)dμ_{t}.

Plug the Equation (5.3) for \frac{d}{dt}ρ_{y_{0},t_{0}}, use the identity (5.6) with g = ψ(1 − η), and complete square we get

\begin{matrix}\frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t} \\ = ∫ [\frac{d}{dt}(ψ(1 − η)) − Δ(ψ(1 − η))] ρ_{y_{0},t_{0}} dμ_{t} \\ − ∫ ψ(1 − η)ρ_{y_{0},t_{0}} [|H + \frac{1}{2(t_{0} − t)}F^{⊥}|^{2} \\ + (H + \frac{1}{2(t_{0} − t)}F^{⊥}) · E] dμ_{t}. \end{matrix} (5.7)

Now

\frac{d}{dt}(ψ(1 − η)) − Δ(ψ(1 − η))

= ψ (−\frac{d}{dt}η + Δη) + (\overline{∇}ψ · H)(1 − η) + 2∇ψ · ∇η − Δψ(1 − η)

where we use \frac{d}{dt}ψ = \overline{∇}ψ · H.

Integration by parts gives

∫ [2∇ψ · ∇η − Δψ(1 − η)]ρ_{y_{0},t_{0}}dμ_{t}

= ∫ [∇ψ · ∇η ρ_{y_{0},t_{0}} + ∇ψ · ∇ρ_{y_{0},t_{0}} (1 − η)]dμ_{t}.

320 mu-tao wang

Therefore, we have

\frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t}

= − ∫ ψηρ_{y_{0},t_{0}}[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] dμ_{t}

− ∫ ψ(1 − η)ρ_{y_{0},t_{0}} |H + \frac{1}{2(t_{0} − t)}F^{⊥} + \frac{E}{2} |^{2} dμ_{t}

+ ∫ ψ(1 − η)ρ_{y_{0},t_{0}} \frac{|E|^{2}}{4} dμ_{t}

+ ∫ [(\overline{∇}ψ · H)(1 − η)ρ_{y_{0},t_{0}} + ∇ψ · ∇η ρ_{y_{0},t_{0}}

+ ∇ψ · ∇ρ_{y_{0},t_{0}} (1 − η)]dμ_{t}.

Since |E| and ∫ ρ_{y_{0},t_{0}}dμ_{t} are both bounded,

\frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t}

≤ C − ∫ ψηρ_{y_{0},t_{0}}[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] dμ_{t}

+ ∫ [(\overline{∇}ψ · H)(1 − η)ρ_{y_{0},t_{0}} + ∇ψ · ∇η ρ_{y_{0},t_{0}}
+ ∇ψ · ∇ρ_{y_{0},t_{0}} (1 − η)]dμ_{t}.

The last term is also bounded by the following computation: ∫ ∇ψ · ∇ρ_{y_{0},t_{0}} (1 − η) dμ_{t} ≤ C ∫_{B\B_{\frac{1}{2} r}(y_{0})} |∇ρ_{y_{0},t_{0}}|dμ_{t}.

Since ∇ρ_{y_{0},t_{0}} = −ρ_{y_{0},t_{0}} \frac{∇ |F −y_{0}|^{2}}{4(t_{0}−t)} and

|∇ |F − y_{0}|^{2}| ≤ |\overline{∇} |F − y_{0}|^{2}|| ≤ 2|F − y_{0}|,

we have

∫ ∇ψ · ∇ρ_{y_{0},t_{0}} (1 − η) dμ_{t}

≤ C ∫_{B\B_{\frac{1}{2} r}(y_{0})} \frac{1}{(t_{0} − t)^{\frac{n}{2} +1}} exp(\frac{−\frac{1}{4}r^{2}}{4(t_{0} − t)})dμ_{t}.

The last expression approaches zero as t → t_{0}.

mean curvature flow 321

Therefore

\frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t}

≤ C − ∫ ψηρ_{y_{0},t_{0}}[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] dμ_{t}

+ ∫ (\overline{∇}ψ · H)(1 − η)ρ_{y_{0},t_{0}}dμ_{t} + ∫ ∇ψ · ∇η ρ_{y_{0},t_{0}}dμ_{t}.

The term ∫ ∇ψ · ∇η ρ_{y_{0},t_{0}} dμ_{t} can be written in the following

∫ (\frac{∇ψ}{\sqrt{ψ}} \sqrt{ρ_{y_{0},t_{0}}}) · (∇η \sqrt{ψ} \sqrt{ρ_{y_{0},t_{0}}}) dμ_{t}

≤ \frac{1}{4ϵ^{2}} ∫ \frac{|\overline{∇}ψ|^{2}}{ψ} ρ_{y_{0},t_{0}} dμ_{t} + ϵ^{2} ∫ |∇η|^{2}ψρ_{y_{0},t_{0}}dμ_{t}

where we use |∇ψ|^{2} ≤ |\overline{∇}ψ|^{2}.

In a normal coordinate system, we compute ∇η, again use the basis in Equation (4.1).

\begin{matrix}∂_{k}η = & ∂_{k} (\frac{ω(∂_{1}, ∂_{2})}{\sqrt{det g}}) \\ = & ∂_{k}(ω(∂_{1}, ∂_{2})) \\ = & ω(A(∂_{k}, ∂_{1}), ∂_{2}) + ω(∂_{1}, A(∂_{k}, ∂_{2})) \\ = & h_{α1k}ω_{α2} + h_{α2k}ω_{1α} \\ = & \sqrt{1 − η^{2}}(h_{41k} + h_{32k}). \end{matrix} (5.8)

Therefore |∇η|^{2} ≤ (1 − η^{2})(h_{41k} + h_{32k})^{2} and thus ∫ ∇ψ · ∇η ρ_{y_{0},t_{0}} dμ_{t}

≤ \frac{1}{4ϵ^{2}} ∫ \frac{|\overline{∇}ψ|^{2}}{ψ} ρ_{y_{0},t_{0}} dμ_{t} + ϵ^{2} ∫ (1 − η^{2})(h_{41k} + h_{32k})^{2}ψρ_{y_{0},t_{0}}dμ_{t}.

Likewise since |H|^{2} ≤ 2[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}], we have

∫ (\overline{∇}ψ · H)(1 − η)ρ_{y_{0},t_{0}}dμ_{t}

≤ \frac{1}{4ϵ^{2}} ∫ \frac{|\overline{∇}ψ|^{2}}{ψ} ρ_{y_{0},t_{0}} dμ_{t} + 2ϵ^{2} ∫ (1 − η)^{2}[(h_{31k} − h_{42k})^{2}

+ (h_{41k} + h_{32k})^{2}]ψρ_{y_{0},t_{0}}dμ_{t}.

Since ψ is of compact support, by Lemma 6.6 in [5], \frac{|\overline{∇}ψ|^{2}}{ψ} ≤ 2 max |\overline{∇}\overline{∇}ψ| is bounded.

Since η > δ, we can choose ϵ small enough so that

\begin{matrix}\frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t} \\ ≤ C − C_{δ} ∫ ψρ_{y_{0},t_{0}}[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] dμ_{t} \end{matrix} (5.9)

where C_{δ} is a constant that depends on δ.

From this we see that lim_{t→t_{0}} ∫ ψ(1 − η)ρ_{y_{0},t_{0}}dμ_{t} exists.

For λ > 1, let’s study the flow S^{λ} ⊂ ℝ^{N} × [−λ^{2}t_{0}, 0). Let ρ_{0,0,}^{λ}(y, s) be the backward heat kernel at (0, 0) and ψ^{λ}(F_{s}^{λ}(x)) = ψ(F_{t}(x)). Recall that t = t_{0} + \frac{s}{λ^{2}} . Thus

\frac{d}{ds} ∫ ψ^{λ}(1 − η^{λ})ρ_{0,0}^{λ} dμ_{s}^{λ}

= \frac{1}{λ^{2}} \frac{d}{dt} ∫ ψ(1 − η)ρ_{y_{0},t_{0}} dμ_{t}

≤ \frac{C}{λ^{2}} − \frac{C_{δ}}{λ^{2}} ∫ ψρ_{y_{0},t_{0}}[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] dμ_{t}.

We notice that η is a scaling invariant quantity therefore η^{λ} = η. It is not hard to check that

\frac{1}{λ^{2}} ∫ ψρ_{y_{0},t_{0}}[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] dμ_{t}

= ∫ ψ^{λ}ρ_{0,0}^{λ}[(h_{31k}^{λ} − h_{42k}^{λ})^{2} + (h_{32k}^{λ} + h_{41k}^{λ})^{2}] dμ_{s}^{λ}.

This is because ρ_{y_{0},t_{0}}dμ_{t} is invariant under the parabolic scaling and the norm of second fundamental form scales like the inverse of the distance.

Therefore

\begin{matrix}\frac{d}{ds} ∫ ψ^{λ}(1 − η^{λ})ρ_{0,0}^{λ} dμ_{s}^{λ} \\ ≤ \frac{C}{λ^{2}} − C_{δ} ∫ ψ^{λ}ρ_{0,0}^{λ}[(h_{31k}^{λ} − h_{42k}^{λ})^{2} + (h_{32k}^{λ} + h_{41k}^{λ})^{2}] dμ_{s}^{λ}. \end{matrix}

Compare with Equation (5.9) and we see this reflects the correct scaling for the parabolic blow-up.

Take any τ > 0 and integrate from −1 − τ to −1.

\begin{matrix}C_{δ} ∫_{−1−τ}^{−1} ∫ ψ^{λ}ρ_{0,0}^{λ} [(h_{31k}^{λ} − h_{42k}^{λ})^{2} + (h_{32k}^{λ} + h_{41k}^{λ})^{2}] dμ_{s}^{λ}ds \\ ≤ ∫ ψ^{λ}(1 − η^{λ})ρ_{0,0}^{λ} dμ_{−1}^{λ} − ∫ ψ^{λ}(1 − η^{λ})ρ_{0,0}^{λ} dμ_{−1−τ}^{λ} + \frac{C}{λ^{2}} . \end{matrix} (5.10)

Notice that

∫ ψ^{λ}(1 − η^{λ})ρ_{0,0}^{λ}dμ_{s}^{λ} = ∫ ψ(1 − η)ρ_{y_{0},t_{0}}dμ_{t_{0}+ \frac{s}{λ^{2}}} .

This equality means the quantity ∫ ψ(1 − η)ρ_{y_{0},t_{0}}dμ_{t} is invariant under parabolic scaling. This fact is extremely important in applying the Monotonicity formula. Recall the natural Monotonicity formula for the volume

\frac{d}{dt} ∫ dμ_{t} = − ∫ |H|^{2}dμ_{t}.

But ∫ dμ_{s}^{λ} = λ^{2} ∫ dμ_{t} is not scaling invariant. This deteriorates the usefulness of the formula in the blow-up analysis.

Now the right hand side in Equation (5.10) tends to zero as λ → ∞. For any sequence λ_{i} → ∞, we can choose s_{i} → −1 such that

∫ ψ^{λ_{i}}ρ_{0,0}^{λ_{i}} [(h_{31k}^{λ_{i}} − h_{42k}^{λ_{i}})^{2} + (h_{32k}^{λ_{i}} + h_{41k}^{λ_{i}})^{2}] dμ_{s_{i}}^{λ_{i}} → 0

as i → ∞.

It is not hard to compute that

|A|^{2}(S_{s}^{λ_{i}}) = \frac{1}{λ_{i}^{2}} |A|^{2} (S_{t_{0}+_{\frac{s}{λ^{2}_{i}}}} ) = (\frac{−1}{s}) (t_{0} − t_{i})|A|^{2}(S_{t_{i}}).

The assumption implies each Σ_{s}^{λ_{i}} has uniformly bounded second fundamental form. By the same method used in [3], any higher covariant derivatives of the second fundamental form of S_{s}^{λ_{i}} is bounded. Therefore the convergence S_{s_{i}}^{λ_{i}} → S_{−1}^{∞} is smooth.

We may assume each S_{t} is connected by taking connected components. Therefore we have (h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2} = 0 for S_{−1}^{∞}. This implies ∇η = 0 and H = 0. Applying the same argument to the monotonicity formula for ∫ ψρ_{y_{0},t_{0}}dμ_{t} gives H + \frac{1}{2}F^{⊥} = 0 for S_{−1}^{∞}. To sum up, we get F^{⊥} = 0 and ∇η = 0 for S_{−1}^{∞}. The first condition implies S_{−1}^{∞} is a plane with multiplicity one. On the other hand,

lim_{t_{i}→t_{0}} ∫ ρ_{y_{0},t_{0}}dμ_{t_{i}} = lim_{i→∞} ∫ \frac{1}{4π(−s_{i})} exp (\frac{−|F_{s_{i}}^{λ_{i}}|^{2}}{4(−s_{i})} ) dμ_{s_{i}}^{λ_{i}}

= ∫ \frac{1}{4π} exp (\frac{−|F_{−1}^{∞}|^{2}}{4} ) dμ_{−1}^{∞}

where λ_{i} = \sqrt{\frac{−s_{i}}{t_{0}−t_{i}}}.

The last Gaussian integral for a plane can be calculated directly and is equal to 1. By White’s theorem, (y_{0}, t_{0}) is a regular point. q.e.d.

We recall the following definition of type I singularities for the mean curvature flow.

Definition 5.2. A singularity at t_{0} is called type I if there exists a C such that |A|^{2} ≤ \frac{C}{t_{0}−t}.

Recall a Kähler manifold M is called Kähler-Einstein if Ric = cg for some constant c. In this case, the scalar curvature s of M is a constant and c = \frac{s}{4}. A immersion F : Σ → M is symplectic if η > 0.

Theorem A. Let M be a four-dimensional Kähler-Einstein manifold, then a symplectic surface remains symplectic along the mean curvature flow and the flow does not develop any type I singularities.

Proof. Since Ric(J·, ·) = ω(·, ·) by Proposition 4.1, the equation of η = ∗ω now becomes

\frac{d}{dt}η = Δη + η [(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2} + c(1 − η^{2})] . (5.11)

The first assertion follows from maximum principle for parabolic equations. Actually, when c ≥ 0, i.e., the nonnegative scalar curvature case, the function min_{Σ} η_{t} is a non-decreasing function of t. In any case, by comparison theorem for parabolic equations, η has a positive lower bound at any finite time and Proposition 5.2 is applicable. q.e.d.

Remark 5.1. The same argument can be used to prove there is no type I singularity for the mean curvature flow of an almost calibrated Lagrangian submanifolds in a Calabi-Yau manifold M. Here the almost calibrated condition is ∗Ω > 0 where Ω is the real part of the canonical form on M. In fact, ∗Ω satisfies

\frac{d}{dt} ∗ Ω = Δ ∗ Ω + |H|^{2} ∗ Ω.

A smooth blow-up limit satisfies H + \frac{1}{2}F^{⊥} = 0 and H = 0 and is thus a linear subspace.

6. Long time existence and convergence

In this section, we study the problem of long time existence. The main result is the following.

Proposition 6.1. Let M be an oriented four-dimensional compact manifold. Let ω′ and ω′′ be two parallel calibrating form such that ω′ is self-dual and ω′′ is anti-self-dual. Let (Σ_{0}, dμ) ↪ M be an compact surface with orientation dμ. Let F : Σ × [0, t_{0}) → M be the mean curvature flow of Σ_{0} such that there exist a δ > 0 with ∗ω′ > δ and ∗ω′′ > δ on F_{t}(Σ) for 0 ≤ t < t_{0}. Then F can be extended smoothly to Σ × [0, t_{0}) for some t_{0} > t_{0}.

Proof. The assumption implies (ω′)^{2} and (ω′′)^{2} determine different orientations on M. Choose an orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}} for T M with {e_{1}, e_{2}} a basis for T Σ such that (ω′)^{2}(e_{1}, e_{2}, e_{3}, e_{4}) > 0 and dμ(e_{1}, e_{2}) > 0.

Both ω′ and ω′′ are parallel calibrating forms and Proposition 4.1 is applicable. Therefore,

\begin{aligned}\frac{d}{dt}η′ = & Δη′ + η′[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] \\ & + (1 − (η′)^{2}) Ric(J_{1}(e_{1}), e_{2}). \end{aligned}

On the other hand, by switching e_{3} and e_{4},

\begin{aligned}\frac{d}{dt}η′′ = & Δη′′ + η′′[(h_{41k} − h_{32k})^{2} + (h_{42k} + h_{31k})^{2}] \\ & + (1 − (η′′)^{2}) Ric(J_{2}(e_{1}), e_{2}). \end{aligned}

Adding these two equations and denote η′ + η′′ by μ, we get

\begin{matrix}\frac{d}{dt}μ = & Δμ + μ|A|^{2} + 2(η′ − η′′)h_{32k}h_{41k} − 2(η′ − η′′)h_{31k}h_{42k} \\ + (1 − (η′)^{2}) Ric(J_{1}(e_{1}), e_{2}) + (1 − (η′′)^{2}) Ric(J_{2}(e_{1}), e_{2}). \end{matrix}

Write μ = 2(min{η′, η′′}) + |η′ − η′′|, then μ ≥ 2δ + |η′ − η′′|. After completing square, μ satisfies the following inequality:

\frac{d}{dt}μ ≥ Δμ + 2δ|A|^{2} − C

where −C is the lower bound of the Ricci curvature of M, Ric ≥ −Cg.

As before, we can isometrically embed M into ℝ^{N}. To detect a possible singularity at a point (y_{0}, t_{0}), where y_{0} ∈ M ↪ ℝ^{N} and t_{0} < ∞, take a ball B of radius r about y_{0} ∈ ℝ^{N} and ψ a cut-off function as in the proof of Proposition 5.2. A similar argument yields the following inequality:

\frac{d}{dt} ∫ ψ(2 − μ)ρ_{y_{0},t_{0}} dμ_{t} ≤ C − C_{δ} ∫ ψρ_{y_{0},t_{0}}|A|^{2} dμ_{t}

where C_{δ} is a constant depend on δ.

Therefore lim_{t→t_{0}} ∫ ψρ_{y_{0},t_{0}}(2 − μ)dμ_{t} exists. Let S^{λ_{i}} be a blow-up sequence at (y_{0}, t_{0}) that converges to S^{∞}. As in the previous section we can show for a fixed τ > 0,

∫_{−1−τ}^{−1} ∫ ψ^{λ_{j}}ρ_{0,0}^{λ_{j}}|A|^{2}dμ_{s}^{λ_{j}}ds ≤ C(j)

where C(j) → 0 as λ_{j} → ∞.

Choose τ_{j} → 0 such that \frac{C(j)}{τ_{j}} → 0 and s_{j} ∈ [−1 − τ_{j}, −1] so that

∫ ψ^{λ_{j}}ρ_{0,0}^{λ_{j}}|A|^{2}dμ_{s_{j}}^{λ_{j}} ≤ \frac{C(j)}{τ_{j}}.

We investigate this inequality more carefully. Notice that ψ^{λ_{j}} is supported in B_{λ_{j}r}(0) ⊂ ℝ^{N} and ψ^{λ_{j}} ≡ 1 in B_{\frac{λ_{j} r}{2}} (0). Also

ρ_{0,0}^{λ_{j}}(F_{s_{j}}^{λ_{j}}) = \frac{1}{4π(−s_{j})} exp (\frac{−|F_{s_{j}}^{λ_{j}}|^{2}}{4(−s_{j})} ) .

If we consider for any R > 0, the ball of radius R, B_{R}(0) ⊂ ℝ^{N}, when j is large enough, we may assume \frac{λ_{j}r}{2} > R and −1 < s_{j} < −\frac{1}{2}, then

∫ ψ^{λ_{j}}ρ_{0,0}^{λ_{j}}|A|^{2}dμ_{s_{j}}^{λ_{j}} ≥ \frac{1}{2π} exp (\frac{−R^{2}}{2} ) ∫_{Σ_{s_{j}}^{λ_{j}} ∩B_{R}(0)} |A|^{2}dμ_{s_{j}}^{λ_{j}}.

This implies for any compact set K ⊂ ℝ^{N},

∫_{Σ_{s_{j}}^{λ_{j}} ∩K} |A|^{2}dμ_{s_{j}}^{λ_{j}} → 0 as j → ∞.

Now we claim this together with the fact that μ has a positive lower bound imply lim_{j→∞} ∫ ρ_{y_{0},t_{0}}dμ_{t_{0}+ \frac{s_{j}}{λ^{2}_{j}}} = lim_{j→∞} ∫ ρ_{0,0}dμ_{s_{j}}^{λ_{j}} ≤ 1. We may

assume the origin in ℝ^{N} is a limit point of Σ_{s_{j}}^{λ_{j}}, otherwise the limit is zero.

We notice that ω′ +ω′′ is a parallel two form with λ_{1} = 2 and λ_{2} = 0 from the last paragraph in §3. Therefore the holonomy group of M splits into SO(2) × SO(2) and M is locally a Riemannian product. For simplicity, we shall assume M is a product Σ_{1} ×Σ_{2} such that \frac{1}{2}(ω′ +ω′′) is the volume form of Σ_{1}. In fact, we can choose local coordinates (x^{1}, y^{1}) on Σ_{1} and (x^{2}, y^{2}) on Σ_{2} so that ω′ = dx^{1} ∧ dy^{1} + dx^{2} ∧ dy^{2}, ω′′ = dx^{1} ∧ dy^{1} − dx^{2} ∧ dy^{2} and μ = 2(dx^{1} ∧ dy^{1}).

Let π_{1} : Σ_{1} × Σ_{2} ↦ Σ_{1} be the projection. \frac{μ}{2} is in fact the Jacobian of the projection π_{1} when restricted to Σ_{t} and the restriction π_{1}|_{Σ_{t}} is a covering map. Now take any neighborhood Ω of π_{1}(y_{0}) ∈ Σ_{1} and consider π_{1}^{−1}(Ω) ∩ Σ_{t}. Take any component and denote it by S_{t}. S is an unparametrized flow. Each S_{t} can be written as the graph of a map u_{t} : Ω ↦ Σ_{2} with uniformly bounded |du_{t}| since μ_{t} has a uniform lower bound. Since y_{0} is a limit point of Σ_{t_{0}+ \frac{s_{j}}{λ^{2}_{j}}}, by choosing Ω small enough,

we may assume the graph of u_{j} = u_{t_{0}+ \frac{s_{j}}{λ^{2}_{j}}} lies in B.

Now we consider the parabolic blow up of the graph of u_{j} in ℝ^{N} by λ_{j}. This is the graph of the map ũ_{j} from λ_{j}Ω to λ_{j}Σ_{2}. It corresponds to a part of Σ_{s_{j}}^{λ_{j}}. By the assumption that the origin is a limit point of Σ_{s_{j}}^{λ_{j}} and |dũ_{j}| is uniformly bounded, we may assume ũ_{j} → ũ_{∞} in C^{α} on compact sets. ũ_{∞} is an entire graph defined on ℝ^{2}.

On the other hand,

|A|_{j} ≤ |∇dũ_{j}| ≤ (\sqrt{1 + |dũ_{j}|^{2}})^{3}|A|_{j}

where |A|_{j} is the norm of the second fundamental form of S_{s_{j}}^{λ_{j}} and |∇dũ_{j}| is the norm of the covariant derivatives of dũ_{j}. Now we identify Ω with an open set in ℝ^{2}. Therefore for any B_{ρ} ⊂ ℝ^{2}, ũ_{j} satisfies

|Dũ_{j}| ≤ C, ∫_{B_{ρ}} |D^{2}ũ_{j}|^{2} → 0

where Dũ_{j} and D^{2}ũ_{j} are the usual derivatives with respect to coordinate variables on ℝ^{2}. Denote v_{j} = \frac{∂ũ_{j}}{∂x_{k}}, then |v_{j}| ≤ C and ∫_{B_{ρ}} |Dv_{j}|^{2} → 0.

Let c_{j} = \frac{1}{Vol(B_{ρ})} ∫ v_{j}, then we can choose a convergent subsequence c_{j} → c. By Poincare inequality,

∫ |v_{j} − c_{j}|^{2} ≤ λ ∫ |Dv_{j}|^{2} → 0.

Therefore \frac{∂ũ_{j}}{∂x^{k}} → c_{k} in L^{2}. Since we may assume ũ_{j} → ũ_{∞} in C^{α} ∩ W_{loc}^{1,2}, this implies S_{s_{j}}^{λ_{j}} → S_{−1}^{∞} as Radon measures and S_{−1}^{∞} is the graph of a linear function. Therefore

lim_{j→∞} ∫ ρ_{0,0}dμ_{s_{j}}^{λ_{j}} = ∫ ρ_{0,0}dμ_{−1}^{∞} = 1.

By White’s theorem again, we have regularity at the point (y_{0}, t_{0}). q.e.d.

Now we prove Theorem B.

Theorem B. Let M be an oriented four-dimensional Einstein manifold with two parallel calibrating forms ω′, ω′′ such that ω′ is self-dual and ω′′ is anti-self-dual. If Σ is a compact oriented surface immersed in M such that ∗ω′, ∗ω′′ > 0 on Σ. Then the mean curvature flow of Σ exists smoothly for all time.

Proof. ∗ω′ and ∗ω′′ have positive lower bound for any finite time by Equation (5.11), therefore the assumption in Proposition 6.1 is satisfied.

q.e.d.

7. Convergence at infinity

In this section we study the convergence of the mean curvature flow at infinity. The key point is to show uniform boundedness of |A|^{2} in space and time. We first compute the evolution of the second fundamental form. Let Σ → M^{n} be an isometric immersion. We choose an orthonormal basis {e_{i}} for T Σ and {e_{α}} for NΣ. Recall the convention for indexes are A, B, C ⋯ = 1 ⋯ n, i, j, k ⋯ for tangent indexes, and α, β, γ ⋯ for normal indexes. Now denote the coefficient of the second fundamental form by h_{αij} = ⟨A(∂_{i}, ∂_{j}), e_{α}⟩. The covariant derivative of A is defined as

(\overline{∇}_{∂_{l}}A)(∂_{i}, ∂_{j}) = (\overline{∇}_{∂_{l}}A(∂_{i}, ∂_{j}))^{N} − A((\overline{∇}_{∂_{l}}∂_{i})^{T} , ∂_{j}) − A(∂_{i}, (\overline{∇}_{∂_{l}}∂_{j})^{T}).

We denote

h_{αij,k} = ⟨(\overline{∇}_{∂_{k}}A)(∂_{i}, ∂_{j}), e_{α}⟩
h_{αij,kl} = ⟨(\overline{∇}_{∂_{l}} \overline{∇}_{∂_{k}}A)(∂_{i}, ∂_{j}), e_{α}⟩.

Let Δh_{αij} = g^{kl}h_{αij,kl} be the Laplacian of h_{αij}.

Proposition 7.1. For a mean curvature flow F : Σ×[0, t_{0}) → M of any dimension, the second fundamental form h_{αij} satisfies the following equation.

\begin{matrix}\frac{d}{dt}h_{αij} = & Δh_{αij} + (\overline{∇}_{∂_{k}}K)_{αijk} + (\overline{∇}_{∂_{j}}K)_{αkik} \\ − 2K_{lijk}h_{αlk} + 2K_{αβjk}h_{βik} + 2K_{αβik}h_{βjk} \\ − K_{lkik}h_{αlj} − K_{lkjk}h_{αli} + K_{αkβk}h_{βij} \\ − h_{αim}(h_{γmj}h_{γ} − h_{γmk}h_{γjk}) \\ − h_{αmk}(h_{γmj}h_{γik} − h_{γmk}h_{γij}) \\ − h_{βik}(h_{βlj}h_{αlk} − h_{βlk}h_{αlj}) \\ − h_{αjk}h_{βik}h_{β} + h_{βij}⟨e_{β}, \overline{∇}_{H}e_{α}⟩ \end{matrix} (7.1)

where K_{ABCD} is the curvature tensor and \overline{∇} is the covariant derivative of M.

In particular, |A|^{2} satisfies the following equation along the mean curvature flow.

\begin{matrix}\frac{d}{dt}|A|^{2} = & Δ|A|^{2} − 2|∇A|^{2} + 2[(\overline{∇}_{∂_{k}}K)_{αijk} + (\overline{∇}_{∂_{j}}K)_{αkik}]h_{αij} \\ − 4K_{lijk}h_{αlk}h_{αij} + 8K_{αβjk}h_{βik}h_{αij} \\ − 4K_{lkik}h_{αlj}h_{αij} + 2K_{αkβk}h_{βij}h_{αij} \\ + 2 ∑_{α,γ,i,m}(∑_{k} h_{αik}h_{γmk} − h_{αmk}h_{γik})^{2} \\ + 2 ∑_{i,j,m,k}(∑_{α} h_{αij}h_{αmk})^{2}. \end{matrix} (7.2)

Proof. We first derive Equation (7.2) from Equation (7.1). Since |A|^{2} = g^{ik}g^{jl}h_{αij}h_{αkl}, calculate using a normal coordinate sys-

tem near a point p we have

\frac{d}{dt}|A|^{2} = 2 (\frac{d}{dt}g^{ik}) h_{αij}h_{αkj} + 2 (\frac{d}{dt}h_{αij}) h_{αij}.

Recall \frac{d}{dt}g_{ik} = 2h_{β}h_{βik} and plug in Equation (7.1) to get

\frac{d}{dt}|A|^{2} =4h_{β}h_{βik}h_{αij}h_{αkj}

+ 2h_{αij}[Δh_{αij} + (\overline{∇}_{∂_{k}}K)_{αijk} + (\overline{∇}_{∂_{j}}K)_{αkik} − 2K_{lijk}h_{αlk} + 2K_{αβjk}h_{βik} + 2K_{αβik}h_{βjk} − K_{lkik}h_{αlj} − K_{lkjk}h_{αli} + K_{αkβk}h_{βij} − h_{αim}(h_{γmj}h_{γ} − h_{γmk}h_{γjk}) − h_{αmk}(h_{γmj}h_{γik} − h_{γmk}h_{γij}) − h_{βik}(h_{βlj}h_{αlk} − h_{βlk}h_{αlj}) − h_{αjk}h_{βik}h_{β} + h_{βij}⟨e_{β}, \overline{∇}_{H}e_{α}⟩].

The first term on the right hand side 4h_{β}h_{βik}h_{αij}h_{αkj} cancels with two later terms. They are so-called “metric” terms and vanish if we choose a orthonormal frame in our computation.

The last term on the right hand side 2h_{αij}h_{βij}⟨e_{β}, \overline{∇}_{H}e_{α}⟩ is zero by symmetry.

Now use Δh_{αij}^{2} = 2|∇A|^{2} + 2h_{αij}Δh_{αij}. Therefore we get

\frac{d}{dt}|A|^{2} =Δ|A|^{2} − 2|∇A|^{2} + 2h_{αij}[(\overline{∇}_{∂_{k}}K)_{αijk} + (\overline{∇}_{∂_{j}}K)_{αkik}

− 2K_{lijk}h_{αlk} + 2K_{αβjk}h_{βik} + 2K_{αβik}h_{βjk} − K_{lkik}h_{αlj} − K_{lkjk}h_{αli} + K_{αkβk}h_{βij} + h_{αim}h_{γmk}h_{γjk} − h_{αmk}(h_{γmj}h_{γik} − h_{γmk}h_{γij}) − h_{βik}(h_{βlj}h_{αlk} − h_{βlk}h_{αlj})].

The fourth order terms can be calculated as the following:

h_{αij}h_{αim}h_{γmk}h_{γjk} − h_{αij}h_{αmk}(h_{γmj}h_{γik} − h_{γmk}h_{γij})

− h_{αij}h_{βik}(h_{βlj}h_{αlk} − h_{βlk}h_{αlj}) = 2h_{αij}h_{αim}h_{γmk}h_{γkj} − 2h_{αij}h_{αmk}h_{γmj}h_{γik} + h_{αij}h_{αmk}h_{γmk}h_{γij}.

The first two terms can be completed to square:

\begin{matrix}2h_{αij}h_{αim}h_{γmk}h_{γkj} − 2h_{αij}h_{αmk}h_{γmj}h_{γik} \\ = 2h_{αij}h_{αik}h_{γmk}h_{γmj} − 2h_{αij}h_{αmk}h_{γmj}h_{γik} \\ = 2h_{αij}h_{γmj}(h_{αik}h_{γmk} − h_{αmk}h_{γik}) \\ = h_{αij}h_{γmj}(h_{αik}h_{γmk} − h_{αmk}h_{γik}) \\ + h_{αmj}h_{γij}(h_{αmk}h_{γik} − h_{αik}h_{γmk}) \\ = ∑_{α,γ,i,m} (∑_{k} h_{αik}h_{γmk} − h_{αmk}h_{γik})^{2} . \end{matrix} (7.3)

Now we calculate Equation (7.1). First the Laplacian of h_{αij} is the following.

\begin{matrix}Δh_{αij} = & h_{α,ij} − (∇_{∂_{k}}K)_{αijk} − (∇_{∂_{j}}K)_{αkik} \\ + 2K_{lijk}h_{αlk} − 2K_{αβjk}h_{βik} − 2K_{αβik}h_{βjk} − K_{αijβ}h_{β} \\ + K_{lkik}h_{αlj} + K_{lkjk}h_{αli} − K_{αkβk}h_{βij} \\ + h_{αim}(h_{γmj}h_{γ} − h_{γmk}h_{γjk}) \\ + h_{αmk}(h_{γmj}h_{γik} − h_{γmk}h_{γij}) \\ + h_{βik}(h_{βlj}h_{αlk} − h_{βlk}h_{αlj}) \end{matrix} (7.4)

where h_{α,ij} = ⟨\overline{∇}_{∂_{j}}^{N} \overline{∇}_{∂_{i}}^{N}H, e_{α}⟩.

In the codimension one case, this equation reduces to

Δh_{ij} =H_{,ij} − (∇_{∂_{k}}K)_{Nijk} − (∇_{pj}K)_{Nkik}
+ 2K_{lijk}h_{lk} − K_{Nijn+1}H
+ K_{lkik}h_{lj} + K_{lkjk}h_{li} − K_{NkNk}h_{ij}
+ h_{im}h_{mj}H − h_{mk}^{2}h_{ij}.

This recovers Equation (1.20) in [6].

Equation (7.4) is computed using the Codazzi equation and the commutation formula.

\begin{matrix}h_{αkj,i} = h_{αki,j} + K_{αkij} \\ h_{αij,kl} = h_{αij,lk} − h_{αim}R_{mjlk} − h_{αmj}R_{milk} − h_{βij}R_{βαlk} \end{matrix}

where R_{mjlk} is the curvature of T Σ and R_{αβlk} is the curvature of NΣ. We start the computation with h_{α,ij} = h_{αkk,ij}.

h_{αkk,ij} = (h_{αki,k} + K_{αkik})_{j}
= h_{αki,kj} + K_{αkik,j}
= h_{αik,jk} − h_{αim}R_{mkjk} − h_{αmk}R_{mijk}
− h_{βik}R_{βαjk} + K_{αkik,j}
= (h_{αij,k} + K_{αijk})_{k} − h_{αim}R_{mkjk} − h_{αmk}R_{mijk}
− h_{βik}R_{βαjk} + K_{αkik,j}.

By the Gauss and Ricci equation, we have

R_{mkjk} = K_{mkjk} + h_{γmj}h_{γkk} − h_{γmk}h_{γkj}
R_{mijk} = K_{mijk} + h_{γmj}h_{γik} − h_{γmk}h_{γij}
R_{βαjk} = K_{βαjk} + h_{βlj}h_{αlk} − h_{βlk}h_{αlj}.

Therefore

Δh_{αij} =h_{α,ij} − K_{αijk,k} − K_{αkik,j} + h_{αim}K_{mkjk}
+ h_{αmk}K_{mijk} + h_{βik}K_{βαjk}
+ h_{αim}(h_{γmj}h_{γ} − h_{γmk}h_{γjk})
+ h_{αmk}(h_{γmj}h_{γik} − h_{γmk}h_{γij})
+ h_{βik}(h_{βlj}h_{αlk} − h_{βlk}h_{αlj}).

The covariant derivative term can be calculated as follows:

K_{αijk,k} =(\overline{∇}_{∂_{k}}K)_{αijk} − K_{lijk}h_{αlk} + K_{αβjk}h_{βik}
+ K_{αiβk}h_{βjk} + K_{αijβ}h_{βkk}
K_{αkik,j} =(\overline{∇}_{∂_{j}}K)_{αkik} − K_{lkik}h_{αlj} + K_{αβik}h_{βkj}
+ K_{αkβk}h_{βij} + K_{αkiβ}h_{βkj}.

Note that K_{αijk} is considered as a section of the bundle NΣ ⊗ T Σ ⊗ T Σ⊗T Σ in taking covariant derivatives . We collect all the embient curvature term and use the first Bianchi identity K_{αiβk} + K_{αkiβ} = −K_{αβki} to get Equation (7.4).

Next we calculate the equation for h_{αij} = ⟨\overline{∇}_{∂_{j}}∂_{i}, e_{α}⟩.

\begin{aligned}\frac{d}{dt}h_{αij} & = ⟨\overline{∇}_{H} \overline{∇}_{∂_{j}}∂_{i}, e_{α}⟩ + ⟨\overline{∇}_{∂_{j}}∂_{i}, \overline{∇}_{H}e_{α}⟩ \\ & = ⟨\overline{∇}_{∂_{j}} \overline{∇}_{H}∂_{i}, e_{α}⟩ − ⟨K(H, ∂_{j})∂_{i}, e_{α}⟩ + ⟨\overline{∇}_{∂_{j}}∂_{i}, \overline{∇}_{H}e_{α}⟩. \end{aligned}

By breaking \overline{∇}_{∂_{j}} \overline{∇}_{∂_{i}}H into normal and tangent parts, we get

\begin{aligned}⟨\overline{∇}_{∂_{j}} \overline{∇}_{∂_{i}}H, e_{α}⟩ & = ⟨\overline{∇}_{∂_{j}} [(\overline{∇}_{∂_{i}}H)^{T} + (\overline{∇}_{∂_{i}}H)^{N}] , e_{α}⟩ \\ & = ⟨\overline{∇}_{∂_{j}}^{N} \overline{∇}_{∂_{i}}^{N}H, e_{α}⟩ − ⟨(\overline{∇}_{∂_{i}}H)^{T} , \overline{∇}_{∂_{j}}e_{α}⟩. \end{aligned}

Therefore,

\frac{d}{dt}h_{αij} = h_{α,ij} − h_{β}K_{βjiα} − ⟨(\overline{∇}_{∂_{i}}H)^{T} , \overline{∇}_{∂_{j}}e_{α}⟩ + ⟨\overline{∇}_{∂_{j}}∂_{i}, \overline{∇}_{H}e_{α}⟩

where h_{α,ij} = ⟨\overline{∇}_{∂_{j}}^{N} \overline{∇}_{∂_{i}}^{N}H, e_{α}⟩.

The term ⟨(\overline{∇}_{∂_{i}}H)^{T} , \overline{∇}_{∂_{j}}e_{α}⟩ is equal to h_{β}h_{βik}h_{αjk}. Also since we choose a normal coordinate in our computation, (\overline{∇}_{∂_{i}}∂_{j})^{T} = 0 and ⟨\overline{∇}_{∂_{i}}∂_{j}, \overline{∇}_{H}e_{α}⟩ = h_{βij}⟨e_{β}, \overline{∇}_{H}e_{α}⟩.

\frac{d}{dt}h_{αij} = h_{α,ij} − h_{β}h_{βik}h_{αjk} − h_{β}K_{βjiα} + h_{βij}⟨e_{β}, \overline{∇}_{H}e_{α}⟩. (7.5)

Combine Equation (7.4) and (7.5), we get the parabolic equation for h_{αij}. q.e.d.

The following proposition provides a uniform bound of the second fundamental form when ∗ω′ and ∗ω′′ are both close to one.

Proposition 7.2. Let M be a compact four-dimensional manifold with bounded geometry. Let ω′ and ω′′ be two parallel calibrating forms such that ω′ ∧ ω′ and ω′′ ∧ ω′′ determine opposite orientation for M. Let Σ be an oriented immersed surface in M. There exists a constant 1 > ϵ > 0 such that if ∗ω′ > 1 − ϵ and ∗ω′′ > 1 − ϵ on F_{t}(Σ) for t ∈ [0, T], then the norm of the second fundamental form of F_{t}(Σ) is uniformly bounded in [0, T].

Proof. The fourth order term in Equation (7.2) can be calculated explicitly in the four-dimensional case:

\begin{matrix}∑_{α,γ,i,m} & (∑_{k} h_{αik}h_{γmk} − h_{αmk}h_{γik}) & 2 \\ = ∑_{i,m} (∑_{k} h_{3ik}h_{4mk} − h_{3mk}h_{4ik})^{2}+ ∑_{i,m} (∑_{k} h_{4ik}h_{3mk} − h_{4mk}h_{3ik})^{2} \\ = 2 ∑_{i,m} (∑_{k} h_{3ik}h_{4mk} − h_{3mk}h_{4ik})^{2} \\ = 2 [(∑_{k} h_{31k}h_{42k} − h_{32k}h_{41k})^{2} + (∑_{k} h_{41k}h_{32k} − h_{42k}h_{31k})^{2}] \\ = 4 (∑_{k} h_{31k}h_{42k} − h_{32k}h_{41k})^{2} \\ = 4 [\frac{1}{2}(h_{31k} + h_{42k})^{2} + \frac{1}{2}(h_{32k} − h_{41k})^{2} − \frac{1}{2}|A|^{2}]^{2} \\ = [(h_{31k} + h_{42k})^{2} + (h_{32k} − h_{41k})^{2} − |A|^{2}]^{2} . \end{matrix} (7.6)

By the Schwarz inequality,

∑_{i,j,m,k} (∑_{α} h_{αij}h_{αmk})^{2} ≤ ∑_{i,j,m,k} (∑_{α} h_{αij}^{2}) (∑_{α} h_{αmk}^{2}) ≤ |A|^{4}.

Therefore

\frac{d}{dt}|A|^{2} ≤ Δ|A|^{2} − 2|∇A|^{2} + 4|A|^{4} + K_{1}|A|^{2} + K_{2} (7.7)

where K_{1} and K_{2} are constants that depend on the curvature tensor and covariant derivatives of the curvature tensor of M.

Again we consider μ = η′ + η′′. Since η′ ≥ 1 − ϵ and η′′ ≥ 1 − ϵ, we have μ ≥ 2 − 2ϵ and |η′ − η′′| ≤ ϵ ≤ \frac{ϵ}{2−2ϵ}μ.

\frac{d}{dt}μ ≥Δμ + μ|A|^{2} + 2(η′ − η′′)h_{32k}h_{41k} − 2(η′ − η′′)h_{31k}h_{42k} − Cμ
≥Δμ + μ (|A|^{2} − 2\frac{ϵ}{2 − 2ϵ}|h_{32k}h_{41k}| − 2\frac{ϵ}{2 − 2ϵ}|h_{31k}h_{42k}|) − Cμ
≥Δμ + μ [\frac{2 − 3ϵ}{2 − 2ϵ}|A|^{2} + \frac{ϵ}{2 − 2ϵ}(|A|^{2} − 2|h_{32k}h_{41k}| − 2|h_{31k}h_{42k}|)]
− Cμ.

The term (|A|^{2}−2|h_{32k}h_{41k}|−2|h_{31k}h_{42k}|) is a complete square and thus nonnegative.

Therefore

\frac{d}{dt}μ ≥ Δμ + c_{1}(ϵ)μ|A|^{2} − Cμ

where c_{1}(ϵ) = \frac{2−3ϵ}{2−2ϵ}, notice that c_{1}(ϵ) → 1 as ϵ → 0.

Let p > 1 be an integer to be determined, we calculate the equation for μ^{p}.

\frac{d}{dt}μ^{p} = pμ^{p−1} \frac{d}{dt}μ ≥ pμ^{p−1}(Δμ + c_{1}(ϵ)μ|A|^{2} − Cμ).

Use the identity Δμ^{p} = p(p − 1)μ^{p−2}|∇μ|^{2} + pμ^{p−1}Δμ, the differential inequality for μ^{p} becomes

\frac{d}{dt}μ^{p} ≥ Δμ^{p} − p(p − 1)μ^{p−2}|∇μ|^{2} + p c_{1}(ϵ)μ^{p}|A|^{2} − Cpμ^{p}.

Now we estimate the term |∇μ|^{2}.

We calculate as in Equation (5.8):

\begin{aligned}|∇η′|^{2} & ≤ 2(1 − (η′)^{2})(h_{41k}^{2} + h_{32k}^{2}) \\ |∇η′′|^{2} & ≤ 2(1 − (η′′)^{2})(h_{31k}^{2} + h_{42k}^{2}). \end{aligned}

mean curvature flow 335

Since η′ ≥ 1 − ϵ, 1 − (η′)^{2} ≤ 1 − (1 − ϵ)^{2} ≤ 2ϵ. Therefore

|∇μ|^{2} ≤ 2(|∇η′|^{2} + |∇η′′|^{2}) ≤ 8ϵ|A|^{2} ≤ \frac{8ϵ}{(2 − 2ϵ)^{2}} μ^{2}|A|^{2}.

Denote \frac{8ϵ}{(2−2ϵ)^{2}} = c_{2}(ϵ).

Thus

\frac{d}{dt}μ^{p} ≥ Δμ^{p} + p[c_{1}(ϵ) − (p − 1)c_{2}(ϵ)]μ^{p}|A|^{2} − Cpμ^{p}.

Plug in f = |A|^{2} and g = μ^{p} in the identity

\frac{d}{dt} (\frac{f}{g} ) = Δ (\frac{f}{g} ) + 2\frac{∇g}{g} · ∇ (\frac{f}{g} )

+ \frac{1}{g^{2}} [(\frac{d}{dt}f − Δf) g − (\frac{d}{dt}g − Δg) f] .

Therefore

\frac{d}{dt} (\frac{|A|^{2}}{μ^{p}} ) ≤ Δ (\frac{|A|^{2}}{μ^{p}} ) + 2∇ (\frac{|A|^{2}}{μ^{p}} ) · \frac{∇μ^{p}}{μ^{p}}

+ \frac{1}{μ^{2p}} \{[−2|∇A|^{2} + 4|A|^{4} + K_{1}|A|^{2} + K_{2}]μ^{p}

− [p(c_{1}(ϵ) − (p − 1)c_{2}(ϵ))μ^{p}|A|^{2} − Cpμ^{p}]|A|^{2}}.

The last term is less than

[4 − p(c_{1}(ϵ) − (p − 1)c_{2}(ϵ))]\frac{|A|^{4}}{μ^{p}} + (K_{1} + Cp)\frac{|A|^{2}}{μ^{p}} + K_{2} \frac{1}{μ^{p}} .

Recall that c_{1}(ϵ) → 1 and c_{2}(ϵ) → 0 as ϵ → 0. Choose p large enough and then ϵ small enough so that

(2 − 2ϵ)^{p}[4 − p(c_{1}(ϵ) − (p − 1)c_{2}(ϵ))] ≤ −C_{1}

for some C_{1} > 0.

Then

\begin{matrix}[4 − p(c_{1}(ϵ) − (p − 1)c_{2}(ϵ))]\frac{|A|^{4}}{μ^{p}} = & [4 − p(c_{1}(ϵ) − (p − 1)c_{2}(ϵ))]μ^{p} \frac{|A|^{4}}{μ^{2p}} \\ ≤ & − C_{1} \frac{|A|^{4}}{μ^{2p}} . \end{matrix}

Denote f = \frac{|A|^{2}}{μ^{p}}, then f satisfies

\frac{d}{dt}f ≤ Δf + V · ∇f − C_{1}f^{2} + C_{2}f + C_{3}.

Now we apply the maximum principle for parabolic equations and conclude the \frac{|A|^{2}}{μ} is uniformly bounded, thus |A|^{2} is also bounded.

q.e.d.

Now we prove Theorem C.

Theorem C. Under the same assumption as in Theorem B. When M has nonnegative curvature, there exists a constant ϵ > 0 such that if Σ is a compact oriented surface immersed in M with ∗ω′, ∗ω′′ > 1−ϵ on Σ, the mean curvature flow of Σ converges smoothly to a totally geodesic surface at infinity.

Proof. In these cases, ∗ω′ and ∗ω′′ are both non-decreasing. Proposition 7.2 is applicable and |A|^{2} is uniformly bounded in space and time.

Integrating Equation (7.7) and we see

\frac{d}{dt} ∫_{Σ_{t}} |A|^{2}dμ_{t} ≤ C. (7.8)

Recall in this case \frac{d}{dt}μ ≥ Δμ + c_{1}(ϵ)μ|A|^{2} and η has a positive lower bound, thus

∫_{0}^{∞} ∫_{Σ_{t}} |A|^{2}dμ_{t}dt ≤ ∞. (7.9)

Equation (7.8) and (7.9) together implies

∫_{Σ_{t}} |A|^{2}dμ_{t} → 0.

By the small ϵ regularity theorem in [4], sup_{Σ_{t}} |A|^{2} → 0 uniformly as t → ∞.

Since the mean curvature flow is a gradient flow and the metrics are analytic, by the theorem of Simon [7], we get convergence at infinity.

q.e.d.

8. Applications

In the following we apply the previous theorem to the case when M is a product S^{2} × S^{2}. Denote their Kähler forms by ω_{1} and ω_{2} respectively.

Let ω′ = ω_{1} + ω_{2} and ω′′ = ω_{1} − ω_{2}, then (ω′)^{2} = 2ω_{1} ∧ ω_{2} and (ω′′)^{2} = −2ω_{1} ∧ ω_{2} determine opposite orientations on M. Both ω′ and ω′′ are parallel calibrating form and they define integrable almost complex structures with opposite orientations.

Theorem D. Let M = (S^{2}, ω_{1}) × (S^{2}, ω_{2}). If Σ is a compact oriented surface embedded in M such that ∗ω_{1} ≥ | ∗ ω_{2}| and the strict inequality holds at at least one point. Then the mean curvature flow of Σ exists for all time and converges smoothly to a S^{2} × {p}.

Proof. We notice the statement is a little bit different from the one given in the introduction. The difference is resolved by the considering the maximum principle for η′ = ∗ω′ and η′′ = ∗ω′′. It is not hard to see the assumption implies ∗ω_{1} > | ∗ ω_{2}| holds everywhere at a later time. By the equation of η′ and η′′,

\frac{d}{dt}η′ = Δη′ + η′[(h_{31k} − h_{42k})^{2} + (h_{32k} + h_{41k})^{2}] + η′(1 − (η′)^{2})
\frac{d}{dt}η′′ = Δη′′ + η′′[(h_{41k} − h_{32k})^{2} + (h_{42k} + h_{31k})^{2}] + η′′(1 − (η′′)^{2})

we see that as t → ∞, they both approach 1. By Theorem A, we have existence for all time. Also the assumption on Proposition 7.2 is satisfied and the second fundamental form is uniformly bounded in space and time. Since the mean curvature flow is a gradient flow and the metrics are analytic, we can apply Simon’s theorem [7] to conclude convergence at infinity. The limiting submanifold has ∗ω_{1} = 1 identically and thus is of the form S^{2} × p. q.e.d.

Corollary D now follows from this since the condition ∗ω_{1} > | ∗ ω_{2}| on the graph of a map f is equivalent to the Jacobian of f being less than one.

We conclude this section by the following two remarks:

Remark 8.1. When M is locally a product of two Riemann surfaces of nonpositive curvature, the method in [8] can be used to prove uniform convergence of the flow. The limit is totally geodesic and the corresponding map converges to one ranged in a lower dimensional submanifold. Notice that the convergence in Theorem C is a stronger smooth convergence under the closeness assumption. Such results are generalized to arbitrary dimension and codimension in [9].

Remark 8.2. The case when ∗ω_{1} > 0 and ∗ω_{2} = 0 corresponds to Σ is Lagrangian surface with respect to the symplectic form ω_{2}. If Σ is the graph of a map f, then f is indeed an area preserving diffeomorphism. This case and the application to the structure of the diffeomorphism groups of compact Riemann surfaces are discussed in [8].

References

[1] K. Ecker & G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 No. 3 (1989) 453–471.

[2] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 No. 1 (1984) 237–266.

[3] , Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 No. 1 (1990) 285–299.

[4] T. Ilmanen, Singularities of mean curvature flow of surfaces, Preprint, 1997.

[5] , Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520,

[6] R. Schoen, L. Simon, & S. T. Yau, Curvature estimates for minimal hypersurfaces Acta Math. 134 No. 3-4 (1975) 275–288.

[7] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 No. 3 (1983) 525–571.

[8] M-T. Wang, Deforming area preserving diffeomorphism of surfaces by mean curvature flow, Preprint, 2000.

[9] , Stability of graphic mean curvature flow in higher codimension, Preprint, 2000.

[10] B. White, The size of the singular set in mean curvature flow of mean convex surfaces, Preprint.

[11] , Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997) 1–35.

[12] , A local regularity theorem for classical mean curvature flow, Preprint, 2000.

Stanford University