j. differential geometry

57 (2001) 409-441

THE WILLMORE FLOW WITH SMALL INITIAL ENERGY

ERNST KUWERT & REINER SCHÄTZLE

Abstract

We consider the L^{2} gradient flow for the Willmore functional. In [5] it was proved that the curvature concentrates if a singularity develops. Here we show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an L^{∞} estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its L^{2} norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results we obtain long time existence and convergence to a round sphere if the total energy is initially small.

1. Introduction

For a closed, immersed surface f : Σ → ℝ^{n} the Willmore functional (as introduced initially by Thomsen [11]) is

W(f) = ∫_{Σ} |A^{∘}|^{2} dμ, (1)

where A^{∘} = A − \frac{1}{2}g ⊗ H denotes the tracefree part of the second fundamental form A = D^{2}f^{⊥} and μ is the induced area measure. The associated Euler-Lagrange operator is

W(f) = ΔH + Q(A^{∘})H. (2)

Here H is the mean curvature vector and Q(A^{∘}) acts linearly on normal vectors along f by the formula (using summation with respect to a g-orthonormal basis {e_{1}, e_{2}})

Q(A^{∘})ϕ = A^{∘}(e_{i}, e_{j})⟨A^{∘}(e_{i}, e_{j}), ϕ⟩. (3)

Received October 12, 1999.

In (2) the Laplace operator Δϕ = −∇^{∗}∇ϕ is understood with respect to the connection ∇_{X}ϕ = (D_{X}ϕ)^{⊥} on normal vector fields along f, where ∇^{∗} denotes the formal adjoint of ∇.

In this paper we continue our study from [5] of the L^{2} gradient flow for (1), briefly called the Willmore flow, which is the fourth order, quasilinear geometric evolution equation

∂_{t}f = −W(f). (4)

As a main result we have shown in [5] that the existence time is bounded from below in terms of the concentration of the measure f_{#}(μ⌞|A|^{2}) in ℝ^{n} at time t = 0. Here we study the operator (2) and the flow (4) under the assumption that W(f) is — either locally or globally — small. This condition is natural from the variational point of view and may be interpreted geometrically by saying that the deviation of f from being round is small in an averaged sense. One of our results is:

Theorem 5.1. There exists ε_{0}(n) > 0 such that if at time t = 0 we have W(f_{0}) < ε_{0}, then the Willmore flow exists smoothly for all times and converges to a round sphere.

The smallness assumption implies, if ε_{0} is not too big, that Σ is topologically a sphere and that f is an embedding (see [13] for the case n = 3). Moreover, any sequence f_{k} with W(f_{k}) → 0 subconverges, after appropriate translation and rescaling, to some round sphere in the sense of both Hausdorff distance and measure [8]. However the f_{k} need not be graphs over the limit sphere, as can be seen by modifying Example 1 in [12]. At present we do not know an example ruling out the possibility of dropping the smallness condition in Theorem 5.1 entirely; in any case it is desirable to replace the number ε_{0} by a more explicit constant.^{1}

The statement of the theorem was recently proved in [9] under the stronger assumption that f_{0} is close to a round sphere in the C^{2,α}- topology, using a center manifold analysis which gives related stability results for a couple of other flows; see [2] for an overview. Our method, which is (and has to be) entirely different, involves deriving a priori estimates from the equation satisfied by the curvature, somewhat analogous to the work of Huisken [3, 4]. However, in our problem the crucial estimates are of integral type and the smallness condition is essential in ruling out possible concentrations related to the scale invariance of the functional.

1Note added in proof: a numerical example of a singularity was recently contributed by U. Mayer & G. Simonett: http://www.math.utah.edu/∼mayer/math /numerics.html.

2. Estimates for surfaces with locally small Willmore energy

Here we derive some bounds for immersed surfaces f : Σ → ℝ^{n} depending on the L^{2} norms of their curvature A and of their Willmore gradient W(f) = ΔH + Q(A^{∘})H, under the assumption that the L^{2} norm of A^{∘}, the tracefree part of the curvature, is locally small.

Recall the equations of Mainardi-Codazzi, Gauß and Ricci:

(5) (∇_{X}A)(Y, Z) = (∇_{Y} A)(X, Z); ∇H = −∇^{∗}A = −2∇^{∗}A^{∘},
(6)  K = \frac{1}{4}|H|^{2} − \frac{1}{2}|A^{∘}|^{2},
(7) R^{⊥}(X, Y)ϕ = A^{∘}(e_{i}, X)⟨A^{∘}(e_{i}, Y), ϕ⟩ − A^{∘}(e_{i}, Y)⟨A^{∘}(e_{i}, X), ϕ⟩.

Note ⟨R^{⊥}(X, Y)ϕ, ϕ⟩ = 0 and in particular R^{⊥} = 0 for n = 3, i.e., codimension one. The Codazzi equations imply that ∇A and ∇^{2}A can be expressed by ∇A^{∘} and ∇^{2}A^{∘}, respectively. In particular one has inequalities

|∇A| ≤ c |∇A^{∘}|, |∇^{2}A| ≤ c |∇^{2}A^{∘}|. (8)

Lemma 2.1. For any p-linear form ϕ along f we have

((∇∇^{∗} − ∇^{∗}∇)ϕ)(X_{1}, …, X_{p}) (9)

= Kϕ(X_{1}, …, X_{p}) + K ∑_{k=2}^{p} ϕ(X_{k}, X_{2}, …, X_{1}, …, X_{p})

− K ∑_{k=2}^{p} g(X_{1}, X_{k}) ϕ(e_{i}, X_{2}, …, e_{i}, …, X_{p})

+R^{⊥}(e_{i}, X_{1}) ϕ(e_{i}, X_{2}, …, X_{p}) − (∇^{∗}T)(X_{1}, …, X_{p}).

Here the tensor T is given by

T (X_{0}, X_{1}, …, X_{p}) = (∇_{X_{0}}ϕ)(X_{1}, X_{2}, …, X_{p})−(∇_{X_{1}} ϕ)(X_{0}, X_{2}, …, X_{p}).

Proof. From the proof of Lemma 2.1 in [5] we have

((∇∇^{∗} − ∇^{∗}∇)ϕ)(X_{1}, …, X_{p})

= (R^{p}(e_{i}, X_{1}) ϕ)(e_{i}, X_{2}, …, X_{p}) − (∇^{∗}T)(X_{1}, …, X_{p}).

Now the curvature operator R^{p} is given by

(R^{p}(e_{i}, X_{1})ϕ)(e_{i}, X_{2}, …, X_{p})

= R^{⊥}(e_{i}, X_{1}) ϕ(e_{i}, X_{2}, …, X_{p}) − ϕ(R(e_{i}, X_{1}) e_{i}, X_{2}, …, X_{p})

− ∑_{k=2}^{p} ϕ(e_{i}, X_{2}, …, R(e_{i}, X_{1})X_{k}, …, X_{p})

= R^{⊥}(e_{i}, X_{1}) ϕ(e_{i}, X_{2}, …, X_{p}) − K ϕ(g(X_{1}, e_{i})e_{i}, X_{2}, …, X_{p}) + K ϕ(g(e_{i}, e_{i})X_{1}, X_{2}, …, X_{p})

− K ∑_{k=2}^{p} ϕ(e_{i}, X_{2}, …, g(X_{1}, X_{k})e_{i}, …, X_{p})

+ K ∑_{k=2}^{p} ϕ(g(e_{i}, X_{k})e_{i}, X_{2}, …, X_{1}, …, X_{p}).

Inserting yields the desired formula. q.e.d.

We will need three different choices for ϕ in (9). Taking first ϕ = A yields T = 0 and ∇^{∗}ϕ = −∇H by (5), and we get Simons’ identity ([10])

ΔA = ∇^{2}H + 2KA^{∘} + R^{⊥}(e_{i}, ) A(e_{i}, ).

To bring this in a more useful form, let us denote by S^{∘}(B) the symmetric, tracefree part of any bilinear form with normal values along f. In particular, we have

S^{∘}(∇^{2}H) = ∇^{2}H − \frac{1}{2} g( , ) ΔH − \frac{1}{2} R^{⊥}( , ) H.

Now Δ (\frac{1}{2} g( , ) H) = \frac{1}{2} g( , ) ΔH and R^{⊥}(e_{i}, X)\frac{1}{2} g(e_{i}, Y) H = −\frac{1}{2}R^{⊥}(X, Y) H, which implies, using (6) and (7),

ΔA^{∘} = S^{∘}(∇^{2}H) + \frac{1}{2}|H|^{2} A^{∘} + A^{∘} ∗ A^{∘} ∗ A^{∘}. (10)

Here and in the following we denote by A ∗ B any universal, linear combination of tensors obtained by tensor product and contraction from A and B. Our second choice in (9) is ϕ = ∇H, where now

T (X, Y) = ∇_{X,Y}^{2} H − ∇_{Y,X}^{2}H = R^{⊥}(X, Y) H.

the willmore flow 413

Using again (5), (6) and (7), we infer

∇^{∗}(∇^{2}H) = ∇(∇^{∗}∇H) − \frac{1}{4}|H|^{2}∇H + A ∗ A^{∘} ∗ ∇A^{∘}. (11)

Finally taking ϕ = ∇A^{∘} in (9) yields

T (X, Y, Z, V) = (R^{2}(X, Y) A^{∘}) (Z, V) = (A ∗ A ∗ A)(X, Y, Z, V),
∇^{∗}T = A ∗ A ∗ ∇A^{∘}.

Thus we obtain from (9) and (6), (7) ∇^{∗}(∇^{2}A^{∘}) = ∇(∇^{∗}∇A^{∘}) + A ∗ A ∗ ∇A^{∘}. (12) We now convert (10), (11) and (12) into integral estimates.

Lemma 2.2. If f : Σ → ℝ^{n} is an immersion with W(f) = W and γ ∈ C_{c}^{1}(Σ) satisfies |∇γ| ≤ Λ, then

∫ |∇A|^{2}γ^{2} dμ ≤ \frac{c}{Λ^{2}} ∫ |W |^{2}γ^{4} dμ + c ∫ |A^{∘}|^{4}γ^{2} dμ + cΛ^{2} ∫_{[γ>0]} |A|^{2} dμ. (13)

Proof. Multiply (10) by γ^{2}A^{∘} and integrate by parts to obtain, after applying (5),

∫ |∇A^{∘}|^{2}γ^{2} dμ + \frac{1}{2} ∫ |H|^{2} |A^{∘}|^{2} γ^{2} dμ

≤ \frac{1}{2} ∫ |∇H|^{2} γ^{2} dμ + c ∫ |A^{∘}|^{4} γ^{2} dμ + ∫ γ ∇γ ∗ A^{∘} ∗ ∇A^{∘} dμ.

Using the equation ΔH + Q(A^{∘})H = W we have

\frac{1}{2} ∫ |∇H|^{2} γ^{2} dμ = −\frac{1}{2} ∫ ⟨H, ΔH⟩γ^{2} dμ + ∫ γ ∇γ ∗ A ∗ ∇A^{∘} dμ

= −\frac{1}{2} ∫ ⟨H, W ⟩γ^{2} dμ + \frac{1}{2} ∫ ⟨H, Q(A^{∘})H⟩γ^{2} dμ

+ ∫ γ ∇γ ∗ A ∗ ∇A^{∘} dμ

≤ \frac{c}{Λ^{2}} ∫ |W |^{2} γ^{4} dμ + cΛ^{2} ∫_{[γ>0]} |H|^{2} dμ

+ \frac{1}{2} ∫ ⟨H, Q(A^{∘})H⟩γ^{2} dμ + ∫ γ ∇γ ∗ A ∗ ∇A^{∘} dμ.

It is easy to see the inequality

0 ≤ ⟨Q(A^{∘})H, H⟩ ≤ |A^{∘}|^{2} |H|^{2}. (14)

Furthermore we have

∫ γ ∇γ ∗ A ∗ ∇A^{∘} dμ ≤ \frac{1}{2} ∫ |∇A^{∘}|^{2} γ^{2} dμ + cΛ^{2} ∫_{[γ>0]} |A|^{2} dμ.

Inserting these inequalities, absorbing and recalling (8) proves the claim. q.e.d.

Lemma 2.3. Under the assumptions of Lemma 2.2 we have for η = γ^{4}

∫ |∇^{2}H|^{2} η + ∫ |A|^{2} |∇A|^{2}η dμ + ∫ |A|^{4} |A^{∘}|^{2} η dμ (15)

≤ c ∫ |W |^{2} η dμ + c ∫ (|A^{∘}|^{2} |∇A^{∘}|^{2} + |A^{∘}|^{6})η dμ

+ c Λ^{4} ∫_{[γ>0]} |A|^{2} dμ.

Proof. We start multiplying (11) by η ∇H and integrating by parts. This yields

∫ |∇^{2}H|^{2} η dμ + \frac{1}{4} ∫ |H|^{2} |∇H|^{2} η dμ

≤ ∫ |ΔH|^{2} η dμ + ∫ A ∗ A^{∘} ∗ ∇A^{∘} ∗ ∇A^{∘}η dμ

+ ∫ γ^{3} ∇γ ∗ ∇H ∗ ∇^{2}H dμ

≤ c ∫ |W |^{2}η dμ + c ∫ |A^{∘}|^{4} |H|^{2}η dμ

+ ε ∫ |H|^{2} |∇A^{∘}|^{2}η dμ + c(ε) ∫ |A^{∘}|^{2} |∇A^{∘}|^{2}η dμ

+ \frac{1}{2} ∫ |∇^{2}H|^{2}η dμ + c Λ^{2} ∫ |∇H|^{2} γ^{2} dμ.

Now by (13) we can estimate

∫ |∇H|^{2} γ^{2} dμ ≤ \frac{c}{Λ^{2}} ∫ |W |^{2}η dμ + \frac{c}{Λ^{2}} ∫ |A^{∘}|^{6}η dμ + c Λ^{2} ∫_{[γ>0]} |A|^{2} dμ.

the willmore flow 415

Using the inequality c |A^{∘}|^{4} |H|^{2} ≤ ε |H|^{4}|A^{∘}|^{2} +c(ε) |A^{∘}|^{6} and rearranging, we arrive at

∫ |∇^{2}H|^{2}η dμ + ∫ |H|^{2} |∇H|^{2} η dμ (16)

≤ c ∫ |W |^{2} η dμ + c Λ^{4} ∫_{[γ>0]} |A|^{2} dμ

+ c(ε) ∫ |A^{∘}|^{2} |∇A^{∘}|^{2} η dμ + c(ε) ∫ |A^{∘}|^{6} η dμ + ε ∫ |H|^{4} |A^{∘}|^{2}η dμ + ε ∫ |H|^{2} |∇A^{∘}|^{2} η dμ.

Next we use (10) to compute

∫ |H|^{2} |∇A^{∘}|^{2} η dμ

= − ∫ |H|^{2}⟨A^{∘}, ΔA^{∘}⟩η dμ + ∫ H ∗ ∇H ∗ A^{∘} ∗ ∇A^{∘} η dμ

+ ∫ |H|^{2} A^{∘} ∗ ∇A^{∘} ∗ ∇η dμ

= − ∫ |H|^{2}⟨A^{∘}, ∇^{2}H + \frac{1}{2}|H|^{2} A^{∘} + A^{∘} ∗ A^{∘} ∗ A^{∘}⟩η dμ

+ ∫ H ∗ ∇H ∗ A^{∘} ∗ ∇A^{∘}η dμ + ∫ |H|^{2} A^{∘} ∗ ∇A^{∘} ∗ ∇η dμ

≤ \frac{1}{2} ∫ |H|^{2} |∇H|^{2}η dμ + ∫ H ∗ ∇H ∗ A^{∘} ∗ ∇A^{∘} η dμ

+ ∫ |H|^{2} A^{∘} ∗ ∇A^{∘} ∗ γ^{3} ∇γ dμ

− \frac{1}{2} ∫ |H|^{4} |A^{∘}|^{2} η dμ + c ∫ |H|^{2} |A^{∘}|^{4}η dμ

≤ (\frac{1}{2} + δ) ∫ |H|^{2} |∇H|^{2}η dμ + c(δ) ∫ |A^{∘}|^{2} |∇A^{∘}|^{2}η dμ

+ δ ∫ |H|^{2} |∇A^{∘}|^{2} η dμ + c(δ) Λ^{2} ∫ |H|^{2} |A^{∘}|^{2} γ^{2} dμ

− \frac{1}{2} ∫ |H|^{4} |A^{∘}|^{2} η dμ + c ∫ |H|^{2} |A^{∘}|^{4}η dμ.

416 e. kuwert & r. schätzle

From the inequalities

c ∫ |H|^{2} |A^{∘}|^{4} η dμ ≤ δ ∫ |H|^{4} |A^{∘}|^{2} η dμ + c(δ) ∫ |A^{∘}|^{6}η dμ,
c(δ) Λ^{2} ∫ |H|^{2} |A^{∘}|^{2} γ^{2} dμ ≤ δ ∫ |H|^{4} |A^{∘}|^{2}η dμ + c(δ) Λ^{4} ∫_{[γ>0]} |A^{∘}|^{2} dμ,

we see that

(1 − δ) ∫ |H|^{2}|∇A^{∘}|^{2} η dμ + (\frac{1}{2} − 2δ) ∫ |H|^{4} |A^{∘}|^{2}η dμ (17)

≤ (\frac{1}{2} + δ) ∫ |H|^{2} |∇H|^{2} η dμ

+ c(δ) (∫ |A^{∘}| |∇A^{∘}|^{2}η dμ + ∫ |A^{∘}|^{6}η dμ + Λ^{4} ∫_{[γ>0]} |A|^{2} dμ) .

Adding the inequalities (16) and (17) yields

∫ |∇^{2}H|^{2}η dμ + (\frac{1}{2} − δ) ∫ |H|^{2}|∇H|^{2}η dμ

+ (1 − δ − ε) ∫ |H|^{2} |∇A^{∘}|^{2}η dμ + (\frac{1}{2} − 2δ − ε) ∫ |H|^{4} |A^{∘}|^{2}η dμ

≤ c(δ, ε) (∫ |A^{∘}|^{2} |∇A^{∘}|^{2}η dμ + ∫ |A^{∘}|^{6}η dμ + Λ^{4} ∫_{[γ>0]} |A|^{2} dμ)

+ c ∫ |W |^{2}η dμ. The claim of the Lemma follows by choosing ε = δ = \frac{1}{8}. q.e.d.

Proposition 2.4. If f : Σ → ℝ^{n} is an immersion with W(f) = W and η = γ^{4}, where γ ∈ C_{c}^{1}(Σ) satisfies |∇γ| ≤ Λ, then ∫ |∇^{2}A|^{2}η dμ + ∫ |A|^{2} |∇A|^{2}η dμ + ∫ |A|^{4} |A^{∘}|^{2}η dμ

≤ c ∫ |W |^{2}η dμ + c Λ^{4} ∫_{[γ>0]} |A|^{2} dμ + c ∫ (|A^{∘}|^{2} |∇A^{∘}|^{2} + |A^{∘}|^{6})η dμ.

Proof. Multiply (12) by η ∇A^{∘}, integrate by parts and apply (10) to get

∫ |∇^{2}A^{∘}|^{2}η dμ ≤ ∫ |ΔA^{∘}|^{2}η dμ + c ∫ |A|^{2} |∇A|^{2}η dμ

+ ∫ γ^{3}∇γ ∗ ∇A^{∘} ∗ ∇^{2}A^{∘} dμ

≤ c ∫ |∇^{2}H|^{2}η dμ

+c ∫ |A|^{2} |∇A|^{2} dμ + c ∫ |A|^{4} |A^{∘}|^{2}η dμ

+\frac{1}{2} ∫ |∇^{2}A^{∘}|^{2}η dμ + c Λ^{2} ∫ |∇A^{∘}|^{2}γ^{2} dμ.

The claim now follows from Lemma 2.2 and Lemma 2.3, recalling (8).

q.e.d.

We next need a multiplicative Sobolev inequality. Lemma 2.5. Under the assumptions of Proposition 2.4 we have

∫ (|A^{∘}|^{2} |∇A^{∘}|^{2} + |A^{∘}|^{6})η dμ (18)

≤ c ∫_{[γ>0]} |A^{∘}|^{2} dμ · ∫ (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2})η dμ

+ c Λ^{4} ( ∫_{[γ>0]} |A^{∘}|^{2} dμ)^{2} .

Proof. Recall the Michael-Simon Sobolev inequality ([7])

(∫_{Σ} u^{2} dμ)^{\frac{1}{2}} ≤ c (∫_{Σ} |∇u| dμ + ∫_{Σ} |H| |u| dμ) , (19)

with c = c(n). Letting u = |A^{∘}| |∇A^{∘}| γ^{2} we obtain

∫ |A^{∘}|^{2} |∇A^{∘}|^{2}η dμ

≤ c (∫ |A^{∘}| |∇^{2}A^{∘}| γ^{2} dμ)^{2} + c (∫ |∇A^{∘}|^{2} γ^{2} dμ)^{2}

+ c (∫ |A| |A^{∘}| |∇A^{∘}| γ^{2} dμ)^{2} + c (∫ |A^{∘}| |∇A^{∘}| γ |∇γ| dμ)^{2}

≤ c ∫_{[γ>0]}|A^{∘}|^{2} dμ ∫ (|∇^{2}A^{∘}|^{2} + |A|^{2} |∇A^{∘}|^{2})η dμ

+ c Λ^{4}( ∫_{[γ>0]}|A^{∘}|^{2} dμ)^{2} + c (∫ |∇A^{∘}|^{2} γ^{2} dμ)^{2} .

In the last term, we integrate by parts to get

(20) ∫ |∇A^{∘}|^{2} γ^{2} dμ ≤ c ∫ |A^{∘}| |∇^{2}A^{∘}| γ^{2} dμ + c Λ ∫ |A^{∘}| |∇A^{∘}| γ dμ

≤ c ( ∫_{[γ>0]} |A^{∘}|^{2} dμ)^{\frac{1}{2}} · (∫ |∇^{2}A^{∘}|^{2} η dμ)^{\frac{1}{2}}

+ \frac{1}{2} ∫ |∇A^{∘}|^{2} γ^{2} dμ + cΛ^{2} ∫_{[γ>0]}|A^{∘}|^{2} dμ.

Absorbing and inserting proves the claimed inequality for the first term in (18). For the other term, choose u = |A^{∘}|^{3} γ^{2} in (19) and compute

∫ |A^{∘}|^{6} η dμ

≤ c ( ∫ |A^{∘}|^{2} |∇A^{∘}| γ^{2} dμ + ∫ |A| |A^{∘}|^{3} γ^{2} dμ + c Λ ∫ |A^{∘}|^{3} γ dμ)^{2}

≤ c ( ∫ |∇A^{∘}|^{2} γ^{2} dμ)^{2} + c ∫_{[γ>0]} |A^{∘}|^{2} dμ · ∫ |A|^{2}|A^{∘}|^{4} η dμ

+ c Λ^{4}( ∫_{[γ>0]} |A^{∘}|^{2} dμ)^{2}.

Combining with (20) proves the estimate for the second term on the left of (18). q.e.d.

Proposition 2.6. Let f : Σ → ℝ^{n} be an immersed surface, and let Λ = ∥∇γ∥_{L^{∞}}, where γ has compact support on Σ. There exists a constant ε_{0} = ε_{0}(n) > 0 such that if

∫_{[γ>0]} |A^{∘}|^{2} dμ < ε_{0},

then we have for a constant c = c(n) < ∞

∫ (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2}) γ^{4} dμ

≤ c ∫ |W(f)|^{2}γ^{4} dμ + c Λ^{4} ∫_{[γ>0]}|A|^{2} dμ.

This is an immediate consequence of Proposition 2.4 and Lemma 2.5. As a first application we deduce the following result.

Theorem 2.7 (Gap Lemma). Let f : Σ → ℝ^{n} be a properly immersed (compact or noncompact) Willmore surface, and let Σ_{ϱ}(0) = f^{−1}(B_{ϱ}(0)). If

lim inf_{ϱ→∞} \frac{1}{ϱ^{4}} ∫_{Σ_{ϱ}(0)}|A|^{2} dμ = 0, and

∫_{Σ} |A^{∘}|^{2} dμ < ε_{0} = ε_{0}(n),

then f is an embedded plane or sphere.

Proof. We take γ(p) = φ(\frac{1}{ϱ} |f(p)|), where φ ∈ C^{1}(ℝ) satisfies φ(s) = 1 for s ≤ \frac{1}{2}, φ(s) = 0 for s ≥ 1 and φ ≥ 0. Then we have Λ = c/ϱ in Proposition 2.6. Since W(f) = 0 by assumption, we can let ϱ → ∞ and conclude A^{∘} ≡ 0. This implies, by a standard result of differential geometry [13], that f maps into a fixed, round 2-sphere or plane S ⊂ ℝ^{n}. As f is complete, it follows that f : (Σ, g) → S is a global isometry. q.e.d.

We shall now derive an L^{∞} bound for A^{∘} from Proposition 2.6.

Lemma 2.8. For γ ∈ C_{c}^{1}(Σ) with |∇γ| ≤ Λ and any normal p-form ϕ along f we have the inequality

∥γ^{2} ϕ∥_{L^{∞}}^{4} ≤ c ∥γ^{2} ϕ∥_{L^{2}}^{2} [∫ (|∇^{2}ϕ|^{2} + |H|^{4} |ϕ|^{2}) γ^{4} dμ + Λ^{4} ∫_{[γ>0]} |ϕ|^{2} dμ] .

Proof. This is Lemma 4.3 in [5], except that there a bound on the second derivatives of γ was assumed. Letting ψ = γ^{2}ϕ we apply Theorem 5.6 in [5] to obtain

∥ψ∥_{L^{∞}}^{6} ≤ c ∥ψ∥_{L^{2}}^{2} (∥∇ψ∥_{L^{4}}^{4} + ∥H ψ∥_{L^{4}}^{4}) (21)

≤ c ∥ψ∥_{L^{2}}^{2} (∫ γ^{8}|∇ϕ|^{4} dμ + Λ^{4} ∫ γ^{4}|ϕ|^{4} dμ + ∫ |H|^{4} |ψ|^{4} dμ) .

The three integrals on the right are estimated as follows (starting with the third):

(22)  ∫ |H|^{4} |ψ|^{4} dμ ≤ ∥ψ∥_{L^{∞}}^{2} ∫ |H|^{4} |ϕ|^{2} γ^{4} dμ,
(23)  Λ^{4} ∫ γ^{4}|ϕ|^{4} dμ ≤ ∥ψ∥_{L^{∞}}^{2} Λ^{4} ∫_{[γ>0]} |ϕ|^{2} dμ.

By partial integration, we infer

∫ |∇ϕ|^{2} γ^{2} dμ ≤ c ∫ |ϕ| |∇^{2}ϕ| γ^{2} dμ + cΛ ∫ |ϕ| |∇ϕ| γ dμ

≤ \frac{c}{Λ^{2}} ∫ |∇^{2}ϕ|^{2} γ^{4} dμ + cΛ^{2} ∫_{[γ>0]}|ϕ|^{2} dμ + \frac{1}{2} ∫ |∇ϕ|^{2} γ^{2} dμ.

Using again integration by parts and Cauchy-Schwarz

∫ |∇ϕ|^{4} γ^{8} dμ ≤ c( ∫ |ϕ| |∇ϕ|^{2} |∇^{2}ϕ| γ^{8} dμ + Λ ∫ |ϕ| |∇ϕ|^{3} γ^{7} dμ)

≤ c ∥ψ∥_{L^{∞}} ( ∫ |∇ϕ|^{4} γ^{8} dμ)^{\frac{1}{2}} ( ∫ |∇^{2}ϕ|^{2} γ^{4} dμ)^{\frac{1}{2}}

+ cΛ ∥ψ∥_{L^{∞}} ( ∫ |∇ϕ|^{4} γ^{8} dμ)^{\frac{1}{2}} ( ∫ |∇ϕ|^{2} γ^{2} dμ)^{\frac{1}{2}} .

Combining the last two inequalities, we get

∫ |∇ϕ|^{4} γ^{8} dμ ≤ c ∥ψ∥_{L^{∞}}^{2}( ∫ |∇^{2}ϕ|^{2} γ^{4} dμ + c Λ^{4} ∫_{[γ>0]}|ϕ|^{2} dμ). (24)

Inserting (22)–(24) into (21) proves the claim. q.e.d.

Combining Proposition 2.6 and Lemma 2.8, where ϕ = A^{∘} and γ is a cutoff function depending on extrinsic distance as in Theorem 2.7, we obtain the following “partial” curvature estimate.

Theorem 2.9 (Tracefree Curvature Estimate). Let f : Σ → ℝ^{n} be an immersed surface with Σ_{ϱ} = f^{−1}(B_{ϱ}(x_{0})) ⊂⊂ Σ, and suppose that

∫_{Σ_{ϱ}} |A^{∘}|^{2} dμ < ε_{0},

where ε_{0} = ε_{0}(n) > 0 is a fixed constant. Then

∥A^{∘}∥_{L^{∞}(Σ_{ϱ/2})}^{2} ≤ c (∥W(f)∥_{L^{2}(Σ_{ϱ})} + \frac{1}{ϱ^{2}} ∥A∥_{L^{2}(Σ_{ϱ})}) ∥A^{∘}∥_{L^{2}(Σ_{ϱ})}. (25)

Assuming smallness of the full second fundamental form A, one easily adapts the arguments above to also prove the following:

Theorem 2.10 (Curvature Estimate). Let f : Σ → ℝ^{n} be an immersed surface, Σ_{ϱ} = f^{−1}(B_{ϱ}(x_{0})) ⊂⊂ Σ and suppose

∫_{Σ_{ϱ}} |A|^{2} dμ < ε_{0},

where ε_{0} = ε_{0}(n) is a fixed constant. Then

∥A∥_{L^{∞}(Σ_{ϱ/2})}^{2} ≤ c (∥W(f)∥_{L^{2}(Σ_{ϱ})} + \frac{1}{ϱ^{2}} ∥A∥_{L^{2}(Σ_{ϱ})}) ∥A∥_{L^{2}(Σ_{ϱ})}. (26)

Remark 2.11. The statements of the Theorems 2.7, 2.9 and 2.10 clearly also hold with the extrinsic distance sets Σ_{ϱ}(x_{0}) replaced by distance sets with respect to the intrinsic distance function, since only a bound on the first derivatives of the cutoff function was needed.

3. Local estimates for the flow

We now consider solutions f : Σ × [0, T) → ℝ^{n} to the gradient flow for the Willmore integral,

∂_{t}f = −W(f).

We abbreviate W(f) =: W in the following and compute first a precise formula for the evolution of the energy density. Recall from [5]:

(27)  ∂_{t}(dμ) = ⟨H, W ⟩ dμ,
(28)  ∂_{t}^{⊥}H = − (ΔW + Q(A^{∘})W + \frac{1}{2}H⟨H, W ⟩) ,
(29) ∂_{t}^{⊥}A(X, Y) = −∇_{X,Y}^{2} W + \frac{1}{2} g(X, Y) [Q(A^{∘})W + \frac{1}{2}H⟨H, W ⟩]

+ \frac{1}{2}H⟨A^{∘}(X, Y), W ⟩ + \frac{1}{2}A^{∘}(X, Y)⟨H, W ⟩

+ \frac{1}{2}R^{⊥}(X, Y)W.

Here we used (2.18), (2.6) and (2.3) from [5]. Furthermore, using (2.15) in [5] we infer

∂_{t}^{⊥} (\frac{1}{2} g(X, Y) H) = −\frac{1}{2} g(X, Y) (ΔW + Q(A^{∘})W + \frac{1}{2}H⟨H, W ⟩)

+ ⟨A^{∘}(X, Y), W ⟩H + \frac{1}{2}g(X, Y)H ⟨H, W ⟩,

and subtracting this from (29) yields

(30) ∂_{t}^{⊥}A^{∘}(X, Y) = −S^{∘}(∇^{2}W) + g(X, Y) Q(A^{∘})W

+ \frac{1}{2} A^{∘}(X, Y)⟨H, W ⟩ − \frac{1}{2} H⟨A^{∘}(X, Y), W ⟩.

Recall that S^{∘}(…) denotes the symmetric, tracefree component. We compute separately for H and A^{∘}. By (27) and (28)

∂_{t} (\frac{1}{2} |H|^{2} dμ)

= − ⟨ΔW + Q(A^{∘})W + \frac{1}{2} H⟨H, W ⟩, H⟩ dμ + \frac{1}{2}|H|^{2} ⟨H, W ⟩ dμ

= −⟨ΔH + Q(A^{∘}) H, W ⟩ dμ + (⟨ΔH, W ⟩ − ⟨H, ΔW ⟩) dμ

= −|W |^{2} dμ + ∇_{e_{i}}(⟨∇_{e_{i}} H, W ⟩ − ⟨H, ∇_{e_{i}} W ⟩) dμ,

whence

∂_{t} (\frac{1}{2} |H|^{2} dμ) + |W |^{2} dμ = (∇^{∗}α) dμ, (31)

where α is the 1-form given by

α(X) = ∇_{X} ⟨H, W ⟩ − 2⟨∇_{X} H, W ⟩. (32)

In order to compute for A^{∘}, we first have (using again (2.15) in [5]) for a g-orthonormal basis

g(∂_{t}e_{i}, e_{j}) + g(e_{i}, ∂_{t}e_{j}) = ∂_{t}(g(e_{i}, e_{j})) − (∂_{t}g)(e_{i}, e_{j})
= −2⟨A(e_{i}, e_{j}), W ⟩
= −2⟨A^{∘}(e_{i}, e_{j}), W ⟩ − δ_{ij}⟨H, W ⟩.

This implies further

⟨A^{∘}(∂_{t}e_{i}, e_{k}), A^{∘}(e_{i}, e_{k})⟩ = g(∂_{t}e_{i}, e_{j})⟨A^{∘}(e_{j}, e_{k}), A^{∘}(e_{i}, e_{k})⟩

= − (⟨A^{∘}(e_{i}, e_{j}), W ⟩ + \frac{1}{2} δ_{ij}⟨H, W ⟩) ⟨A^{∘}(e_{i}, e_{k}), A^{∘}(e_{j}, e_{k})⟩

= −⟨A^{∘}(e_{i}, e_{k})⟨A^{∘}(e_{i}, e_{j}), W ⟩, A^{∘}(e_{j}, e_{k})⟩ − \frac{1}{2} |A^{∘}|^{2}⟨H, W ⟩

= − ⟨\frac{1}{2} g(e_{j}, e_{k})Q(A^{∘})W, A^{∘}(e_{j}, e_{k})⟩ − \frac{1}{2} |A^{∘}|^{2}⟨H, W ⟩

= −\frac{1}{2}|A^{∘}|^{2}⟨H, W ⟩,

where we used (2.5) from [5]. We use this and (30) to compute

∂_{t} (|A^{∘}|^{2} dμ)
= 2⟨(∂_{t} A^{∘}) (e_{i}, e_{k}), A^{∘}(e_{i}, e_{k})⟩dμ

+ 2⟨A^{∘}(∂_{t}e_{i}, e_{k}) + A^{∘}(e_{i}, ∂_{t}e_{k}), A^{∘}(e_{i}, e_{k})⟩ dμ + |A^{∘}|^{2} ⟨H, W ⟩dμ

= −2⟨∇^{2}W, A^{∘}⟩dμ + |A^{∘}|^{2} ⟨H, W ⟩dμ
− ⟨A^{∘}(e_{i}, e_{k}), W ⟩ ⟨A^{∘}(e_{i}, e_{k}), H⟩dμ
− 2|A^{∘}|^{2}⟨H, W ⟩dμ + |A^{∘}|^{2} ⟨H, W ⟩dμ
= −2⟨∇^{2}W, A^{∘}⟩dμ − ⟨Q(A^{∘})H, W ⟩dμ
= ( − 2 ∇_{e_{i}}⟨∇_{e_{j}}W, A^{∘}(e_{i}, e_{j})⟩ + ⟨∇_{e_{j}}W, ∇_{e_{j}}H⟩)dμ
− ⟨Q(A^{∘})H, W ⟩dμ
= ( − 2 ∇_{e_{i}}⟨∇_{e_{j}}W, A^{∘}(e_{i}, e_{j})⟩ + ∇_{e_{i}}⟨W, ∇_{e_{i}}H⟩)dμ
− ⟨W, ΔH⟩dμ − ⟨Q(A^{∘})H, W ⟩dμ.

Thus we have shown ∂_{t}(|A^{∘}|^{2} dμ) + |W |^{2} dμ = (∇^{∗}β) dμ, (33) where β is the 1-form defined by β(X) = 2⟨∇_{e_{j}}W, A^{∘}(X, e_{j})⟩ − ⟨∇_{X}H, W ⟩. (34)

Lemma 3.1. If f is a Willmore flow, then for any function η and W = W(f) we have:

(35) ∂_{t} ∫ \frac{1}{2}|H|^{2}η dμ + ∫ |W |^{2}η dμ

= ∫ (\frac{1}{2}|H|^{2}∂_{t}η − ⟨H, W ⟩Δη − 2⟨∇_{grad η} H, W ⟩) dμ,

(36) ∂_{t} ∫ |A^{∘}|^{2}η dμ + ∫ |W |^{2}η dμ

= ∫ (|A^{∘}|^{2} ∂_{t}η − 2⟨A^{∘}(e_{i}, e_{j}), W ⟩∇_{e_{i},e_{j}}^{2}η − 2⟨∇_{grad η} H, W ) dμ.

Proof. Formula (35) is immediate from (31) and (32). For (36) we compute for β as in (34):

∫ η ∇^{∗}β dμ = ∫ (2⟨∇_{e_{j}}W, A^{∘}(grad η, e_{j})⟩ − ⟨∇_{grad η} H, W ⟩)dμ
= − ∫ 2⟨(∇_{e_{j}} A^{∘}) (e_{j}, grad η), W ⟩dμ

− ∫ 2⟨A^{∘}(∇_{e_{j}} grad η, e_{j}), W ⟩dμ − ∫ ⟨∇_{grad η} H, W ⟩dμ

= − ∫ 2 ∇_{e_{i},e_{j}}^{2}η ⟨A^{∘}(e_{i}, e_{j}), W ⟩dμ − ∫ 2⟨∇_{grad η} H, W ⟩dμ,

which, together with (33), proves (36). q.e.d.

In controlling the energy density in time, difficulties arise because of the dependence of ∂_{t}η and ∇^{2}η on f, and since W(f) differs from ΔH by the term Q(A^{∘})H. For a ball B_{ϱ} = B_{ϱ}(x_{0}) ⊂ ℝ^{n} and f : Σ → ℝ^{n} we adopt as in Section 2 the notation

Σ_{ϱ}(x_{0}) = f^{−1}(B_{ϱ}(x_{0}))

and consider a cutoff function γ̃ ∈ C_{c}^{1}(B_{ϱ}), γ̃ ≥ 0, such that

|D γ̃| ≤ \frac{c}{ϱ}, |D^{2} γ̃| ≤ \frac{c}{ϱ^{2}} . (37)

We put γ = γ̃ ∘ f and observe

(38)  ∇γ = (D γ̃ ∘ f) · Df
∇^{2}γ = (D^{2} γ̃ ∘ f) (Df, Df) + (D γ̃ ∘ f) · A( , ).

the willmore flow 425

Lemma 3.2. If f : Σ × [0, T) → ℝ^{n} is a Willmore flow and η = γ^{4} for γ = γ̃ ∘ f with γ̃ as in (37), then we have for W = W(f)

(39)  ∂_{t} ∫ |A^{∘}|^{2}η dμ + \frac{1}{2} ∫ |W |^{2}η dμ

≤ \frac{c}{ϱ^{2}} ∫ |A|^{2} |A^{∘}|^{2} γ^{2} dμ + \frac{c}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ

(40)  ∂_{t} ∫ |A|^{2}η dμ + ∫ |W |^{2}η dμ

≤ \frac{c}{ϱ^{2}} ∫ |A|^{4} γ^{2} dμ + \frac{c}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ.

Proof. We estimate the terms in (36) and (35). We have

∫ γ^{2}|∇H|^{2} dμ = − ∫ γ^{2}⟨H, ΔH⟩dμ + \frac{c}{ϱ} ∫ γ|H| |∇H| dμ
≤ − ∫ γ^{2}⟨H, W ⟩dμ + c ∫ γ^{2}|A^{∘}|^{2} |H|^{2}dμ

+\frac{1}{2} ∫ γ^{2}|∇H|^{2}dμ + \frac{c}{ϱ^{2}} ∫_{[γ>0]}|H|^{2}dμ.

As

∫ |∇η| |∇H| |W |dμ ≤ ε ∫ |W |^{2}η dμ + \frac{c(ε)}{ϱ^{2}} ∫ γ^{2} |∇H|^{2}dμ,

we obtain by combining

− ∫ 2⟨∇_{grad η} H, W ⟩dμ ≤ ε ∫ |W |^{2}ηdμ + \frac{c(ε)}{ϱ^{2}} ∫ γ^{2}|A^{∘}|^{2} |H|^{2}dμ

+\frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|H|^{2}dμ.

Next using (38)

− ∫ 2⟨A^{∘}(e_{i}, e_{j}), W ⟩∇_{e_{i},e_{j}}^{2}η dμ ≤ c ∫ |A^{∘}| |W | (\frac{1}{ϱ^{2}} γ^{2} + \frac{1}{ϱ}γ^{3}|A^{∘}|) dμ

≤ ε ∫ |W |^{2}η dμ + \frac{c(ε)}{ϱ^{2}} ∫ |A^{∘}|^{4}γ^{2} dμ

+ \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A^{∘}|^{2} dμ.

Finally

∫ |A^{∘}|^{2} ∂_{t}η dμ ≤ \frac{c}{ϱ} ∫ |A^{∘}|^{2} |W | γ^{3} dμ
≤ ε ∫ |W |^{2} γ^{4} dμ + \frac{c(ε)}{ϱ^{2}} ∫ |A^{∘}|^{4}γ^{2} dμ.

Combining the three estimates and absorbing for ε > 0 small, we obtain (39). The estimate (40) follows analogously from (35). q.e.d.

Lemma 3.3. Let f : Σ × [0, T) → ℝ^{n} be a Willmore flow. If

(41)  ∫_{Σ_{ϱ}(x_{0})} |A^{∘}|^{2} dμ < ε_{0} at some time t ∈ [0, T),

then for a constant c_{0} > 0 we have at time t

(42) ∂_{t} ∫ |A^{∘}|^{2}γ^{4} dμ + c_{0} ∫ (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2})γ^{4} dμ

≤ \frac{c}{ϱ^{4}} ∫_{Σ_{ϱ}(x_{0})} |A|^{2} dμ,

and

(43) ∂_{t} ∫ |H|^{2}γ^{4} dμ + c_{0} ∫ (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2})γ^{4} dμ

≤ \frac{c}{ϱ^{4}} ∫_{Σ_{ϱ}(x_{0})} |A|^{2} dμ.

Proof. (42) follows by combining (39) with Proposition 2.6, after estimating

\frac{c}{ϱ^{2}} ∫ |A|^{2} |A^{∘}|^{2} γ^{2} dμ ≤ ε ∫ |A|^{4} |A^{∘}|^{2} γ^{4} dμ + \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A^{∘}|^{2} dμ.

For the other bound we must go back to (35), estimating the three terms on the right hand side. We have

∫ \frac{1}{2}|H|^{2} ∂_{t} η dμ = − ∫ \frac{1}{2}|H|^{2} D η̃ ∘ f · (ΔH + Q(A^{∘}) H) dμ.

By Young’s inequality with 4 and 4/3, we have

\frac{1}{ϱ} ∫ |A^{∘}|^{2} |A|^{3} γ^{3} dμ ≤ ε ∫ |A^{∘}|^{8/3} |A|^{10/3} γ^{4} dμ + \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ. (44)

Using integration by parts, we infer

∫ H ∗ H⟨D η̃ ∘ f, ΔH⟩ dμ

≤ \frac{c}{ϱ} ∫ |H| |∇H|^{2} γ^{3} dμ + \frac{c}{ϱ^{2}} ∫ |H|^{2} |∇H| γ^{2} dμ

≤ ε ∫ |H|^{2} |∇H|^{2} γ^{4} dμ + \frac{c(ε)}{ϱ^{2}} ∫ |∇H|^{2} γ^{2} dμ

+ \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|H|^{2} dμ.

In the proof of Lemma 3.2, we have already shown by partial integration that

∫ γ^{2}|∇H|^{2} dμ ≤ δϱ^{2} ∫ |W |^{2} γ^{4} dμ + \frac{c(δ)}{ϱ^{2}} ∫_{[γ>0]}|A|^{2} dμ

+δϱ^{2} ∫ |A|^{2} |A^{∘}|^{4} γ^{4} dμ,

so that by combining we obtain

(45)  ∫ H ∗ H⟨D η̃ ∘ f, ΔH⟩ dμ

≤ ε ∫ (|W |^{2} + |A|^{2} |∇A|^{2} + |A|^{2} |A^{∘}|^{4}) γ^{4}dμ

+ \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ.

Thus (44) and (45) estimate the first of the three terms on the right hand side of (35). For the second we use

− ∫ ⟨H, W ⟩ Δη dμ

≤ − ∫ ⟨H, ΔH⟩ ⟨D η̃ ∘ f, H⟩dμ + \frac{c}{ϱ^{2}} ∫ |H| |Δ H| γ^{2} dμ

+ \frac{c}{ϱ} ∫ |A^{∘}|^{2} |A|^{3} γ^{3} dμ + \frac{c}{ϱ^{2}} ∫ |A^{∘}|^{2} |A|^{2}γ^{2} dμ.

The first integral is estimated by (45), the third integral by (44). Furthermore

\frac{c}{ϱ^{2}} ∫ |H| |Δ H| γ^{2} dμ ≤ ε ∫ |∇^{2}A|^{2} γ^{4} dμ + \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ

\frac{c}{ϱ^{2}} ∫ |A^{∘}|^{2} |A|^{2} γ^{2} dμ ≤ ε ∫ |A^{∘}|^{4} |A|^{2} γ^{4} dμ + \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ.

The third integral on the right of (35) satisfies

∫ |∇η| |∇H| |W | dμ ≤ ε ∫ |W |^{2}γ^{4} dμ + \frac{c(ε)}{ϱ^{2}} ∫ |∇H|^{2} γ^{2} dμ

and the right hand side is already estimated. Thus putting things together we have shown

∂_{t} (∫ \frac{1}{2}|H|^{2} η dμ) + \frac{3}{4} ∫ |W |^{2}η dμ

≤ ε ∫ (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2})η dμ + \frac{c(ε)}{ϱ^{4}} ∫_{[γ>0]}|A|^{2} dμ.

Now (43) follows from Proposition 2.6. q.e.d.

Proposition 3.4. Let f : Σ × [0, T) → ℝ^{n} be a Willmore flow with ∫_{Σ} |A|^{2} dμ ≤ κ. There exist constants ε_{1} = ε_{1}(n) > 0 and c_{1} = c(n)/κ > 0, such that if ϱ > 0 is chosen with

(46) ∫_{Σ_{ϱ}} |A^{∘}|^{2} dμ ≤ ε < ε_{1} at time t = 0 for all Σ_{ϱ} = Σ_{ϱ}(x_{0}) ⊂ ℝ^{n},

then for any time 0 ≤ t < t_{1} = min{c_{1} ϱ^{4}, T } we have

(47) ∫_{Σ_{ϱ}} |A^{∘}|^{2} dμ + ∫_{0}^{t} ∫_{Σ_{ϱ}} (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2})dμ dτ
≤ c(ε + κϱ^{−4} t),

∫_{0}^{t} ∥A^{∘}∥_{L^{∞}(Σ_{ϱ})}^{4}dτ ≤ c(ε + κ ϱ^{−4} t). (48)

Moreover, for 0 < σ ≤ ϱ and τ < min{c_{1} σ^{4}, T } we then also have

∫_{Σ_{σ/2}(x)} |A|^{2} dμ|_{t=τ} ≤ ∫_{Σ_{σ}(x)} |A|^{2} dμ|_{t=0} + c κ σ^{−4} τ ∀x ∈ ℝ^{n}. (49)

Proof. Let N = N(n) be the number of balls B_{ϱ/2} ⊂ ℝ^{n} needed to cover B_{ϱ} ⊂ ℝ^{n} and choose ε_{1} ≤ \frac{ε_{0}}{4N}, where ε_{0} > 0 is as in Lemma 3.3. Assume (41) is satisfied on [0, t] for all B_{ϱ} ⊂ ℝ^{n} , and integrate (42) to obtain using (46)

∫_{Σ_{ϱ/2}} |A^{∘}|^{2} dμ + c_{0} ∫_{0}^{t} ∫_{Σ_{ϱ/2}} (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2}) dμ ds

≤ ε + cκϱ^{−4}t.

Assuming t ≤ c_{1}ϱ^{4} where c_{1} is chosen with 0 < c_{1} ≤ \frac{ε_{0}}{4N c κ}, we conclude

∫_{Σ_{ϱ}} |A^{∘}|^{2} dμ + c_{0} ∫_{0}^{t} ∫_{Σ_{ϱ}} (|∇^{2}A|^{2} + |A|^{2} |∇A|^{2} + |A|^{4} |A^{∘}|^{2}) dμ ds
≤ N(ε + c κ ϱ^{−4} t)
≤ N(ε_{1} + c κ c_{1})
≤ \frac{ε_{0}}{2} .

It follows that (41) holds up to time t = t_{1} for all x_{0} ∈ ℝ^{n}, and (47) follows. In particular ∫_{Σ_{ϱ}} |A^{∘}|^{2} dμ ≤ c (ε_{1} + κc_{1}), whence a covering argument with possibly smaller ε_{1}, c_{1} implies the smallness hypothesis in Theorem 2.9 for any ball B_{2ϱ} ⊂ ℝ^{n} and any t ∈ [0, t_{1}]. Inequality (48) now follows from combining (25) with (47), again using a covering. Finally (49) is obtained by integrating (43) and (42) on [0, t]. q.e.d.

We next state a version of the higher order estimates obtained in [5] which is localized in time.

Theorem 3.5 (Interior Estimates). Let f : Σ × (0, T] → ℝ^{n} be a Willmore flow satisfying the condition

sup_{0<t≤T} ∫_{Σ_{ϱ}(0)} |A|^{2} dμ ≤ ε < ε_{0}(n), (50)

where T ≤ c(n) ϱ^{4}. Then for any k ∈ ℕ_{0} we have at time t ∈ (0, T] the estimates

(51)  ∥∇^{k}A∥_{L^{∞}(Σ_{ϱ/2}(0))} ≤ c(k)\sqrt{ε} t^{− \frac{k+1}{4}}
(52)  ∥∇^{k}A∥_{L^{2}(Σ_{ϱ/2}(0))} ≤ c(k)\sqrt{ε} t^{− \frac{k}{4}} .

Proof. By scaling f_{ϱ}(p, t) = \frac{1}{ϱ}f(p, ϱ^{4}t) we may assume ϱ = 1. Using (4.13) and (4.9) from [5], see also Proposition 4.6 in [5], we obtain on B = B_{3/4}(0) the inequalities

(53)  ∫_{0}^{T} ∫_{Σ_{3/4}} (|∇^{2}A|^{2} + |A|^{6}) dμ dt ≤ c ε,

(54)  ∫_{0}^{T} ∥A∥_{L^{∞}(Σ_{3/4})}^{4} dt ≤ c ε.

Fix a cutoff function γ̃ ∈ C_{c}^{∞}(ℝ^{n}) with χ_{B_{1/2}} ≤ γ̃ ≤ χ_{B} and ∥D γ̃∥_{L^{∞}} + ∥D^{2} γ̃∥_{L^{∞}} ≤ c. Also define cutoff functions in time by

χ_{j}(t) = \begin{cases}0 & for t ≤ (j − 1) \frac{T}{m} \\ \frac{m}{T} (t − (j − 1) \frac{T}{m}) & in between \\ 1 & for t ≥ j \frac{T}{m}, \end{cases}

where 0 ≤ j ≤ m and m ∈ ℕ_{0}. Note χ_{0} ≡ 1 on [0, T], χ_{m}(T) = 1 and

0 ≤ χ̇_{j} ≤ \frac{m}{T} χ_{j−1}. (55)

Introducing the notation α(t)=∥A∥_{L^{∞}(Σ_{3/4})}^{4}, E_{j}(t) = ∫ |∇^{2j}A|^{2} γ^{4j+4}dμ (where γ = γ̃ ∘ f), we have by (4.14) in [5]

\frac{d}{dt}E_{j}(t) + \frac{1}{2}E_{j+1}(t) ≤ c α(t) E_{j}(t) + c (1 + α(t))ε.

Letting e_{j}(t) = χ_{j}(t) E_{j}(t) this implies, using also (55),

(56)  \frac{d}{dt}e_{j}(t) ≤ c α(t) e_{j}(t) − \frac{1}{2}χ_{j}(t) E_{j+1}(t)

+c (1 + α(t)) ε + \frac{m}{T} χ_{j−1}(t) E_{j}(t).

We now prove by induction for 0 ≤ j ≤ m and all t ∈ (0, T] the inequality

e_{j}(t) + \frac{1}{2} ∫_{0}^{t} χ_{j}(s) E_{j+1}(s) ds ≤ \frac{c(m) ε}{T^{j}}.

For j = 0 this follows from assumption (50) and estimate (53). Integrating (56) on [0, T] yields, for j ≥ 1,

e_{j}(t) + \frac{1}{2} ∫_{0}^{t} χ_{j}(s) E_{j+1}(s) ds

≤ c ∫_{0}^{t} α(s) e_{j}(s) ds + c ε ∫_{0}^{t} (1 + α(s))ds

+ \frac{m}{T} ∫_{0}^{t} χ_{j−1}(s) E_{j}(s) ds.

Now since ∫_{0}^{T} α(t) dt ≤ c ε by (54), we may apply Gronwall’s inequality to get

e_{j}(t) + \frac{1}{2} ∫_{0}^{t} χ_{j}(s) E_{j+1}(s) ds ≤ c ε + \frac{c m}{T} \frac{c(m) ε}{T^{j−1}}
≤ \frac{c(m) ε}{T^{j}},

as T ≤ c(n) by assumption. Thus we have at time t = T

∫ |∇^{2m}A|^{2} γ^{4m+4} dμ ≤ \frac{c(m)ε}{T^{m}}.

The estimate for ∇^{2m+1}A follows by interpolation as in Lemma 5.1 of [5], taking r = 1, p = q = 2, α = 1, β = 0, s = 4m + 6 and t = \frac{1}{s} ∈ [−\frac{1}{2}, \frac{1}{2}] there and using again T ≤ c(n). Renaming T into t, the L^{2}-estimate (52) is proved. Using (4.9) and (4.7) in [5], the L^{∞}-estimate (51) follows.

q.e.d.

4. Construction of the blowup

In this section we rescale the Willmore flow at an assumed singularity at finite or infinite time, thereby constructing a static Willmore surface as a limit. We shall need the following local area bound due to L. Simon [8].

Lemma 4.1. Let f : Σ → ℝ^{n} be a properly immersed surface. Then for 0 < σ ≤ ϱ < ∞ and Σ_{ϱ} = Σ_{ϱ}(x_{0}) one has

\frac{μ(Σ_{σ})}{σ^{2}} ≤ c (\frac{μ(Σ_{ϱ})}{ϱ^{2}} + ∫_{Σ_{ϱ}} |H|^{2} dμ) .

In particular if Σ is compact without boundary

\frac{μ(Σ_{σ})}{σ^{2}} ≤ c (W(f) + 4πχ(Σ)).

The following compactness theorem, whose proof is omitted, is a localized version of the result of J. Langer [6].

Theorem 4.2. Let f_{j} : Σ_{j} ⟶ ℝ^{n} be a sequence of proper immersions, where Σ_{j} is a two-dimensional manifold without boundary. Let

Σ_{j}(R) = {p ∈ Σ_{j} : |f_{j}(p)| < R}

and assume the bounds

μ_{j}(Σ_{j}(R)) ≤ c(R) for any R > 0,
∥∇^{k}A_{j}∥_{L^{∞}} ≤ c(k) for any k ∈ ℕ_{0}.

Then there exists a proper immersion f̂: Σ̂ → ℝ^{n}, where Σ̂ is again a two-manifold without boundary, such that after passing to a subsequence we have a representation

f_{j} ∘ φ_{j} = f̂+ u_{j} on Σ̂(j) = {p ∈ Σ̂ : |f̂(p)| < j} (57)

with the following properties:

φ_{j} : Σ̂(j) → U_{j} ⊂ Σ_{j} is diffeomorphic,
Σ_{j}(R) ⊂ U_{j} if j ≥ j(R),
u_{j} ∈ C^{∞}(Σ̂(j), ℝ^{n}) is normal along f̂,
∥ ˆ∇^{k}u_{j}∥_{L^{∞}(Σ̂(j))} → 0 as j → ∞, for any k ∈ ℕ_{0}.

Roughly speaking, the theorem says that on any ball B_{R}(0) the immersion f_{j} can be written as a normal graph f̂+ u_{j} with small norm for j large over a limit immersion f̂, after suitably reparametrizing with φ_{j}.

Now let f : Σ × [0, T) → ℝ^{n} be a smooth Willmore flow defined on a closed surface Σ, where 0 < T ≤ ∞. Define

κ(r, t) = sup_{x∈ℝ^{n}} ∫_{Σ_{r}(x)} |A(t)|^{2} dμ_{t}.

Choose an arbitrary sequence r_{j} ↘ 0 and assume concentration in the sense that for all j

t_{j} = inf{t ≥ 0 : κ(r_{j}, t) > ε_{1}} < T, (58) where ε_{1} = ε_{0}/c and ε_{0}, c are the constants from Theorem 1.2 in [5]. Clearly

∫_{Σ_{r_{j}} (x)} |A(t_{j})|^{2} dμ_{t_{j}} ≤ ε_{1} for any x ∈ ℝ^{n}.

On the other hand, choosing an appropriate sequence of balls at times τ_{ν} ↘ t_{j}, we find a point x_{j} ∈ ℝ^{n} satisfying

∫_{f^{−1}(\overline{B_{r_{j}} (x_{j})})}|A(t_{j})|^{2} dμ_{t_{j}} ≥ ε_{1}.

Now we rescale by considering

\begin{matrix}f_{j} : Σ × [−r_{j}^{−4}t_{j}, r_{j}^{−4}(T − t_{j})) → ℝ^{n}, \\ f_{j}(p, t) = \frac{1}{r_{j}} (f(p, t_{j} + r_{j}^{4}t) − x_{j}). \end{matrix} (59)

By the above we have κ_{j}(1, t) ≤ ε_{1} for t ≤ 0 and

∫_{f^{−1}(\overline{B_{1}(0)})} |A_{j}(0)|^{2} dμ_{j} ≥ ε_{1}. (60)

Furthermore Theorem 1.2 of [5] yields r_{j}^{−4}(T − t_{j}) ≥ c_{0} and in fact

κ_{j}(1, t) ≤ ε_{0} for 0 < t ≤ c_{0}.

We may now apply Theorem 3.5 on parabolic cylinders B_{1}(x) × (t−1, t] to obtain

∥∇^{k}A_{j}(t)∥_{L^{∞}} ≤ c(k) for −r_{j}^{−4}t_{j} + 1 ≤ t ≤ c_{0}. (61)

Furthermore Lemma 4.1 yields

\frac{μ_{j}(t)(Σ_{R}(0))}{R^{2}} ≤ c(W(f_{0}) + 4π χ(Σ)) < ∞.

We apply Theorem 4.2 to the sequence f_{j} = f_{j}( · , 0 ) : Σ → ℝ^{n}, thus obtaining a limit immersion f̂_{0} : Σ̂ → ℝ^{n}. Let φ_{j} : Σ̂(j) → U_{j} ⊂ Σ be as in (57). Then the reparametrization

f_{j}(φ_{j}, ·) : Σ̂(j) × [0, c_{0}] → ℝ^{n} (62) is a Willmore flow with initial data

f_{j}(φ_{j}, 0) = f̂_{0} + u_{j} : Σ̂(j) → ℝ^{n}. (63)

The flows (62) satisfy the curvature bounds (61) and have initial data converging locally in C^{k} to the immersion f̂_{0} : Σ → ℝ^{n}. By standard estimates for geometric evolution equations, see (4.24)–(4.28) in [5], we deduce the locally smooth convergence

f_{j}(φ_{j}, ·) → f̂, (64)

where f̂: Σ̂ × [0, c_{0}] → ℝ^{n} is a Willmore flow with initial data f_{0}. But on the other hand we have

∫_{0}^{c_{0}} ∫_{Σ̂(j)} |W(f_{j}(φ_{j}, t))|^{2} dμ_{f_{j}(φ_{j}, ·)}dt
= ∫_{0}^{c_{0}} ∫_{U_{j}} |W(f_{j}( · , t))|^{2} dμ_{j}dt
≤ ∫_{Σ} |A_{j}(c_{0})|^{2} dμ_{j} − ∫_{Σ} |A_{j}(0)|^{2} dμ
= ∫_{Σ} |A(t_{j} + r_{j}^{4} c_{0})|^{2} dμ − ∫_{Σ} |A(t_{j})|^{2} dμ,

which converges to zero as j → ∞. This implies that W(f̂) ≡ 0 which means that f̂( · , t) ≡ f̂_{0} is an immersed Willmore surface, which is independent of time. Furthermore (60) implies, because of the smooth convergence in (64),

∫_{f̂^{−1}(\overline{B_{1}(0)})} |Â|^{2} dμ̂ ≥ ε_{1} > 0. (65)

Thus f̂ is not a union of planes.

Lemma 4.3. Let f̂ : Σ̂ ⟶ ℝ^{n} be the blowup constructed above. If Σ̂ contains a compact component C, then in fact Σ̂ = C and Σ is diffeomorphic to C.

Proof. For j sufficiently large, φ_{j}(C) is open and closed in Σ. Hence by connectedness of Σ we have Σ = φ_{j}(C) and thus Σ̂ = C. q.e.d.

Theorem 4.4 (Nontriviality of the Blowup). Let f̂ : Σ̂ → ℝ^{n} be the blowup of a Willmore flow as constructed above. Then none of the components of f̂ parametrizes a round sphere. In particular, the blowup has a component which is a nonumbilic (compact or noncompact) Willmore surface.

Proof. Otherwise, Lemma 4.3 implies that the blowup surface f̂: Σ̂ → ℝ^{n} is an embedded round sphere, i.e., has no further components. It follows that, up to the diffeomorphism φ_{j} : Σ̂ → Σ, the map f_{j}( · , 0) is C^{k}-close to a round sphere and therefore

∫_{Σ} |A^{∘}(t_{j})|^{2} dμ = ∫_{Σ} |A_{j}^{∘}(0)|^{2}dμ_{j} → 0,
μ(t_{j})(Σ) = r_{j}^{2} μ_{j}(0)(Σ) → 0.

This contradicts the lower area bound which will be proved in Theorem 5.2. q.e.d.

5. Small initial energy

In this section we finally prove our main result:

Theorem 5.1 (Global Existence and Convergence for Small Initial Energy). There exists an ε_{0} = ε_{0}(n) > 0 such that if at time t = 0 there holds

W(f_{0}) = ∫_{Σ} |A^{∘}|^{2} dμ < ε_{0},

then the Willmore flow exists smoothly for all times and converges exponentially to a round sphere as t → ∞.

We split the proof into several steps. The first step was already used in Theorem 4.4 and is of independent interest.

Theorem 5.2 (Area Estimate). Let f : Σ × [0, T) → ℝ^{n} be a Willmore flow with W(f) = ∫_{Σ} |A^{∘}|^{2} dμ ≤ ε < ε_{1}, where ε_{1} = ε_{1}(n) is as in Proposition 3.4. Then

(66)  (1 − cε) μ_{0}(Σ) ≤ μ(Σ) ≤ (1 + cε) μ_{0}(Σ)

(67)  ∫_{0}^{t} ∫_{Σ}(|∇A|^{2} + |A|^{2} |A^{∘}|^{2}) dμ ds ≤ cε μ_{0}(Σ).

Proof. We have

\frac{d}{dt} μ(Σ) = − ∫_{Σ} |∇H|^{2} dμ + ∫_{Σ}⟨Q(A^{∘})H, H⟩dμ.

Multiplying Simons’ identity (10) by A^{∘} and integrating yields (cf. Lemma 2.2):

(68)  2 ∫_{Σ}|∇A^{∘}|^{2} dμ + ∫_{Σ} |H|^{2} |A^{∘}|^{2} dμ
= ∫_{Σ} |∇H|^{2} dμ + ∫_{Σ} A^{∘} ∗ A^{∘} ∗ A^{∘} ∗ A^{∘} dμ.

As ⟨Q(A^{∘})H, H⟩ ≤ |A^{∘}|^{2} |H|^{2} by (14), we obtain

\frac{d}{dt} μ(Σ) + 2 ∫_{Σ} |∇A^{∘}|^{2}dμ ≤ c ∫_{Σ} |A^{∘}|^{4} dμ
≤ c ∥A^{∘}∥_{L^{∞}}^{4} μ(Σ).

From (48) with ϱ = ∞ we have

∫_{0}^{t} ∥A^{∘}∥_{L^{∞}}^{4} ds ≤ c ε,

and the Gronwall inequality yields

μ(Σ) + 2 ∫_{0}^{t} ∫_{Σ} |∇A^{∘}|^{2} dμ ds ≤ (1 + c ε) μ_{0}(Σ).

On the other hand, by Michael-Simon

∫ |∇H|^{2} dμ ≤ c (∫ (|∇^{2}H| + |H||∇H|) dμ)^{2}
≤ c μ(Σ) ∫_{Σ} (|∇^{2}H|^{2} + |H|^{2}|∇H|^{2}) dμ.

As ⟨Q(A^{∘})H, H⟩ ≥ 0 we obtain

\frac{d}{dt} μ(Σ) ≥ −c μ(Σ) ∫_{Σ}(|∇^{2}H|^{2} + |H|^{2} |∇H|^{2}) dμ.

Using (47) with ϱ = ∞ implies the remaining inequality in (66). In particular we obtain

∫_{0}^{t} ∫_{Σ} |∇A^{∘}|^{2} dμ ds ≤ c ε μ_{0}(Σ),

and

∫_{0}^{t} ∫_{Σ} |A^{∘}|^{4} dμ ds ≤ (1 + cε) μ_{0}(Σ) ∫_{0}^{t} ∥A^{∘}∥_{L^{∞}}^{4}ds
≤ c ε μ_{0}(Σ).

Finally from (68) and Codazzi (5)

∫_{0}^{t} ∫_{Σ} |H|^{2} |A^{∘}|^{2} dμ ds ≤ c ∫_{0}^{t} ∫_{Σ}(|∇A^{∘}|^{2} + |A^{∘}|^{4})dμ ds
≤ c ε μ_{0}(Σ).

This proves the theorem. q.e.d.

Remark. The extrinsic diameter is bounded above and below by the diameter of the initial surface, cf. [8].

Lemma 5.3. Under the assumptions of Theorem 5.1 there exists a radius r_{0} > 0 such that

∫_{Σ_{r_{0}}(x)} |A(t)|^{2} dμ ≤ ε_{1} for all x ∈ ℝ^{n}, t ∈ [0, ∞),

where ε_{1} > 0 is as in (58).

Proof. Otherwise, the blowup construction of Section 4 yields an immersed Willmore surface f̂: Σ̂ → ℝ^{n} with

∫_{f̂^{−1}(\overline{B_{1}(0)})} |Â|^{2} dμ̂ ≥ ε_{1},

whereas

∫Σ̂ |Â^{∘}|^{2} dμ̂ < ε_{0}.

By Theorem 2.7 the surface f̂ must be a union of embedded planes and round spheres, which however contradicts the nontriviality of the blowup, Theorem 4.4. q.e.d.

Lemma 5.4. For any sequence t_{j} → ∞ there exist x_{j} ∈ ℝ^{n} and φ_{j} ∈ Diff (Σ) such that, after passing to a subsequence, the immersions f(φ_{j}, t_{j}) − x_{j} converge smoothly to an embedded round sphere.

Proof. Let x_{j} = f(p, t_{j}) where p ∈ Σ is arbitrary. By the previous lemma and the interior curvature estimates from Theorem 3.5, we have for t_{j} ≥ 1

∥∇^{k}A( ·, t_{j})∥_{L^{∞}} ≤ c(k). (69)

Furthermore, Lemma 4.1 yields the area bound

\frac{μ(t_{j}) (B_{R}(x_{j}))}{R^{2}} ≤ c(W(f( · , t_{j})) + 4πχ(Σ)).

According to Theorem 4.2, there exist a properly immersed surface f̂: Σ̂ ⟶ ℝ^{n} and diffeomorphisms φ_{j} : Σ̂(j) ⟶ U_{j} ⊂ Σ, such that (after selection of a subsequence)

f(φ_{j}, t_{j}) − x_{j} ⟶ f̂

locally in C^{k} on Σ̂. On Σ̂(j) we consider the Willmore flows

g_{j}(p, t) = f(φ_{j}(p), t_{j} + t) − x_{j} (t ≥ −t_{j}).

These satisfy the curvature estimates (69), and the initial data (at t = 0) converge to f̂. Arguing as in (64), we obtain that g_{j} converges locally smoothly on Σ̂ × [0, ∞) to a Willmore flow g : Σ̂ × [0, ∞) → ℝ^{n} with initial data f̂. But now

∫_{0}^{1} ∫_{Σ̂(j)} |W(g_{j})|^{2} dμ_{g_{j}}dt ≤ ∫_{t_{j}}^{t_{j}+1} ∫_{Σ} |W(f)|^{2}dμ dt → 0 as j → ∞.

Therefore we have W(g) ≡ 0 which proves that f̂ is a Willmore surface, and Theorem 2.7 implies that f̂ is a union of embedded planes and round spheres. Using the upper area bound in (66) and excluding several components as in Lemma 4.3 we conclude that f̂must be a round sphere, and that the subconvergence is smooth. q.e.d.

As a consequence of the above, we obtain that

W(f) = ∫_{Σ} |A^{∘}|^{2} dμ → 0 as t → ∞. (70)

Moreover, Theorem 5.2 implies the existence of the nonzero limit

ω = lim_{t→∞} μ(t)(Σ) ∈ (0, ∞). (71)

Finally we now prove exponential decay of the curvature; from this one obtains smooth convergence of f to a round sphere f̂ and thus Theorem 5.1 in a standard way.

Lemma 5.5. As t ↗ ∞, the following asymptotic statements hold, where λ > 0 is a constant:

(72)  ∥∇^{k}A(t)∥_{L^{∞}} ≤ c_{k} e^{−λt} for k ≥ 1,
(73)  ∥A^{∘}∥_{L^{∞}} ≤ c_{0} e^{−λt}.

Proof. For ω as in (71), the previous Lemma implies that the sectional curvature and the mean curvature of f( · , t) satisfy

∥K( ·, t) − \frac{4π}{ω} ∥_{L^{∞}} ⟶ 0,

∥ |H|^{2} − \frac{16π}{ω} ∥_{L^{∞}} ⟶ 0 as t → ∞.

In particular, we may assume after a fixed time translation that

|H|^{2} ≥ c > 0 for all t ≥ 0.

By Lemma 3.3, we then have for all t

∂_{t} ∫_{Σ} |A^{∘}|^{2}dμ + c ∫_{Σ}(|∇^{2}A|^{2} + |∇A|^{2} + |A^{∘}|^{2})dμ ≤ 0,

which implies

∫_{Σ} |A^{∘}|^{2}dμ + ∫_{t}^{∞} ∫_{Σ}(|∇^{2}A|^{2} + |∇A|^{2})dμ ds ≤ c e^{−2λt}, (74)

for a constant λ = λ(n) > 0. From here we easily derive exponential decay of all derivatives of A. Namely, letting ϱ → ∞ in Corollary 3.4 of [5], we have for ϕ = ∇^{m}A (m ≥ 1)

\frac{d}{dt} ∫ |ϕ|^{2}dμ + \frac{3}{4} ∫ |∇^{2}ϕ|^{2}dμ ≤ ∫ (P_{3}^{m+2}(A) + P_{5}^{m}(A)) ∗ ϕ dμ.

Using that A and all its derivatives remain bounded as t → ∞, we can estimate

∫ P_{2}^{0}(A) ∗ ∇^{m+2}A ∗ ϕ dμ ≤ ε ∫ |∇^{2}ϕ|^{2}dμ + c(ε) ∫ |∇^{m}A|^{2}dμ,
∫ (P̂_{3}^{m+2}(A) + P_{5}^{m}(A)) ∗ ϕdμ ≤ c ∑_{j=1}^{m+1} ∫ |∇^{j}A|^{2}dμ.

Here P̂_{3}^{m+2}(A) denotes all terms of type P_{3}^{m+2}(A) that do not contain the (m + 2)-th derivative, and of course c is not a universal constant here. We obtain

\frac{d}{dt} ∫ |ϕ|^{2}dμ + \frac{1}{2} ∫ |∇^{2}ϕ|^{2}dμ ≤ c ∑_{j=1}^{m+1} ∫ |∇^{j}A|^{2}dμ,

and now by induction using (74)

∥∇^{m}A∥_{L^{2}}^{2} + ∫_{t}^{∞} ∥∇^{m+2}A∥_{L^{2}}^{2} ds ≤ c e^{−2λt}.

By a Sobolev inequality, e.g., the Michael-Simon inequality, we deduce

(75)  ∥A^{∘}∥_{L^{∞}} ≤ c_{0} e^{−λt},
(76)  ∥∇^{k}A∥_{L^{∞}} ≤ c_{k} e^{−λt},

which ends the proof of Lemma 5.5 and of Theorem 5.1. q.e.d.

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Albert-Ludwigs-Universität Freiburg Universität Bonn