Local existence and uniqueness of skew mean curvature flow

Chong Song

doi:10.1515/crelle-2021-0023

Published by De Gruyter May 9, 2021

Local existence and uniqueness of skew mean curvature flow

Chong Song

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2021-0023

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Abstract

The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11971400

Funding statement: The work is partially supported by National Natural Science Foundation of China (Grant No. 11971400).

A Appendix

Let F:Σn→ℝm be a compact immersed submanifold. In this appendix, we show that the energy ℰk=vol+∥H∥p2+∥A∥Hk,22 is equivalent to the Sobolev norm of the Gauss map ℰ¯k=∥d⁢ρ∥Wk,22, where the Sobolev norms will be defined later. For a preliminary introduction to the geometry of Grassmannian manifolds and Gauss maps, we refer to [27].

Let D denote the usual connection of the exterior product space Λ:=Λn⁢ℝm, which is induced by the standard derivative on ℝm. Let ∇ denote the Levi-Civita connection of the Grassmannian manifold G:=G⁢(n,m-n) and let Π∈Γ⁢(T*⁢G⊗T*⁢G⊗N⁢G) denote the second fundamental form of G as a submanifold in Λ. We can regard Π as an 1-form Π=Πa⁢d⁢ya on G, where each entry Πa∈Γ⁢(T*⁢G⊗N⁢G) is a linear map from TG to NG.

The Gauss map of F is a map ρ:Σ→G↪Λ. We will still denote the pull-back connections on the pull-back bundles ρ*⁢T⁢Λ⊗(T*⁢Σ)s and ρ*⁢T⁢G⊗(T*⁢Σ)s by D and ∇, respectively. Then applying D on d⁢ρ∈Γ⁢(ρ*⁢T⁢G⊗T*⁢Σ), we have

D⁢d⁢ρ=(D⁢d⁢ρ)⊤+(D⁢d⁢ρ)⊥=∇⁡d⁢ρ+ρ*⁢Π⁢(d⁢ρ),

where ρ*⁢Π=Πa⁢∂i⁡ρa⁢d⁢xi is the pull-back 1-form on Σ. Since we can identify d⁢ρ with the second fundamental form A of the immersion F, we can write the above equality as

D⁢A=∇⁡A+#⁡Π⁢(ρ)A2,

where # denotes linear combinations. Taking once more derivative, we get

D2⁢A=D⁢∇⁡A+D⁢(#⁡Π⁢(ρ)A2)

=(∇2⁡A+#⁡ΠA∇⁡A)+(#⁡D⁢ΠA3+#⁡#⁡ΠD⁢AA)

=∇2⁡A+#⁡ΠA∇⁡A+#⁡(D⁢Π+#⁡ΠΠ)A3.

Inductively, we can derive for any k≥1,

(A.1)Dk⁢A=∇k⁡A+∑J#⁡CJ⁢(Π)∇j1⁡#⁡A⋯∇js⁡A,

or equivalently,

(A.2)Dk⁢d⁢ρ=∇k⁡A+∑J#⁡CJ⁢(Π)∇j1⁡#⁡d⁢ρ⋯∇js⁡d⁢ρ,

where CJ⁢(Π) is a linear combination of Π and its derivatives, and the summation is taken for all indices J=(j1,…,js) with

0≤ji≤k-1 and ∑i(ji+1)=∑iji+s=k+1,

Now for k≥1, define the Sobolev norms

∥A∥Hk,2=(∑l=0k∫|∇l⁡A|2)12,

and

∥d⁢ρ∥Wk,2=(∑l=0k∫|Dl⁢d⁢ρ|2)12.

Theorem A.1.

Let F:Σn→Rm be a compact immersed submanifold, k≥l0:=[n2], and suppose vol+∥H∥p≤B for some p>n. Then there exist two constants C1⁢(k,B) and C2⁢(k,B) which only depend on k and B (and is independent of the submanifold) such that

(A.3)∥A∥Hk,2≤C1⁢(k,B)⁢∑i=1k+1∥d⁢ρ∥Wk,2i

and

(A.4)∥d⁢ρ∥Wk,2≤C2⁢(k,B)⁢∑i=1k+1∥A∥Hk,2i.

Proof.

Since by assumption vol+∥H∥p≤B, we have uniform interpolation inequalities by Theorem 2.4. Then in view of (A.1) and (A.2), the proof follows step by step from [8, Proposition 2.2 ]. ∎

Theorem A.2.

Suppose F:Σn→Rm is a compact immersed submanifold and let k≥l0:=[n2]. Then the energy

ℰk=vol+∥H∥p2+∥A∥Hk,22

is equivalent to

ℰ¯k=∥ρ∥Wk+1,22.

Namely, there exist two functions fk and gk which only depend on k (and is independent of the submanifold) such that

ℰ¯k≤fk⁢(ℰk),ℰk≤gk⁢(ℰ¯k).

Proof.

First note that |ρ|=1 since the image of ρ lies in G which is contained in the unit sphere in Λ. Thus

∫|ρ|2=vol and ∥ρ∥Wk+1,22=vol+∥d⁢ρ∥Wk,22.

If we take B=ℰk in Theorem A.1, then by (A.4),

ℰ¯k=vol+∥d⁢ρ∥Wk,22≤vol+C2⁢(k,B)⁢∑i=1k+1∥A∥Hk,2i.

So it is easy to find the desired function fk such that ℰ¯k≤fk⁢(ℰk).

On the other hand, by letting j=1, k=l0+1, r=∞, q=2 in the universal interpolation inequality of Theorem 2.1, we have

∥A∥p=∥d⁢ρ∥p≤C⁢∥Dl0⁢d⁢ρ∥21l0+1⁢∥ρ∥∞1-1l0+1≤C⁢ℰ¯k1l0+1,

where p=2⁢(l0+1)>n. Therefore, we get

vol+∥H∥p2≤B′:=ℰ¯k+C⁢ℰ¯k2l0+1.

Then by (A.3) in Theorem A.1, there exists a function gk such that ℰk≤gk⁢(ℰ¯k). ∎

Acknowledgements

Part of this work was carried out during a visit at Tsinghua University from September 2017 to February 2018. The author would like to thank Professor Yuxiang Li, Huaiyu Jian and Hongyan Tang for their support and hospitality. He is also grateful to Professor Yu Yuan, Jingyi Chen and Youde Wang for stimulating conversations and their generous help over the past several years.

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Received: 2019-05-01

Revised: 2021-03-13

Published Online: 2021-05-09

Published in Print: 2021-07-01

Local existence and uniqueness of skew mean curvature flow

Abstract

A Appendix

Theorem A.1.

Proof.

Theorem A.2.

Proof.

Acknowledgements

References

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