Vol. 306, No. 1, 2020

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Surface diffusion flow of arbitrary codimension in space forms

Dong Pu and Hongwei Xu

Vol. 306 (2020), No. 1, 291–320
Abstract

We first prove that a properly immersed static n-dimensional submanifold (ΔH = 0) with restricted growth of the curvature at infinity in 𝔽n+p(c)(c 0) is totally umbilical if the Willmore functional is pinched by a positive constant depending only on n. Secondly, we obtain a global rigidity theorem for Willmore surfaces in the sphere. Thirdly, we give a lower bound on the lifespan of the surface diffusion flow in 𝔽2+p(c). Finally, we get the longtime existence and convergence of the surface diffusion flow in 𝔽2+p(c)(c 0) under the small initial Willmore energy condition.

Dedicated to Professor Duanzhuang Qian on his 110th anniversary.

Keywords
surface diffusion, Willmore surface, gap theorem, convergence theorem.
Mathematical Subject Classification 2010
Primary: 53C24, 53C44
Milestones
Received: 10 April 2019
Revised: 11 December 2019
Accepted: 20 December 2019
Published: 14 June 2020
Authors
Dong Pu
Center of Mathematical Sciences
Zhejiang University
Hangzhou
China
Hongwei Xu
Center of Mathematical Sciences
Zhejiang University
Hangzhou
China