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Abstract
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We first prove that a properly immersed static
-dimensional
submanifold
with restricted growth of the curvature at infinity in
is totally
umbilical if the Willmore functional is pinched by a positive constant depending only
on
.
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
.
Finally, we get the longtime existence and convergence of the surface diffusion flow in
under
the small initial Willmore energy condition.
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Dedicated to Professor Duanzhuang Qian
on his 110th anniversary.
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Keywords
surface diffusion, Willmore surface, gap theorem,
convergence theorem.
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Mathematical Subject Classification 2010
Primary: 53C24, 53C44
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Milestones
Received: 10 April 2019
Revised: 11 December 2019
Accepted: 20 December 2019
Published: 14 June 2020
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