A Mobius rigidity of compact Willmore hypersurfaces in Sn+1
A Mobius rigidity of compact Willmore hypersurfaces in Sn+1
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Sn 1 中紧致 Willmore 超曲面的莫比乌斯刚度
DOI：
10.1016/j.jmaa.2019.123707
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发表时间：
2020
影响因子：
1.3
通讯作者：
Li Tongzhu
中科院分区：
文献类型：
--
作者：
Lin Limiao;Li Tongzhu
令x：Mn → Sn + 1是一个无脐点的浸入超曲面，则可以定义超曲面Mn上的Möbius度量g、Möbius第二基本形式B和Blaschke张量A，它们在Sn + 1的Möbius变换群下是不变的.一个超曲面称为Willmore超曲面，如果它是Mn的体积泛函关于莫比乌斯度量g的临界点。本文证明了：若超曲面x是无脐点的紧致Willmore超曲面，则（<$A <$+ B <$2 − n− 1 n）d M g≥ 0，等式成立当且仅当超曲面M n是与Willmore环面W n之一等价的莫比乌斯，m= S m（n− m n）× S n− m（m n）<$S n+ 1，1≤ m≤ n− 1，其中张量A = A+ B 2− 1 n [tr（A）+ n− 1 n] g。
Let x: M n→ S n+ 1 be an immersed hypersurface without umbilical point, then one can define the Möbius metric g, the Möbius second fundamental form B and the Blaschke tensor A on the hypersurface M n which are invariant under the Möbius transformation group of S n+ 1. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of M n with respect to the Möbius metric g. In this paper, we prove that if the hypersurface x is a compact Willmore hypersurface without umbilical point, then∫ M n (‖ A˜+ B‖ 2− n− 1 n) d M g≥ 0, the equality holds if and only if the hypersurface M n is Möbius equivalent to one of the Willmore tori W n, m= S m (n− m n)× S n− m (m n)↪ S n+ 1, 1≤ m≤ n− 1, where the tensor A˜= A+ B 2− 1 n [t r (A)+ n− 1 n] g.