| A Geometric Problem and the Hopf Lemma. Ⅱ |
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| 作者姓名： | YanYan LI Louis NIRENBERG |
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| 摘 要： |
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| 关 键 词： | 引理 最大值原理 移动面 对称性 均值曲率 |
| 收稿时间： | 29 January 2006 |
| 修稿时间： | 2/6/2004 12:00:00 AM |
A Geometric Problem and the Hopf Lemma. II |
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| YanYan LI,Louis NIRENBERG.A Geometric Problem and the Hopf Lemma. II[J].Chinese Annals of Mathematics,Series B,2006,27(2):193-218. |
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| Authors: | YanYan LI and Louis NIRENBERG |
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| Institution: | (1) Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA;(2) Courant Institute, 251 Mercer Street, New York, NY 10012, USA |
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| Abstract: | Abstract A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1 =constant in case M satisfies: for any two points (X′,Xn+1),
on M, with
, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper,
we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems
are described. Part I dealt with corresponding one dimensional problems.
(Dedicated to the memory of Shiing-Shen Chern)
* Partially supported by NSF grant DMS-0401118. |
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| Keywords: | Hopf Lemma Maximum principle Moving planes Symmetry Mean curvature |
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