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| A Geometric Problem and the Hopf Lemma. II |
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Citation： |
YanYan LI,Louis NIRENBERG.A Geometric Problem and the Hopf Lemma. II[J].Chinese Annals of Mathematics B,2006,27(2):193~218 |
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Net amount： 881 |
Authors： |
YanYan LI; Louis NIRENBERG |
Foundation： |
Partially supported by NSF grant DMS-0401118. |
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| Abstract： |
A classical result of A. D. Alexandrov states that a connected
compact smooth $n$-dimensional manifold without boundary, embedded
in $\mathbb R^{n+1}$, and such that its mean curvature is
constant, is a sphere. Here we study the problem of symmetry of
$M$ in a hyperplane $X_{n+1}=$constant in case $M$ satisfies: for
any two points $(X', X_{n+1})$, $(X', \widehat X_{n+1})$ on $M$,
with $X_{n+1}>\widehat X_{n+1}$, 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. |
Keywords： |
Hopf Lemma, Maximum principle, Moving planes, Symmetry, Mean
curvature |
Classification： |
35J60, 53A05 |
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