TY - JOUR
T1 - A geometric problem and the hopf lemma. II
AU - Li, Yan Yan
AU - Nirenberg, Louis
N1 - Funding Information:
Manuscript received January 29, 2006. Revised February 4, 2006. ∗Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA. E-mail: [email protected] ∗∗Courant Institute, 251 Mercer Street, New York, NY 10012, USA. E-mail: [email protected] ∗∗∗Partially supported by NSF grant DMS-0401118.
PY - 2006/4
Y1 - 2006/4
N2 - A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in 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′,X̂ n+1), on M, with X n+1 > 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.
AB - A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in 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′,X̂ n+1), on M, with X n+1 > 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.
KW - Hopf Lemma
KW - Maximum principle
KW - Mean curvature
KW - Moving planes
KW - Symmetry
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U2 - 10.1007/s11401-006-0037-3
DO - 10.1007/s11401-006-0037-3
M3 - Article
AN - SCOPUS:33745677002
SN - 0252-9599
VL - 27
SP - 193
EP - 218
JO - Chinese Annals of Mathematics. Series B
JF - Chinese Annals of Mathematics. Series B
IS - 2
ER -