A Geometric Problem and the Hopf Lemma. II

Abstract

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′,X _n+1), ${\left( {{X}\ifmmode{'}\else$'$\fi,\ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n + 1}} } \right)}$ on M, with $X_{{n + 1}} > \ifmmode\expandafter\hat\else\expandafter\^\fi{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 / 35J60 / 53A05

References

[1] Agmon Comm. Pure Appl. Math., 1963, 16: 121

[2] Alexandrov Vestnik Leninggrad. Univ., 1958, 13: 5

[3] Bao, J. G., Private communication.

[4] Caffarelli, L., Nirenberg, L. and Spruck, J., Nonlinear second order elliptic equations IV. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equations, Y. Ohya, K. Kasahara and N. Shimakura (eds.), Kinokunize Co., Tokyo, 1986, 1-26.

[5] Hartman Arch. Rational Mech. Anal., 1980, 74: 319

[6] Li Contemporary Mathematics, AMS, 1997, 205: 203

[7] Li, Y. Y. and Nirenberg, L., A geometric problem and the Hopf Lemma. I, J. Eur. Math. Soc., to appear.

[8] Nirenberg Comm. Pure Appl. Math., 1963, 16: 331