A geometric problem and the Hopf Lemma. I

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 $X_{n+1}=\mathrm{const}$ in case $M$ satisfies: for any two points $(X^{\prime },X_{n+1})$, $(X^{\prime },{\hat{X}}_{n+1})$ on $M$, with $X_{n+1}>{\hat{X}}_{n+1}$, 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 condition for $n=1$. Some variations of the Hopf Lemma are also presented. Part II [Y.Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193–218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.