JournalsjemsVol. 8, No. 2pp. 317–339

A geometric problem and the Hopf Lemma. I

  • YanYan Li

    Rutgers University, Piscataway, United States
  • Louis Nirenberg

    New York University, United States
A geometric problem and the Hopf Lemma. I cover
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Abstract

A classical result of A.~D. Alexandrov states that a connected compact smooth nn-dimen\-sional manifold without boundary, embedded in Rn+1{\mathbb R}^{n+1}, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of MM in a hyperplane Xn+1=constX_{n+1}={\rm const} in case MM satisfies: for any two points (X,Xn+1)(X', X_{n+1}), (X,X^n+1)(X', \widehat X_{n+1}) on MM, with Xn+1>X^n+1X_{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 condition for n=1n=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.

Cite this article

YanYan Li, Louis Nirenberg, A geometric problem and the Hopf Lemma. I. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 317–339

DOI 10.4171/JEMS/55