Geodesic
A geometric problem and the Hopf Lemma. I
Journal of the European Mathematical Society, Tome 8 (2006) no. 2, pp. 317-339.

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A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in Rn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1​=const in case M satisfies: for any two points (X′,Xn+1​), (X′,Xn+1​) on M, with Xn+1​>Xn+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.
DOI : 10.4171/jems/55
Classification : 35-XX, 00-XX 
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YanYan Li; Louis Nirenberg. A geometric problem and the Hopf Lemma. I. Journal of the European Mathematical Society, Tome 8 (2006) no. 2, pp. 317-339. doi : 10.4171/jems/55. http://geodesic.mathdoc.fr/articles/10.4171/jems/55/

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