Symmetry of hypersurfaces with ordered mean curvature in one direction

Abstract

For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in ${\mathbb{R}}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a one-directional analog of this result: if every pair of points $(x^{\prime },a),(x^{\prime },b)\in M$ with $a<b$ has ordered mean curvature $H(x^{\prime },b)\le H(x^{\prime },a)$, then $M$ is symmetric about some hyperplane $x_{n+1}=c$ under some additional conditions. Their proof was done by the moving plane method and some variations of the Hopf Lemma. We obtain the symmetry of $M$ under some weaker assumptions using a variational argument, giving a positive answer to the conjecture given by Li and Nirenberg.

Additional references added. To appear in Calc. Var. Partial Differential Equations