Symmetry of hypersurfaces with ordered mean curvature in one direction

TL;DR

This paper proves that hypersurfaces with ordered mean curvature in one direction are symmetric, using a variational approach, thus confirming a conjecture by Li and Nirenberg under weaker conditions.

Contribution

It introduces a variational method to establish symmetry of hypersurfaces with ordered mean curvature, relaxing previous assumptions and confirming a conjecture.

Findings

Hypersurfaces with ordered mean curvature are symmetric about a hyperplane.

The variational approach replaces the moving plane method.

The result confirms Li and Nirenberg's conjecture under weaker assumptions.

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 $(x',a), (x',b)\in M$ with $a<b$ has ordered mean curvature $H(x',b)\leq H(x',a)$, 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.