Foliation by free boundary constant mean curvature leaves

TL;DR

This paper proves the existence of a smooth foliation near a boundary point in a Riemannian manifold, where leaves are constant mean curvature submanifolds meeting the boundary perpendicularly, under a nondegeneracy condition.

Contribution

It establishes a new local foliation result for constant mean curvature leaves near boundary points with nondegenerate mean curvature criticality.

Findings

Existence of a smooth foliation near boundary points with prescribed properties.

Leaves are of constant mean curvature and meet the boundary orthogonally.

The result applies under a nondegeneracy condition on the mean curvature function.

Abstract

Let $M$ be a Riemannian manifold of dimension $n+1$ with smooth boundary and $p\in \partial M$. We prove that there exists a smooth foliation around $p$ whose leaves are submanifolds of dimension $n$, constant mean curvature and its arrive perpendicular to the boundary of M, provided that $p$ is a nondegenerate critical point of the mean curvature function of $\partial M$.