A splitting theorem for capillary graphs under Ricci lower bounds

TL;DR

This paper proves a splitting theorem for capillary graphs with constant mean curvature on manifolds with nonnegative Ricci curvature, leading to classification results over certain domains in product spaces.

Contribution

It introduces a new splitting theorem for capillary graphs under Ricci lower bounds and classifies such graphs over specific domains in product manifolds.

Findings

Splitting theorem for capillary graphs on manifolds with nonnegative Ricci curvature.

Classification of capillary graphs over Lipschitz epigraphs and slabs in product spaces.

A new gradient estimate for positive CMC graphs on Ricci-bounded manifolds.

Abstract

In this paper, we study capillary graphs defined on a domain $\Omega$ of a complete Riemannian manifold $M$, where a graph is said to be capillary if it has constant mean curvature and locally constant Dirichlet and Neumann conditions on $\partial \Omega$. Our main result is a splitting theorem both for $\Omega$ and for the graph function on a class of manifolds with nonnegative Ricci curvature. As a corollary, we classify capillary graphs over domains that are globally Lipschitz epigraphs or slabs in a product space $M = N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci curvature, including the case $M = \mathbb{R}^2,\mathbb{R}^3$. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds.