Gradient Estimates of Mean Curvature Equation with Neumann Boundary Condition in $ M^n\times \mathbb{R}$

TL;DR

This paper derives boundary gradient estimates for solutions to the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifolds, leading to an existence result.

Contribution

It establishes boundary gradient estimates for the mean curvature equation with Neumann conditions on Riemannian products, a novel extension in geometric analysis.

Findings

Boundary gradient estimates are obtained using the maximum principle.

Existence of solutions is proved under the studied conditions.

Results extend previous work to Riemannian product manifolds.

Abstract

In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $ M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.