Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold $M^{n}\times\mathbb{R}$

TL;DR

This paper proves the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary conditions in a product manifold, extending understanding of geometric flows in curved spaces.

Contribution

It establishes the existence of solutions for mean curvature flow with specific boundary conditions in a product manifold setting, a novel extension in geometric analysis.

Findings

Existence of translating solutions proven

Applicable to manifolds with nonnegative Ricci curvature

Extends previous results to nonzero boundary data

Abstract

In this paper, we can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold $M^{n}\times\mathbb{R}$, where $M^{n}$ is an $n$-dimensional ($n\geq2$) complete Riemannian manifold with nonnegative Ricci curvature, and $\mathbb{R}$ is the Euclidean $1$-space.