Non-parametric mean curvature flow with prescribed contact angle in Riemannian products

TL;DR

This paper proves convergence of solutions to non-parametric mean curvature flow with prescribed contact angles in Riemannian products, extending existing results to non-Euclidean geometries under curvature bounds.

Contribution

It establishes convergence to a translating soliton in Riemannian products and generalizes recent Euclidean results to non-Euclidean settings with curvature conditions.

Findings

Graphical solutions converge to a translating soliton plus linear term as time approaches infinity.

Generalization of existence results to non-Euclidean geometries with curvature bounds.

Provides conditions under which the mean curvature flow stabilizes in Riemannian product spaces.

Abstract

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $\Omega$ and Ricci curvature in $\Omega$.