Dirichlet problem for $f$-minimal graphs

TL;DR

This paper investigates the existence and asymptotic behavior of $f$-minimal graphs in Cartan-Hadamard manifolds, extending the understanding of these hypersurfaces and their boundary conditions under curvature and decay assumptions.

Contribution

It proves existence results for $f$-minimal graphs with prescribed boundary data and constructs solutions to the asymptotic Dirichlet problem under decay conditions on $f$.

Findings

Existence of $f$-minimal graphs with prescribed boundary behavior.

Construction of solutions to the asymptotic Dirichlet problem under decay conditions.

Results are nearly optimal given the curvature and decay assumptions.

Abstract

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with prescribed boundary behavior on a bounded domain $\Omega \subset M$ under suitable assumptions on $f$ and the boundary of $\Omega$. In the second part, we consider the asymptotic Dirichlet problem. Provided that $f$ decays fast enough, we construct solutions to the problem. Our assumption on the decay of $f$ is linked with the sectional curvatures of $M$. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.