Optimal regularity of minimal graphs in the hyperbolic space

TL;DR

This paper establishes the optimal regularity of solutions to the minimal graph equation in hyperbolic space with boundary conditions, showing they are Hölder continuous with an exponent depending on boundary curvature.

Contribution

It proves the optimal Hölder regularity of minimal graph solutions in hyperbolic space under certain boundary curvature conditions, extending previous results.

Findings

Solutions are in C^{1/(n+1)}(ar{A}) for boundaries with nonnegative mean curvature.

The Hölder exponent can be improved if specific principal curvatures of the boundary do not vanish.

The regularity result is optimal under the given geometric conditions.

Abstract

We discuss the global regularity of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space when the boundary of the domain $\Omega\subset\mathbb R^n$ has a nonnegative mean curvature and prove an optimal regularity $f\in C^{\frac{1}{n+1}}(\bar{\Omega})$. We can improve the H\"older exponent for $f$ if certain combinations of principal curvatures of the boundary do not vanish, a phenomenon observed by F.-H. Lin.