The convergence of boundary expansions and the analyticity of minimal surfaces in the hyperbolic space

TL;DR

This paper proves that minimal surfaces in hyperbolic space are analytic up to the boundary when the boundary data is analytic, by establishing convergence of boundary expansions for solutions to the Dirichlet problem.

Contribution

It establishes the local convergence of boundary expansions for minimal graphs in hyperbolic space and confirms a conjecture that such surfaces are analytic up to the boundary under analytic boundary conditions.

Findings

Boundary expansions converge locally for solutions with analytic boundary data.

Minimal graphs are analytic up to the boundary if the boundary is analytic.

Confirmed F.-H. Lin's conjecture on boundary analyticity of minimal surfaces.

Abstract

We study expansions near the boundary of solutions to the Dirichlet problem for minimal graphs in the hyperbolic space and prove the local convergence of such expansions if the boundary is locally analytic. As a consequence, we prove a conjecture by F.-H. Lin that the minimal graph is analytic up to the boundary if the boundary is analytic and the minimal graph is smooth up to the boundary.