Compact minimal vertical graphs with non-connected boundary in $\mathbb{H}^n\times\mathbb{R}$

TL;DR

This paper investigates the existence and uniqueness of compact minimal vertical graphs in hyperbolic space cross the real line, focusing on boundary conditions with non-connected boundaries and establishing nonexistence results for certain configurations.

Contribution

It provides new existence and non-uniqueness results for minimal vertical graphs with complex boundary conditions in hyperbolic product spaces, including nonexistence theorems.

Findings

Nonexistence of certain compact minimal vertical graphs with boundary in two slices.

Existence results under specific boundary data and geometric conditions.

Non-uniqueness of solutions in some boundary configurations.

Abstract

We study the existence and uniqueness problem of compact minimal vertical graphs in $\mathbb{H}^n\times\mathbb{R}$, $n\geq 2$, over bounded domains in the slice $\mathbb{H}^n\times\{0\}$, with non-connected boundary having a finite number of $C^0$ hypersufaces homeomorphic to the sphere $\mathbb{S}^{n-1}$, with prescribed bounded continuous boundary data, under hypotheses relating those data and the geometry of the boundary. We show the nonexistence of compact minimal vertical graphs in $\mathbb{H}^n\times\mathbb{R}$ having the boundary in two slices and the height greater than or equal to $\pi/(2n-2)$.