Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field

TL;DR

This paper studies minimal graphs over non-compact domains in 3-manifolds with a Killing vector field, establishing existence, uniqueness, and regularity results, including in the Heisenberg group setting.

Contribution

It provides the first solutions to the Dirichlet problem for minimal Killing graphs over unbounded domains in such manifolds, with new estimates and singularity removability results.

Findings

Existence of solutions to the Dirichlet problem for minimal Killing graphs.

Uniqueness of solutions in the Heisenberg group for bounded boundary data.

Removability of isolated singularities of Killing graphs.

Abstract

Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\E$. Then, there exists a Killing Submersion from $\E$ onto a connected and orientable surface $M$ whose fibers are the integral curves of the Killing vector field. In this setting, assuming that $M$ is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of $M$, prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable.