The mean curvature equation on semidirect products $\mathbb{R}^2\rtimes_A\mathbb{R}$: Height estimates and Scherk-like graphs

TL;DR

This paper investigates height estimates and existence of Scherk-like minimal graphs in semidirect product spaces with a focus on prescribed mean curvature equations, extending classical results to more general Lie group settings.

Contribution

It establishes uniform height estimates for mean curvature graphs in semidirect product spaces and proves existence results for minimal graphs with prescribed boundary conditions.

Findings

Height estimates depend on domain and parameter alpha.

Oscillation of minimal graphs tends to zero or infinity based on boundary values.

Existence of minimal graphs with prescribed boundary curves is demonstrated.

Abstract

On the ambient space of a Lie group with a left invariant metric that is isometric and isomorphic to a semidirect product $\mathbb{R}^2\rtimes_A\mathbb{R}$, we consider a domain $\Omega\subseteq \mathbb{R}^2\rtimes_A\{0\}$ and vertical $\pi$-graphs over $\Omega$ and study the partial differential equation a function $u:\Omega \rightarrow \mathbb{R}$ must satisfy in order to have prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a $\pi-$graph has non-negative mean curvature, then it satisfy some uniform height estimates that depend on $\Omega$ and on a parameter $\alpha$, given a priori. When $\text{trace}(A) > 0$, these estimates imply that the oscillation of a minimal graph assuming the same constant value $n$ along the boundary tends to zero when $n\rightarrow + \infty$ and goes to $+ \infty$ if $n\rightarrow - \infty$. Furthermore, we use some of the estimates, allied with techniques from Killing graphs, to prove the existence of minimal $\pi-$graphs assuming the value $0$ along a piecewise smooth curve $\gamma$ with endpoints $p_1,\,p_2$ and having as boundary $\gamma \cup (\{p_1\}\times[0,\,+\infty))\cup(\{p_2\}\times[0,\,+\infty))$.