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

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 $\gamma \cup ({p_1}\times[0,\,+\infty))\cup({p_2}\times[0,\,+\infty))$.