The mean curvature equation on semidirect products ${\mathbb{R}}^2{⋊}_A\mathbb{R}$: 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 ${\mathbb{R}}^2{⋊}_A\mathbb{R}$, we consider a domain $\Omega \subseteq {\mathbb{R}}^2{⋊}_A\{0\}$ and vertical $\pi$-graphs over $\Omega$ and study the partial differential equation a function $u:\Omega \to \mathbb{R}$ 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 $n\to +\infty$ and goes to $+\infty$ if $n\to -\infty$. 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 $\gamma \cup (\{p_1\}\times [0,+\infty ))\cup (\{p_2\}\times [0,+\infty ))$.

arXiv comment28 pages, 5 figures

References in corpus (1)

• Periodic minimal surfaces in semidirect products