Mean convex properly embedded $[\phi ,{\vec{e}}_3]$-minimal surfaces in ${\mathbb{R}}^3$

Abstract

We establish curvature estimates and a convexity result for mean convex properly embedded $[\phi ,{\vec{e}}_3]$-minimal surfaces in ${\mathbb{R}}^3$, i.e., $\phi$-minimal surfaces when $\phi$ depends only on the third coordinate of ${\mathbb{R}}^3$. Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana, for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in ${\mathbb{R}}^3$, we use a compactness argument to provide curvature estimates for a family of mean convex $[\phi ,{\vec{e}}_3]$-minimal surfaces in ${\mathbb{R}}^3$. We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded $[\phi ,{\vec{e}}_3]$-minimal surface in ${\mathbb{R}}^3$ with non positive mean curvature when the growth at infinity of $\phi$ is at most quadratic.

25 pages, 0 figures