Stochastic half-space theorems for minimal surfaces and $H$-surfaces of ${\mathbb{R}}^3$

Abstract

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $M\times {\mathbb{R}}_{+}$ equals some $M\times \{s\}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with ${\text{Ric}}_M\ge 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in ${\mathbb{R}}^3$ with bounded curvature if $\sup |H_{{}_M}|<H$, or $\mathrm{dist}(M,N)=0$ when $\sup |H_{{}_M}|=H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.

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• Bernstein and half-space properties for minimal graphs under Ricci lower bounds
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