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

TL;DR

This paper establishes stochastic half-space theorems for minimal and $H$-surfaces in $ ^3$, revealing new geometric constraints and maximum principles for these surfaces under bounded curvature and recurrence conditions.

Contribution

It introduces stochastic half-space theorems for minimal and $H$-surfaces, extending classical results to broader classes with bounded curvature and recurrence assumptions.

Findings

Recurrent minimal surfaces cannot be contained in certain half-spaces.

Complete minimal hypersurfaces with bounded curvature in $M\times \R_+$ are flat slices.

Stochastic completeness prevents $H$-surfaces from lying on the mean convex side of certain other $H$-surfaces.

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 $M\times \R_+$ equals some $M\times \{s\}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $\text{Ric}_M \geq 0$ 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 $\R^3$ with bounded curvature if $\sup \vert H_{_M}\vert < H$, or ${\rm dist}(M,N)=0$ when $\sup \vert H_{_M}\vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.