The half-space property and entire positive minimal graphs in M x R

TL;DR

This paper proves that under certain curvature conditions, the only positive entire minimal graphs in M x R are horizontal slices, extending classical half-space theorems to more general Riemannian manifolds.

Contribution

It establishes new half-space properties for minimal hypersurfaces in product manifolds with specific curvature bounds, generalizing previous results.

Findings

Properly immersed minimal hypersurfaces in M x R_+ are horizontal slices when M is recurrent with bounded curvature.

Positive entire minimal graphs over M are horizontal slices if M has nonnegative Ricci curvature with bounded below curvature.

The results extend classical half-space theorems to broader classes of Riemannian manifolds.

Abstract

We show that a properly immersed minimal hypersurface in M x R_+ equals some M x {c} when M is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. If on the other hand, M has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over M.