Comparison principle, stochastic completeness and half-space theorems

Abstract

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete submanifolds in Riemannian products, Riemannian submersions and wedges. These estimates are applied for obtaining both horizontal and vertical half-space theorems for submanifolds in H n × R ℓ . Keywords comparison principle• mean curvature • stochastic completeness Mathematics Subject Classification (2010) 53C42 • 53C21 G. P. Bessa partially supported by PRONEX/FUNCAP/CNPq J. H. Lira partially supported by PRONEX/FUNCAP/CNPq A. Medeiros partially supported by CNPq

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

“…It then appears that only within the last decade it was realized by Borbély [Bo11] that one can prove bi-halfspace theorems for minimal 2-surface immersions Σ 2 → R 3 , under the assumption that the Omori-Yau principle (so named after [Om67]- [CY75]) is known to be available on the given Σ 2 . This was also expanded by Bessa, de Lira and Medeiros in [BLM13] where they showed Borbély-style "wedge" theorems for stochastically complete minimal surfaces in Riemannian products (M × N, g M ⊕ g N ), where (N, g N ) is complete without boundary. Seeing as the Huisken-Ilmanen metric, in which self-translaters are the minimal surfaces, is not a Riemannian product 1 nor complete, and our surfaces can have boundaries, we will directly take Borbély's method as our point of departure.…”