Half-space theorems for $1$-surfaces of $\mathbb{H}^3$

TL;DR

This paper establishes new half-space theorems and intersection results for 1-surfaces in hyperbolic space and manifolds with Ricci curvature bounds, advancing understanding of their geometric properties and properness criteria.

Contribution

It introduces novel half-space theorems for 1-surfaces in hyperbolic space and Ricci-bounded manifolds, including criteria for properness and splitting results under distance conditions.

Findings

Proved a Frankel's type theorem for 1-surfaces with bounded curvature in Ricci > -2 manifolds.

Established strong half-space theorems for various classes of 1-surfaces in .

Derived a Maximum Principle at Infinity for 1-surfaces in .

Abstract

In this paper we investigate the intersection problem for $1$-surfaces immersed in a complete Riemannian three-manifold $P$ with Ricci curvature bounded from below by $-2$. We first prove a Frankel's type theorem for $1$-surfaces with bounded curvature immersed in $P$ when $\text{\rm Ric}_{P} > -2$. In this setting we also give a criterion for deciding whether a complete $1$-surface is proper. A splitting result is established when the distance between the $1$-surfaces is realized, even if $\text{\rm Ric}_{P} \geq -2$. In the hyperbolic space $\mathbb{H}^3$ we show strong half-space theorems for the classes of complete $1$-surfaces with bounded curvature, parabolic $1$-surfaces, and stochastically complete $H$-surfaces with $H<1$. As a by-product of our techniques a Maximum Principle at Infinity is given for $1$-surfaces in $\mathbb{H}^3.$