Height estimates for Killing graphs

TL;DR

This paper establishes global height estimates for Killing graphs over complete manifolds with boundary, utilizing weighted manifold analysis and potential theory to relate geometric properties to height bounds.

Contribution

It introduces a novel approach linking Killing graph geometry to weighted manifold structures and develops potential theory tools for height estimation.

Findings

Weighted volume estimates for intrinsic balls on Killing graphs

Height bounds for graphs with constant weighted mean curvature

Connection between Killing graph geometry and weighted manifold analysis

Abstract

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled.