General curvature estimates for stable H-surfaces in 3-manifolds and applications

TL;DR

This paper derives curvature estimates for stable H-surfaces in 3-manifolds with bounded curvature, providing bounds based on boundary distance and ambient geometry, with applications to gradient estimates in Killing submersions.

Contribution

It introduces a new curvature estimate for stable H-surfaces that depends on boundary distance and ambient geometry, not on the manifold itself.

Findings

Curvature bounds depend on boundary distance and ambient geometry.

Application of estimates to interior gradient bounds in Killing submersions.

Establishment of a general framework for curvature estimates in 3-manifolds.

Abstract

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bounds on the geometry of the ambient manifold but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.