Low index capillary minimal surfaces in Riemannian $3$-manifolds

TL;DR

This paper establishes rigidity and geometric bounds for low index capillary minimal surfaces in certain Riemannian 3-manifolds, advancing understanding of their structure and limitations.

Contribution

It provides new local rigidity results and bounds on genus, boundary components, and area for low index capillary minimal surfaces in specific Riemannian 3-manifolds.

Findings

Rigidity result for infinitesimally rigid capillary surfaces.

Bounds on genus, boundary components, and area for low index surfaces.

Conditions on ambient curvature affecting surface properties.

Abstract

We prove a local rigidity result for infinitesimally rigid capillary surfaces in some Riemannian $3$-manifolds with mean convex boundary. We also derive bounds on the genus, number of boundary components and area of any compact two-sided capillary minimal surface with low index under certain assumptions on the curvature of the ambient manifold and of its boundary.