Rigidity of 3D spherical caps via $\mu$-bubbles

TL;DR

This paper demonstrates the rigidity of 3D spherical caps under certain geometric perturbations using Gromov's $$-bubble method, with potential generalizations discussed.

Contribution

It introduces a novel application of Gromov's $$-bubble technique to establish rigidity results for 3D spherical caps.

Findings

3D spherical caps are rigid under specific geometric constraints

The method extends to various generalizations

Provides new insights into geometric rigidity

Abstract

By using Gromov's $\mu$-bubble technique, we show that the $3$-dimensional spherical caps are rigid under perturbations that do not reduce the metric, the scalar curvature, and the mean curvature along its boundary. Several generalizations of this result will be discussed.