Mean Curvature Rigidity and Non-rigidity Results on Spherical Caps

TL;DR

This paper investigates the rigidity and non-rigidity of spherical caps under mean curvature constraints, extending known results and demonstrating both unique rigidity and possible perturbations.

Contribution

It extends the mean curvature rigidity phenomenon to spherical caps and shows the existence of nontrivial perturbations, revealing a contrast between rigidity and flexibility.

Findings

Hemisphere as a graph admits no smooth perturbations with H≥1 fixing boundary.

Existence of nontrivial perturbations with H≥1 on spherical caps.

Rigidity and non-rigidity phenomena also occur in 1D and discrete cases.

Abstract

We prove that a hemisphere in the Euclidean space $R^{n+1}$, viewed as the graph of a function, admits no smooth perturbations as graphs with mean curvature $H\ge 1$ whose boundary equator is fixed up to $C^2$. This is an extension of the \emph{Mean Curvature Rigidity} phenomenon discovered by Gromov and Souam on non-compact totally umbilic hypersurfaces in space forms. The proof uses a Tangency Principle. On the other hand, we show that there exist nontrivial smooth perturbations with $H\ge 1$ on a great spherical cap whose boundary is fixed up to $C^2$. Similar results hold true for perturbations decreasing $H$, and for the $r$ mean curvature function $H_r$. This contrast between rigidity and non-rigidity is even true in the 1-dimensional case for circles and for discrete objects (polygons inscribed in a circle).