Symmetry of hypersurfaces with symmetric boundary

TL;DR

This paper proves that minimal and constant mean curvature hypersurfaces with symmetric boundaries inherit the symmetry of their boundary in Euclidean space, using Lie group actions and regularity theory.

Contribution

It establishes symmetry inheritance results for various classes of hypersurfaces with symmetric boundaries, extending previous symmetry results to higher order mean curvature and Helfrich-type hypersurfaces.

Findings

Interior symmetry of minimal hypersurfaces with symmetric boundary

Symmetry inheritance for hypersurfaces of higher order mean curvature

Application of Lie group actions and regularity theory in symmetry proofs

Abstract

Let $G$ be a compact connected subgroup of $SO(n+1)$. In $\mathbb{R}^{n+1}$, we gain interior $G$-symmetry for minimal hypersurfaces and hypersurfaces of constant mean curvature (CMC) which have $G$-invariant boundaries and $G$-invariant contact angles along boundaries. The main ingredients of the proof are to build an associated Cauchy problem based on infinitesimal Lie group actions, and to apply Morrey's regularity theory and the Cauchy-Kovalevskaya Theorem. Moreover, we also investigate the same kind of symmetry inheritance from boundaries for hypersurfaces of constant higher order mean curvature and Helfrich-type hypersurfaces in $\mathbb{R}^{n+1}$.