A modified mean curvature flow in Euclidean space and soap bubbles in symmetric spaces

TL;DR

This paper introduces a modified mean curvature flow approach to construct small spherical soap bubbles in symmetric spaces, revealing their shape and curvature properties, and linking Euclidean flows to symmetric space geometries.

Contribution

It develops a new modified mean curvature flow method to generate soap bubbles in symmetric spaces from Euclidean small spheres, expanding understanding of geometric flows in complex spaces.

Findings

Small spherical soap bubbles are obtained from limits of modified mean curvature flows.

These bubbles are invariant under isotropy subgroup actions.

The shape and mean curvature of the bubbles are characterized.

Abstract

In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, where we note that the small spherical soap bubbles are invariant under the isotropy subgroup action of the isometry group of the symmetric space. Furthermore, we investigate the shape and the mean curvature of the small spherical soap bubbles.