A geometric approach to the equilibrium shapes of self-gravitating fluids

TL;DR

This paper proves that equilibrium shapes of self-gravitating fluids in Riemannian manifolds are isoparametric submanifolds, linking geometric properties with classical physics problems and expanding understanding of their symmetry and shape classification.

Contribution

It establishes that equilibrium shapes are isoparametric submanifolds and explores their geometric properties, connecting them with isoperimetric problems and symmetry groups.

Findings

Equilibrium shapes are isoparametric submanifolds.

Relationship with isoperimetric problem established.

Analysis of symmetry groups of equilibrium shapes.

Abstract

The classification of the possible equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in mathematical physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of the equilibrium shapes are also obtained, specifically the relationship with the isoperimetric problem and the group of isometries of the manifold. This work follows the new geometric approach to the problem developed by the author (see math-ph/0305038)