Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids

TL;DR

This paper investigates how the topology and geometry of a Riemannian manifold influence the shape and structure of solutions to elliptic free-boundary problems, with applications to classifying equilibrium shapes of self-gravitating fluids.

Contribution

It introduces a novel analytical technique based on the fibers' properties to derive topological and geometrical restrictions on solutions, linking PDE solutions to manifold characteristics.

Findings

Fibers must satisfy specific topological constraints depending on the manifold.

The method provides a global analytical characterization of solution fibers.

Results apply to classifying equilibrium shapes of self-gravitating fluids.

Abstract

Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property) for going from the initial PDE to a global analytical characterization of the fibres (the equilibrium partition condition). We study this analytical characterization and obtain several topological and geometrical properties that the fibres of the solutions must possess, depending on the topology of M and the metric tensor g. We apply these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic static self-gravitating fluids. We also suggest a relationship with the isometries of a Riemannian manifold.