Physics – Mathematical Physics
Scientific paper
2003-05-19
An extended and improved version has been published in Commun. Math. Phys. 267 (2006) 93
Physics
Mathematical Physics
36 pages. In this new version the analytic representation hypothesis is proved. Please address all correspondence to D. Peralt
Scientific paper
10.1007/s00220-006-0069-2
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.
Pelayo Alvaro
Peralta-Salas Daniel
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