38 research outputs found
The Isoperimetric Profile of a Noncompact Riemannian Manifold for Small Volumes
In the main theorem of this paper we treat the problem of existence of
minimizers of the isoperimetric problem under the assumption of small volumes.
Applications of the main theorem to asymptotic expansions of the isoperimetric
problem are given.Comment: 33 pages, improved version after the referee comments, (Submitted
First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds
We calculate the first and the second variation formula for the
sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We
consider general variations that can move the singular set of a C^2 surface and
non-singular variation for C_H^2 surfaces. These formulas enable us to
construct a stability operator for non-singular C^2 surfaces and another one
for C2 (eventually singular) surfaces. Then we can obtain a necessary condition
for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in
term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally
we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly
changed and Remark 9.9 adde
Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids
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.Comment: 36 pages. In this new version the analytic representation hypothesis
is proved. Please address all correspondence to D. Peralta-Sala