Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries
Abstract
This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blowup analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the $C^2$norm and the total LeraySchauder degree of all solutions is equal to 1. Then we deduce from this compactness result the existence of at least one solution to our problem.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2001
 arXiv:
 arXiv:math/0104041
 Bibcode:
 2001math......4041F
 Keywords:

 Mathematics  Analysis of PDEs;
 35J60;
 53C21;
 58G30
 EPrint:
 34 pages