Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

Details Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

2001-04-03

arxiv.org/abs/math/0104041v1

Mathematics

Analysis of PDEs

34 pages

Scientific paper

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 blow-up 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 Leray-Schauder 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.

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