Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries
نویسندگان
چکیده
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 -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. MSC classification: 35J60, 53C21, 58G30.
منابع مشابه
CONFORMAL METRICS WITH PRESCRIBED CURVATURE FUNCTIONS ON MANIFOLDS WITH BOUNDARY By BO GUAN Dedicated to Professor Joel Spruck on the occasion of his 60th birthday
We study the Dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on Riemannian manifolds with boundary. As a consequence we prove the existence of a conformal metric, given its value on the boundary as a prescribed metric conformal to the (induced) background metric, with a prescribed curvature function of the Schouten tensor.
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