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

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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 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 C2-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.

Mathematics Subject Classification (2000): 35J60, 53C21, 58G30

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Authors and Affiliations

  1. Scuola Internazionale Superiore di Studi Avanzati (S.I.S.S.A.), via Beirut, 2-4, 34014, Trieste, Italy

    V. Felli

  2. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Beringstrasse 4, 53115, Bonn, Germany

    M. Ould Ahmedou

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  1. V. Felli
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  2. M. Ould Ahmedou
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Felli, V., Ahmedou, M. Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries. Math. Z. 244, 175–210 (2003). https://doi.org/10.1007/s00209-002-0486-7

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  • DOI: https://doi.org/10.1007/s00209-002-0486-7

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