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

By Veronica Felli, M. Ahmedou. Mathematische Zeitschrift, 2001

58 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/7dcc6dc239171228660fc4ff0553d4d7241f83df

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In Applied Mathematics

1. Yanyan Li
2. Lei Zhang
3. S. Almaraz
4. Jingang Xiong
5. Tianling Jin
6. Aobing Li
7. Wolfgang Reichel
8. Luis Caffarelli

In Computational Theory and Mathematics

1. Yanyan Li
2. Lei Zhang
3. S. Almaraz
4. Aobing Li
5. Jingang Xiong
6. Tianling Jin
7. Luis Caffarelli
8. X. Cabré

Citations in the corpus

• An existence theorem of conformal scalar-flat metrics on manifolds with boundary (2009, background): “Other works concerning conformal deformation on manifolds with boundary include [Ahmedou 2003; Ambrosetti et al. 2002; Ben Ayed et al. 2005; Brendle 2002; Djadli et al. 2003; 2004; Escobar 1992b; 1996a; Escobar and Garcia 2004; Felli and Ould Ahmedou 2003; 2005; Han and Li 1999; 2000] We will now di”
• Compactness of solutions to the Yamabe problem (2004, methodology): “For the Nirenberg problem and the Yamabe problem on manifolds with boundary, which are related to the Yamabe problem, compactness of solutions has been studied by Schoen ([81]), Schoen and Zhang ([84]), Chang, Gursky and Yang ([32]), Li ([66]) and [67]), Han and Li ([57]), Chen and Lin ([35]), Felli”
• Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary (2007, unknown): “Other works concerning conformal deformation on manifolds with boundary include [1], [4], [5], [8], [9], [11], [12], [13], [14] and [15].”