A geometric equation with critical nonlinearity on the boundary

• Autores: Veronica Felli , Mohameden Ould Ahmedou
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 218, Nº 1, 2005, págs. 75-100
• Idioma: inglés
• DOI: 10.2140/pjm.2005.218.75
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• Resumen

  • A theorem of Escobar asserts that if a three-dimensional smooth compact Riemannian manifold M with boundary is of positive type and is not conformally equivalent to the standard three-dimensional ball, a necessary and sufficient condition for a C2 function H on M to be the mean curvature of some conformal scalar ?at metric is that H be positive somewhere. We show that, when the boundary is umbilic and the function H is positive everywhere, all such metrics stay in a compact set with respect to the C2 norm and the total degree of all solutions is -1.