On a geometric equation with critical nonlinearity on the boundary

Veronica Felli; Mohameden Ould Ahmedou

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On a geometric equation with critical nonlinearity on the boundary

Veronica Felli·Mohameden Ould Ahmedou

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Abstract

A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a $C^2$ function $H$ to be the mean curvature of some conformal flat metric is that $H$ is 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 $C^2$ norm and the total degree of all solutions is equal to -1.

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Veronica Felli, Mohameden Ould Ahmedou. 2002-05-22. On a geometric equation with critical nonlinearity on the boundary. https://arxiv.org/abs/math/0106219

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