A geometric equation with critical nonlinearity on the boundary

Veronica Felli and Mohameden Ould Ahmedou

Vol. 218 (2005), No. 1, 75–99

DOI: 10.2140/pjm.2005.218.75

Abstract

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 flat 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.

Mathematical Subject Classification 2000

Primary: 35J60, 53C21

Milestones

Received: 26 July 2001

Revised: 9 September 2002

Published: 1 January 2005