The zero scalar curvature Yamabe problem on noncompact manifolds with boundary

Details The zero scalar curvature Yamabe problem on noncompact manifolds with boundary The zero scalar curvature Yamabe problem on noncompact manifolds with boundary

2006-05-24

arxiv.org/abs/math/0605650v1

Indiana Univ. Math. J. 55 (2006), 1449-1460

Mathematics

Differential Geometry

8 pages, to appear in Indiana Math. J

Scientific paper

Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curvature $f$ on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.

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