Some nonuniqueness results for the Yamabe problem

The Yamabe problem consists of finding a complete metric with constant
scalar curvature in a prescribed conformal class. A landmark
achievement of Geometric Analysis is that a solution always exists on
compact manifolds, but the situation is more delicate in the
noncompact case: there are complete noncompact manifolds where no
solutions exist. In this talk, I will discuss how analytic and
topological techniques can be combined to obtain infinitely many
solutions on certain manifolds (both compact and noncompact). As a
consequence, one obtains infinitely many new solutions to the
so-called singular Yamabe problem on spheres. (This is based on joint
works with P. Piccione and B. Santoro)