Blow-up phenomena for the Yamabe equation

Simon Brendle, Stanford University
Fine Hall 314

The Yamabe problem asserts that any Riemannian metric on a compact manifold can be conformally deformed to one of constant scalar curvature. However, this metric is not, in general, unique, and there are examples of manifolds that admit many metrics of constant scalar curvature in a given conformal class. It was conjectured by R. Schoen in the 1980s (and, independently, by Aubin) that the set of all metrics of constant scalar curvature 1 in a given conformal class is compact, except if the underlying manifold is conformally equivalent to the sphere $S^n$ equipped with its standard metric. The significance of Schoen's conjecture is that it would imply Morse inequalities for the total scalar curvature functional. I will discuss counterexamples to this conjecture in dimension 52 and higher. I will also describe joint work with F. Marques, which extends these counterexamples to dimension 25 and higher. The condition $n \geq 25$ turns out to be optimal.