Abstract:
The classic Yamabe problem shows that each conformal class of metrics of a compact manifold with dimension ≥ 3 admits at least a constant scalar curvature metric. Finding such metric is equivalent to solving a semi-linear partial differential equation. A natural question is to ask if the solutions of Yamabe equation are compact. This question has been well studied and it turns out that for non-flat manifolds under dimension 24, the solutions are compact while non-compact counterexamples exist for dimension ≥ 25. In this talk, I will survey some results in this field and in particular I’d like to talk about the counterexample constructed by S. Brendle and F. Marques.