GEOMETRIC ANALYSIS SEMINARS 2016

Abstract: In this on-going scalar curvature seminar series, we last time heard from Claudio Arezzo about the role of scalar curvature in the theory of Sacks-Uhlenbeck/Schoen-Yau concerning existence of harmonic maps and incompressible minimal surfaces. The main conclusion was of the form: For a Riemannian 3-manifold (M³, g), having positive scalar curvature implies a topological phenomenon: That there are no 'holes' inside M³ which 'look like' a genus >=1 surface.

This time around we will look at a different (but related) aspect, using the theory of Kazdan-Warner and Yamabe/Trudinger/Aubin/Schoen: Given a manifold Mⁿ, which functions (and of special interest: the constant ones) can be the scalar curvature of some metric on Mⁿ?". The setting here is to start with a metric g and search only within the same conformal class [g], which is a natural reduction of the problem to "just" one second order nonlinear PDE in one scalar function on M.