Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo
“Weyl curvature, conformal geometry, and uniformization: Part III”
We complete our introduction to conformal geometry this week. We start by completing the proof that, if (M,g) is a compact Riemannian manifold with strictly negative scalar cur- vature, then there exists a conformal metric with constant negative scalar curvature, using the maximum principle, the basic estimates of elliptic regularity, and the continuity method. These are all basic important tools in geometric analysis. Then we use this result to establish the Yamabe theorem in dimension n = 2 for genus k 1. The case k = 0 is slightly different, and we discuss this as well. Finally, we relate these results in dimension n = 2 to the classical uniformization theorem of complex analysis.
MC 5479