Geometric Analysis Working Seminar

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

“Weyl curvature, conformal geometry, and uniformization: Part II”

We continue where we left off last time. First, we complete the derivation of the transfor- mation of the various curvatures under a conformal change of metric. In particular we establish the conformal invariance of the Weyl curvature operator. Then we consider the differential equation for constant scalar curvature S ̃ of the new metric g ̃ = e2ug. When the original metric has strictly negative scalar curvature S < 0, we show that we can always find a new metric in the conformal class that has constant negative scalar curvature, in any dimension. 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