Geometric Analysis Working SeminarExport this event to calendar

Tuesday, January 19, 2016 — 1:00 PM EST

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

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

Before we start going through the paper by Parker and Lee on the Yamabe problem, we will cover some basic material from Riemannian geometry needed to study conformal geometry. First, we will discuss the decomposition of the Riemann curvature tensor into the scalar, traceless Ricci, and Weyl tensors, and the behaviour of these tensors under a conformal change of metric. In particular, we will see that the Weyl tensor is the obstruction to local conformal flatness. Then, we will consider the baby case (n=2) of the Yamabe problem: given a compact oriented Riemannian 2-manifold, we’ll show that we can always find a metric in the same conformal class that has the constant curvature K. [Recall there is only scalar curvature in dimension 2.] This is closely related to the classical uniformization theorem. The case K ¿ 0 has a significantly different proof than the cases K = 0 and K ¡ 0. We may need slightly more than one lecture to get through all of this material.

MC 5479

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