## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

**Returning to in-person experiences in February:** Visit the COVID-19 website for more information.

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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.