# Geometric Analysis and Topology Seminar

#### Conformally Kahler Einstein Metrics and the Bach Tensor

**Speaker:**
Brian Weber, SUNY Stony Brook

**Location:**
Warren Weaver Hall 202

**Date:**
Friday, March 5, 2010, 11 a.m.

**Synopsis:**

In the search for "canonical metrics", such as Einstein metrics, attempts to directly minimize an energy functional have proved difficult to carry out, in part due to collapsing behavior. Certain indirect approaches have been more succesful in special cases. A local obstruction to finding an Einstein metric in a conformal class is the non-vanishing of the Bach tensor, which is defined to be the gradient of the Weyl curvature functional \int |W|^2. On a Kahler manifold there are no other obstructions, and Bach-flat Kahler metrics are locally conformally Einsteinian. The conformal factor is also geometrically interesting and sometimes controllable. This talk will describe the results of a paper with X. Chen and C. LeBrun, where it was shown that a certain Kahler metric on CP^2#2CP^2 is conformal to an Einstein metric, establishing for the first time an Einstein metric on CP^2#2CP^2.