##### Department of Mathematics,

University of California San Diego

****************************

### Math 258 - Differential Geometry

## Ben Weinkove

#### Columbia University

## The J-flow and the Mabuchi energy

##### Abstract:

The J-flow is a parabolic flow on compact Kahler manifolds with two Kahler metrics. It was discovered by S. Donaldson and X. X. Chen independently. Donaldson defined it in the setting of moment maps and symplectic geometry. Chen described the flow as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. The Mabuchi energy is an important functional on the space of Kahler potentials. Its critical points give constant scalar curvature metrics, and its lower boundedness is related to stability in the sense of geometric invariant theory. I will show that under a condition on the initial data, the J-flow converges to a critical metric. I will then explain how this implies the lower boundedness of the Mabuchi energy for an open set of Kahler classes on manifolds with negative first Chern class.

Host: Lei Ni and Ben Chow

### February 20, 2004

### 1:00 PM

### AP&M 6218

****************************