##### Department of Mathematics,

University of California San Diego

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

### Differential Geometry Seminar

## Brett Kotschwar

#### Max Planck Institute for Gravitational Physics, Potsdam

## Ricci flow and the holonomy group

##### Abstract:

I will discuss a "non-contraction" result for the holonomy group of a solution to Ricci flow, namely, that if the reduced holonomy of a complete solution of uniformly bounded curvature is restricted to a subgroup of SO(n) at some non-initial time, it must be restricted to the same subgroup at all previous times; it follows then from existing results that the holonomy group is exactly preserved by the equation. In particular, a solution may be Kahler or locally reducible (as a product) on some time slice only if it is identically so on its entire interval of existence. In contrast to the question of "non-expansion" of holonomy, the problem of non-contraction cannot be reduced completely to an application of the classification and splitting theorems of Berger and De Rham and a series of appeals to a relevant uniqueness theorem (here, backwards-uniqueness). However, with an infinitesimal reformulation, we show that the problem can nevertheless be reduced to one of unique continuation, and specifically to one for a coupled system of partial- and ordinary-differential inequalities of a form amenable to an approach by Carleman inequalities. This reformulation also leads to an alternative and essentially self-contained proof of the non-expansion of holonomy via the analysis of a similar (albeit simpler and strictly parabolic) system by means of the maximum principle.

Host: Lei Ni

### April 13, 2011

### 4:00 PM

### AP&M 5402

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