Abstract

The limiting behaviour of the Ricci flow

Natasa Sesum
Massachusetts Institute of Technology
Mathematics

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. In the talk we will prove that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.
We will also give a brief proof of a convergence of a flow to a soliton, i.e. if $(g_{ij})_t = -2R_{ij} + \frac{1}{\tau}g_{ij}$ with $|\rem| \le C$ and $\diam(M,g(t)) \le C$ for all $t\in[0,\infty)$, then for every sequence of times $t_i\to\infty$ as $i\to\infty$ there exists a subsequence $g(t_i + t)$ converging to metrics $h(t)$ in $C^{\infty}$ norm. Moreover, $h(t)$ is a soliton type solution to the flow.