Abstract
The limiting behaviour of the Ricci flow
Natasa Sesum
University of Pennsylvania
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 convergence of a flow to a soliton. Namely, if
\[
(g_{ij})_t = -2R_{ij} + \frac{1}{\tau} g_{ij}
\]
with \( |\mathrm{Rm}| \le C \) and \( \operatorname{diam}(M, g(t)) \le C \) 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 the \( C^{\infty} \) norm. Moreover, \( h(t) \) is a soliton-type solution to the flow.
We will also give a brief proof of convergence of a flow to a soliton. Namely, if
\[
(g_{ij})_t = -2R_{ij} + \frac{1}{\tau} g_{ij}
\]
with \( |\mathrm{Rm}| \le C \) and \( \operatorname{diam}(M, g(t)) \le C \) 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 the \( C^{\infty} \) norm. Moreover, \( h(t) \) is a soliton-type solution to the flow.
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