Mathematics – Differential Geometry
Scientific paper
2010-06-03
Mathematics
Differential Geometry
23 pages
Scientific paper
We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $\kappa$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's previous result on backward limits of $\kappa$-solutions in dimension 3, in which case that the curvature operator is nonnegative (follows from Hamilton-Ivey curvature pinching estimate). The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.
Cao Xiaodong
Zhang Qi S.
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