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# Communications in Analysis and Geometry

## Volume 26 (2018)

### Number 3

### Pseudo-locality for a coupled Ricci flow

Pages: 585 – 626

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n3.a5

#### Authors

#### Abstract

Let $(M, g, \phi)$ be a solution to the Ricci flow coupled with the heat equation for a scalar field $\phi$. We show that a complete, $\kappa$-noncollapsed solution $(M, g, \phi)$ to this coupled Ricci flow with a Type I singularity at time $T \lt \infty$ will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base point is Type I singular. A key ingredient is a version of Perelman pseudo-locality for the coupled Ricci flow.

#### 2010 Mathematics Subject Classification

53C23, 53C44

This work was supported in part by NSF grant DMS-12-66033.

Received 3 April 2016

Published 27 July 2018