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

Bin Guo (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Zhijie Huang (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Duong H. Phong (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

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

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

Received 3 April 2016

Published 27 July 2018