Locally conformally flat ancient Ricci flows

Catino, Giovanni; Mantegazza, Carlo; Mazzieri, Lorenzo

doi:10.2140/apde.2015.8.365

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Locally conformally flat ancient Ricci flows

Giovanni Catino, Carlo Mantegazza and Lorenzo Mazzieri

Vol. 8 (2015), No. 2, 365–371

DOI: 10.2140/apde.2015.8.365

Abstract

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.

Keywords

Ricci solitons, Ricci flow

Mathematical Subject Classification 2010

Primary: 53C24, 53C25, 53C44

Milestones

Received: 8 May 2014

Accepted: 9 February 2015

Published: 10 May 2015

Authors

Giovanni Catino
Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 I-20133 Milano Italy

Carlo Mantegazza
Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 I-56126 Pisa Italy

Lorenzo Mazzieri
Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 I-56126 Pisa Italy

Vol. 8, No. 2, 2015