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2015 Slowly converging Yamabe flows
Alessandro Carlotto, Otis Chodosh, Yanir Rubinstein
Geom. Topol. 19(3): 1523-1568 (2015). DOI: 10.2140/gt.2015.19.1523

Abstract

We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse-theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is nondegenerate), then we show that the flow converges exponentially fast. In general, we make use of a suitable Łojasiewicz–Simon inequality to prove that the slowest the flow will converge is polynomially. When the limit metric satisfies an Adams–Simon-type condition we prove that there exist flows converging to it exactly at a polynomial rate. We conclude by constructing explicit examples of this phenomenon. These seem to be the first examples of a slowly converging solution to a geometric flow.

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Alessandro Carlotto. Otis Chodosh. Yanir Rubinstein. "Slowly converging Yamabe flows." Geom. Topol. 19 (3) 1523 - 1568, 2015. https://doi.org/10.2140/gt.2015.19.1523

Information

Received: 19 January 2014; Revised: 29 April 2014; Accepted: 29 May 2014; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1326.53089
MathSciNet: MR3352243
Digital Object Identifier: 10.2140/gt.2015.19.1523

Subjects:
Primary: 35K55 , 53C44
Secondary: 58K05 , 58K55

Keywords: constant scalar curvature , Lojasiewicz–Simon inequality , nonintegrable critical points , Polynomial convergence , Yamabe flow

Rights: Copyright © 2015 Mathematical Sciences Publishers

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