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Slowly converging Yamabe flows

Alessandro Carlotto, Otis Chodosh and Yanir A Rubinstein

Geometry & Topology 19 (2015) 1523–1568

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.

Yamabe flow, nonintegrable critical points, polynomial convergence, Lojasiewicz–Simon inequality, constant scalar curvature
Mathematical Subject Classification 2010
Primary: 35K55, 53C44
Secondary: 58K05, 58K55
Received: 19 January 2014
Revised: 29 April 2014
Accepted: 29 May 2014
Published: 21 May 2015
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, John Lott
Alessandro Carlotto
Department of Mathematics
Stanford University
Stanford, CA 94305
Otis Chodosh
Department of Mathematics
Stanford University
Stanford, CA 94305
Yanir A Rubinstein
Department of Mathematics
University of Maryland
College Park, MD 20742