TY - JOUR
T1 - Slowly converging yamabe flows
AU - Carlotto, Alessandro
AU - Chodosh, Otis
AU - Rubinstein, Yanir A.
N1 - Publisher Copyright:
© 2015, Mathematical Sciences Publishers. All rights reserved.
PY - 2015/5/21
Y1 - 2015/5/21
N2 - 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.
AB - 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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U2 - 10.2140/gt.2015.19.1523
DO - 10.2140/gt.2015.19.1523
M3 - Article
AN - SCOPUS:84930658067
SN - 1465-3060
VL - 19
SP - 1523
EP - 1568
JO - Geometry and Topology
JF - Geometry and Topology
IS - 3
ER -