Type II extinction profile of maximal solutions to the Ricci flow in ℝ2

Panagiota Daskalopoulos, Natasa Sesum

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

We consider the Cauchy problem for the logarithmic fast diffusion equation ut = Δ log u in ℝ2 with u(x, 0) = u 0(x) ≥ 0, corresponding to the evolution of the metric gij:= udxi dxj by the Ricci flow. It is well known that the maximal (complete) solution u vanishes identically after time T = 1/4π ∫2 u0. Assuming that u0 is compactly supported, we provide a precise description of the type II vanishing behavior of u at time T: we show the existence of an inner region with exponentially fast vanishing profile, up to proper scaling, a soliton cigar solution, and the existence of an outer region of persistence of a logarithmic cusp. This result recovers rigorously formal asymptotics derived by King (Philos. Trans. R. Soc, Lond. A 343:337-375, 1993).

Original languageEnglish (US)
Pages (from-to)565-591
Number of pages27
JournalJournal of Geometric Analysis
Volume20
Issue number3
DOIs
StatePublished - Jul 1 2010
Externally publishedYes

Fingerprint

Maximal Solution
Ricci Flow
Extinction
Logarithmic
Fast Diffusion Equation
Cusp
Soliton Solution
Persistence
Vanish
Cauchy Problem
Scaling
Metric
Profile

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

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Type II extinction profile of maximal solutions to the Ricci flow in ℝ2. / Daskalopoulos, Panagiota; Sesum, Natasa.

In: Journal of Geometric Analysis, Vol. 20, No. 3, 01.07.2010, p. 565-591.

Research output: Contribution to journalArticle

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