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March 2008 Explicit Yamabe Flow of an Asymmetric Cigar
Almut Burchard, Robert J. McCann, Aaron Smith
Methods Appl. Anal. 15(1): 65-80 (March 2008).


We consider the Yamabe flow of a conformally Euclidean manifold for which the conformal factor’s reciprocal is a quadratic function of the Cartesian coordinates at each instant in time. This leads to a class of explicit solutions having no continuous symmetries (no Killing fields) but which converge in time to the cigar soliton (in two-dimensions, where the Ricci and Yamabe flows coincide) or in higher dimensions to the collapsing cigar. We calculate the exponential rate of this convergence precisely, using the logarithm of the optimal bi-Lipschitz constant to metrize distance between two Riemannian manifolds.


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Almut Burchard. Robert J. McCann. Aaron Smith. "Explicit Yamabe Flow of an Asymmetric Cigar." Methods Appl. Anal. 15 (1) 65 - 80, March 2008.


Published: March 2008
First available in Project Euclid: 10 December 2008

zbMATH: 1172.53042
MathSciNet: MR2482210

Primary: 53C44
Secondary: 35K55 , 58J35

Keywords: attractor , basin of attraction , biLipschitz , cigar soliton , conformally flat non-compact manifold , Exact Yamabe flows , Lyapunov exponent , quadratic conformal factor , rate of convergence , Ricci flow

Rights: Copyright © 2008 International Press of Boston


Vol.15 • No. 1 • March 2008
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