Methods and Applications of Analysis

Volume 15 (2008)

Number 1

Explicit Yamabe Flow of an Asymmetric Cigar

Pages: 65 – 80

DOI: http://dx.doi.org/10.4310/MAA.2008.v15.n1.a7

Authors

Almut Burchard

Robert J. McCann

Aaron Smith

Abstract

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.

Keywords

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

2010 Mathematics Subject Classification

Primary 53C44. Secondary 35K55, 58J35.

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