Methods and Applications of Analysis

Explicit Yamabe Flow of an Asymmetric Cigar

Almut Burchard, Robert J. McCann, and Aaron Smith

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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.

Article information

Methods Appl. Anal., Volume 15, Number 1 (2008), 65-80.

First available in Project Euclid: 10 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 35K55: Nonlinear parabolic equations 58J35: Heat and other parabolic equation methods

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


Burchard, Almut; McCann, Robert J.; Smith, Aaron. Explicit Yamabe Flow of an Asymmetric Cigar. Methods Appl. Anal. 15 (2008), no. 1, 65--80.

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