Methods and Applications of Analysis

Explicit Yamabe Flow of an Asymmetric Cigar

Almut Burchard, Robert J. McCann, and Aaron Smith

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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.

Article information

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

Dates
First available in Project Euclid: 10 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.maa/1228920873

Mathematical Reviews number (MathSciNet)
MR2482210

Zentralblatt MATH identifier
1172.53042

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

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

Citation

Burchard, Almut; McCann, Robert J.; Smith, Aaron. Explicit Yamabe Flow of an Asymmetric Cigar. Methods Appl. Anal. 15 (2008), no. 1, 65--80. https://projecteuclid.org/euclid.maa/1228920873


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