Methods and Applications of Analysis
Volume 15 (2008)
Explicit Yamabe Flow of an Asymmetric Cigar
Pages: 65 – 80
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.
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.