Explicit Yamabe Flow of an Asymmetric Cigar

Abstract

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.Key words. 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.

Continuous maximal regularity on uniformly regular Riemannian manifolds

“…In recent years, these problems have been studied by several mathematicians, including Y. An and L. Ma [7], A. Burchard, R.J. McCann and A. Smith [14]. In comparison with the existing results, we do not ask for a uniform bound on the curvatures of the initial metric and the background manifolds are admitted to be non-complete within certain conformal classes, see [3, 4, Section 1] for justification.…”

“…Less is known about the Yamabe flow on non-compact manifolds. To the best of the authors' knowledge, all available results in this direction require the underlying manifold to be complete and have bounded curvatures, or to be of some explicit expression, see [7] and [14]. We will formulate an existence and regularity result for the Yamabe flow on a manifold, which may not satisfy any of the above conditions.…”

Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

“…Furthermore, it appears that considering immersions into a sufficiently large family of manifolds, which include spaces with constant sectional curvature, is a handy way to expand the previous studies to a broader class of ambient spaces. Warped product metrics explain a natural metric that includes spaces with constant sectional curvature throughout its range [17][18][19][20][21][22][23][24][25][26].…”

Yamabe solitons in contact geometry

“…The evolution of a Riemannian metric g on a smooth manifold M to a metric g(t) in time t through the equation ∂ ∂t g(t) = −r(t)g(t), g(0) = g, where r(t) denotes the scalar curvature of g(t), is called the Yamabe flow, and was introduced by Hamilton [8]. The Yamabe flow is a natural geometric deformation to metrics of constant scalar curvature, and corresponds to the fast diffusion case of the porous medium equation (the plasma equation) in mathematical physics (Burchard et al [3]). Just as a Ricci soliton is a special solution of the Ricci flow, a Yamabe soliton is a special solution of the Yamabe flow that moves by a one parameter family of diffeomorphisms ϕ t generated by a time-dependent vector field W t on M , and homotheties, i.e., g(t) = σ(t)ϕ * t g, where σ is a positive real valued function of the parameter t. Substituting the foregoing equation in the Yamabe flow equation and setting σ(0) = 1, − σ(0) = c gives the equation…”