31 research outputs found
Yamabe flow: Steady solitons and Type II singularities
No full textWe study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time T or as t -> + infinity. (C) 2018 Elsevier Ltd. All rights reserved.
A note on blowup limits in 3d Ricci flow
No full textWe prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.
EVOLUTION OF NONCOMPACT HYPERSURFACES BY INVERSE MEAN CURVATURE
No full textWe study the evolution of complete, noncompact, convex hypersurfaces in Rn+1 by the inverse mean curvature flow. We establish the long-time existence of solutions, and we provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of a result by Huisken and Ilmanen on compact star-shaped solutions.
Diffusion of Biological Organisms: Fickian and Fokker-Planck Type Diffusions
No full textIn this paper we derive diffiusion equations in a heterogeneous environment. We consider a system of discrete kinetic equations that consists of two phenotypes of different turning frequencies. The two phenotypes change their states according to state transition frequencies which depend on the environment. We show that the density of the total population of the two phenotypes converges to the solution of a Fokker-Planck type diffiusion equation if turning frequencies are of higher order than the state transition frequencies. If it is the other way around, i.e., if the state changes many times between each turning, the density converges to the solution of a Fickian diffiusion equation.11Nsciescopu
Convergence of Gauss curvature flows to translating solitons
No full text© 2022 Elsevier Inc.We address the asymptotic behavior of the α-Gauss curvature flow, for α>1/2, with a complete non-compact convex initial hypersurface which is contained in a cylinder of a bounded cross section. We show that the flow converges, as t→+∞, locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface.11Nsciescopu
FINITE-DIMENSIONAL LEADING ORDER DYNAMICS FOR THE FAST DIFFUSION EQUATION NEAR EXTINCTION
No full textThe fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near the vanishing solution to any prescribed convergence rate.
Hitting estimates on Einstein manifolds and applications
No full textWe generalize the Benjamini-Pemantle-Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for two independent Brownian motions on manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio. We also obtain generalizations of (1) and (2) for the manifolds with two-sided Ricci bounds and Einstein manifolds with nonzero Einstein constant.
Inverse Mean Curvature Flow With Singularities
No full textThis paper concerns the inverse mean curvature flow (IMCF) running from the boundary of a convex body that has no regularity assumption. We study the evolution of singularities by looking at the blow-up tangent cone around each singular point. We prove that the cone also evolves by the IMCF and that each singularity is removed when the evolving cone becomes f lat. As a result, we derive the exact waiting time for a weak solution to become a smooth solution. In particular, necessary and sufficient condition for the existence of a smooth classical solution is given.
UNIQUENESS OF ANCIENT SOLUTIONS TO GAUSS CURVATURE FLOW ASYMPTOTIC TO A CYLINDER
No full textWe address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross-section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time -infinity from two opposite ends.
Type II Singularities on complete non-compact Yamabe flow
No full textThis work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact confor-mally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rota-tionally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow
