195,136 research outputs found
Ricci flow coupled with harmonic map flow
No full text07.02.13 KB. Accepted version ok to add to Spiral. SMF/SherpaWe investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of phi a-priori - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for alpha = 0. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times
L-optimal transportation for Ricci flow
No full textWe introduce the notion of L-optimal transportation, and use it to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [13] (both related to entropy and to L-length) and a recent result of McCann and the author [11]
On Type-I singularities in Ricci flow
No full text07.02.13 KB. Accepted version ok to add to Spiral. IP/Sherpa.We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time. We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow
The volume entropy of a surface decreases along the Ricci flow
No full textThe volume entropy, h(g), of a compact Riemannian manifold (M,g) measures the growth rate of the volume of a ball of radius R in its universal cover. Under the Ricci flow, g evolves along a certain path that improves its curvature properties. For a compact surface of variable negative curvature we use a Katok–Knieper–Weiss formula to show that h(gt) is strictly decreasing
Maine Voices piece by Lawrence R. Ricci, director of The Spurwink Child Abuse
No full textMaine Voices piece by Lawrence R. Ricci, director of The Spurwink Child Abuse Program in Portland, on two upcoming meetings that should impact the issue of physician response to child abuse. The first is the Fifth Annual Spurwink Northern New England Conference on Child Maltreatment, which will be held in Portland on Sept. 18-19. The second, which will be held in Salt Lake City on Sept. 21-24, is a meeting of the newly formed Helfer Society, an honorary professional society of physicians dedicated to the appropriate identification, treatment and prevention of child abuse
Non--existence of nodal and one--signed solutions or nonlinear variational equations with special symmetries
No full textIn this paper we extend to a particular class of symmetric operators some non existence results of nodal solutions and of solutions of constant sign of variational problems, extendig some results obtained in a previous paper [Filipppucci, Ghiselli Ricci , Pucci (1994) Archive Rat. Mech. Anal.
Neckpinch singularities of Ricci flow
No full textWe use asymptotic analysis techniques to analyze Ricci flow asymptotic to a cylinder and prove two results. First, we derive the precise asymptotic behavior of a compact rotationally symmetric Ricci flow near a singularity modeled on the round cylinder . Second, we derive partial results about the shape of a 4-dimensional steady gradient Ricci soliton with tangent flow equal to .Ph.D.Includes bibliographical reference
The Riesz transform on a complete Riemannian manifold with Ricci curvature bounded from below
No full textWe study the Riesz transform and Hodge-Dirac operator on a complete Riemannian manifold with Ricci curvature bounded from below. We define the Hodge-Dirac operator ∏ on Lp(ΛTM) as the closure of d + d* on smooth, compactly supported k-forms for 1 < p < ∞. Given the boundedness of the Riesz transform on Lp(ΛTM), we show that ∏ is R-bisectorial on Lp(ΛTM). From this we conclude that ∏ has a bounded H∞-functional calculus on a bisector under mild assumptions which we conjecture to be true when the Ricci curvature is nonnegative. We conclude by showing that from this bounded H∞-functional calculus for the Hodge-Dirac operator we can retrieve the boundedness of the Riesz transform, thus giving us that the mentioned assertions are equivalent when the Ricci curvature is nonnegative.Electrical Engineering, Mathematics and Computer ScienceDelft Institute of Applied Mathematic
On special weakly Ricci symmetric, Ricci bi-symmetric and R-harmonic quasi-Einstein manifolds
No full textIn this paper, we have studied some geometric properties of special weakly Ricci symmetric quasi-Einstein manifold, special weakly Ricci bisymmetric quasi-Einstein manifold and R-harmonic quasi-Einstein manifold
On Uniqueness Theoremsfor Ricci Tensor
No full textsummary:In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold and a symmetric 2-tensor , construct a metric on whose Ricci tensor equals . In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita connection on any compact Einstein manifold with non-negative section curvature. In the present paper we generalize the result of DeTurck and Koiso for a Riemannian manifold with non-negative section curvature. In addition, we extended our result to complete non-compact Riemannian manifolds with nonnegative sectional curvature and with finite total scalar curvature
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