146,337 research outputs found
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]
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
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
Cohomology of D-complex manifolds
No full textIn order to look for a well-behaved counterpart to Dolbeault cohomology in D-complex geometry, we study the de Rham cohomology of an almost D-complex manifold and its subgroups made up of the classes admitting invariant, respectively anti-invariant, representatives with respect to the almost D-complex structure, miming the theory introduced by Li and Zhang (2009) in [20] for almost complex manifolds. In particular, we prove that, on a 4-dimensional D-complex nilmanifold, such subgroups provide a decomposition at the level of the real second de Rham cohomology group. Moreover, we study deformations of D-complex structures, showing in particular that admitting D-Kähler structures is not a stable property under small deformations
A spinorial energy functional: Critical points and gradient flow
No full textOn the universal bundle of unit spinors we study a natural energy
functional whose critical points, if dimM C 3, are precisely the pairs (g,φ) consisting
of a Ricci-flat Riemannian metric g together with a parallel g-spinor φ.
We investigate the basic properties of this functional and study its negative
gradient flow, the so-called spinor flow. In particular, we prove short-time
existence and uniqueness for this flow
Note su problemi e figure della tutela in Sicilia, attraverso il carteggio di Corrado Ricci
No full textATTI DEL CONVEGNO: Corrado Ricci storico dell’arte tra esperienza e progetto, RAVENNA 2001CONFERENCE PROCEEDINGS: Corrado Ricci historian of experience and planning, RAVENNA 200
On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor
No full textIn this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci
solitons to classify -dimensional () complete -flat gradient
steady Ricci solitons. More precisely, we prove that any -dimensional
complete noncompact gradient steady Ricci soliton with vanishing -tensor is
either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof
extends to the shrinking case and the expanding case as well
Ricci solitons on Kenmotsu manifolds under D-homothetic deformation
No full textThe aim of the present paper is to study Ricci solitons in Kenmotsu manifolds under D-homothetic deformation. We analyzed behaviour of Ricci solitons when potential vector field is orthogonal to Reeb vector field and pointwise collinear with Reeb vector field. Further we prove Ricci solitons in D-homothetically transformed Kenmotsu manifolds are shrinking. © 2017 Khayyam Journal of Mathematics
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