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Modified Ricci flow on a principal bundle
Get PDFtextLet M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow. In this thesis, we derive the Ricci Yang-Mills flow and show that solutions exist for a short time and are unique. We study obstructions to the long-time existence of the flow and prove a compactness theorem for pointed solutions. We represent the Ricci Yang-Mills flow as a gradient flow and derive monotonicity formulas that can be used to study breather and soliton solutions. Finally, we use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the Ricci Yang-Mills flow in dimension 2 at Einstein Yang-Mills metrics.Mathematic
L-optimal transportation for Ricci flow
Get PDFWe 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
Get PDF07.02.13 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
Get PDF07.02.13 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
Instantaneously complete Ricci flows on surfaces
Get PDFThe intention of this thesis is to give a survey of instantaneously complete Ricci
flows
on surfaces, focussing on the existence and uniqueness of its Cauchy problem. We
prove a general existence result for instantaneously complete Ricci
flows starting at
an arbitrary Riemannian surface which may be incomplete and may have unbounded
curvature. We give an explicit formula for the maximal existence time, and describe
the asymptotic behaviour in most cases. The issue of uniqueness within this class of
instantaneously complete Ricci
flows is still conjectured but we are going to describe
the progress towards its proof. Finally, we apply that new existence result in order to
construct an immortal complete Ricci
flow which has unbounded curvature for all time
The volume entropy of a surface decreases along the Ricci flow
Get PDFThe 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
Boccaccio letterato, umanista e narratore Intervista di Elisabetta Menetti a Lucia Battaglia Ricci
Get PDFUn dialogo che ripercorre le questioni critiche che hanno mosso gli studi di Lucia Battaglia Ricci sul capolavoro di Giovanni Boccaccio
Leonardo Ricci 100. Scrittura, pittura architettura. 100 note a margine dell’Anonimo del XX secolo
No full textIl catalogo raccoglie saggi di Claudia Conforti, Antonella Greco, Aldo Colonnetti, Giovanni Leoni, Giovanna Uzzani, Luca Barontini, Giovanni Bartolozzi, Paola Ricci, Stefano Lambardi e Ilaria Cattabriga, oltre ai saggi degli stessi curatori. Nel volume sono pubblicati i molti documenti inediti, schizzi, disegni, quadri e fotografie, esposti nella mostra al Refettorio di Santa Maria Novella, organizzati secondo un ritmo scandito dalle parole del libro-diario di Ricci, l’Anonimo del XX secolo
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On the blow-up of four-dimensional Ricci flow singularities
Get PDFtextIn 2002, Feldman, Ilmanen, and Knopf constructed the first example of a non-trivial (i.e. non-constant curvature) complete non-compact shrinking soliton, and conjectured that it models a Ricci flow singularity forming on a closed four-manifold. In this thesis, we confirm their conjecture and, as a consequence, show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature.Mathematic
Ricci solitons in Kenmotsu manifolds.
No full textIn this paper we study Ricci solitons in Kenmotsu manifolds. We consider quasi conformal, conharmonic and projective curvature tensors in a Kenmotsu manifold admitting Ricci solitons and prove the conditions for the Ricci solitons to be shrinking, steady and expanding
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