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Topic: Ricci flow

Related Topics

Ricci curvature

Geometrization conjecture

Curvature of Riemannian manifolds

Einstein manifold

Uniformization theorem

Richard Hamilton (professor)

Riemannian metric

Christoffel symbols

Differential geometry

Grigori Perelman

Soul theorem

Euler characteristic

Clay Mathematics Institute

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	Perelman
		This webpage is meant to be a repository for material related to Perelman's papers on Ricci flow.
		``Geometrization conjecture and Ricci flow'' by Misha Kapovich
		``Ricci flow and the Poincare conjecture'' by John Morgan and Gang Tian
	www.math.lsa.umich.edu /~lott/ricciflow/perelman.html (418 words)

	NationMaster - Encyclopedia: Ricci flow (Site not responding. Last check: 2007-10-29)
		In differential geometry, the Ricci flow is a process which deforms the metric of a Riemannian manifold in a manner formally analogous to the diffusion of heat.
		The Ricci flow was introduced by Richard Hamilton in 1981 in order to gain insight into the geometrization conjecture of William Thurston, which concerns the topological classification of three-dimensional smooth manifolds.
		Ricci curvature also appears in the Ricci flow equation, where a time-dependent Riemannian metric is deformed in the direction of minus its Ricci curvature.
	www.nationmaster.com /encyclopedia/Ricci-flow (2558 words)

	Collected Papers on Ricci Flow by H. Cao
		Ricci flow as an approach to the Geometrization Conjecture has recently received attention in the press with articles appearing in The New York Times and other popular newspapers and magazines.
		In the past two decades the Ricci flow, and particularly Richard Hamilton's work, has received attention as both having a profound influence on geometric evolution equations and as a possible approach to studying Thurston's Geometrization Conjecture.
		The graduate student or researcher unfamiliar with the Ricci flow may use it as an introduction to the Ricci flow quickly leading to current research topics and open problems.
	www.notpricyatall.com /Collected-Papers-on-Ricci-Flow-0_1571461108_1 (226 words)

	alapage.com - Livre Anglais: Hamilton's Ricci Flow
		Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds.
		Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible.
		A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincare conjecture and Thurston's geometrization conjecture.
	www.alapage.com /mx/?tp=F&type=101&l_isbn=0821842315&fulltext=&nopp=1 (225 words)

	Workshop on Geometric and Renormalization Group Flows
		Ricci flow, entropies and the averaging of cosmological spacetimes
		The Ricci flow as bulk renormalization group equation of sigma models; the mean curvature flow as boundary renormalization group equation of sigma models; the Calabi flow as equation for gravitational radiation in general relativity.
		We show that Perelman's extension of the Ricci flow sheds new light on such an issue, in particular we analyze the role that entropy-like quantities may have in controlling the deformation procedure.
	www.aei.mpg.de /~olito/WS/program.html (651 words)

	DTP - Woolgar, Eric (Site not responding. Last check: 2007-10-29)
		His method is based on Hamilton's Ricci flow, which is a sort of heat equation for tensors.
		One example is that the Ricci flow is the renormalization group flow for a nonlinear sigma model (at one loop).
		Another is that Perelman's results are based on a monotonicity theorem for the Ricci flow, which he shows is in fact the "second law" applied to a certain statistical entropy obtained from a formal partition function (for an unknown statistical ensemble).
	www.cap.ca /congress/2004/invAbstracts/woolgar-dtp.html (169 words)

	PlanetMath: Thurston's geometrization conjecture (Site not responding. Last check: 2007-10-29)
		Grigori Perelman sketched a proof of the geometrization conjecture in 2003 using Ricci flow with surgery, which (as of 2006) appears to be essentially correct.
		The Fields Medal was awarded to Thurston in 1982 partially for his proof of the conjecture for Haken manifolds.
		In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse.
	www.planetmath.org /encyclopedia/ThurstonsGeometrizationConjecture.html (767 words)

	Ian's home page (Site not responding. Last check: 2007-10-29)
		Thus, to try to prove geometrization, one could flow the metric on the manifold proportional to the Ricci curvature, and hope that the flow converges to a fixed point (up to scaling), which would be an Einstein metric.
		Of course, things are not this simple, and Hamilton first considered an equation g_t=2/3 R g-2 Ric(g), the traceless Ricci curvature, which was suggested by Eels and Sampson as the gradient flow of the total scalar curvature, but for which he couldn't even prove short-time existence.
		This sort of operation is necessary if one expects to prove geometrization using Ricci flow, since a 3-manifold may be a non-trivial connect sum of 3-manifolds which are not 3-spheres.
	www.math.uic.edu /~agol/blog/030226.html (1292 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The Entropy Formula for the Ricci Flow and its Geometric Applications, arXiv.org, November 11, 2002.
		Ricci Flow with Surgery on Three-Manifolds, arXiv.org, March 10, 2003.
		Ricci Flow and the Poincarand#233; Conjecture, by John Morgan and Gang Tian, arXiv.org, July 25, 2006.
	www.izvestia.ru /forum/board45/topic20188?page=12&act=quote&id_message=322702 (290 words)

	Ricci Flow, 3-Manifolds and Geometry — Program
		The emphasis of this course is Perelman's works on Ricci flow in 3-dimensions and geometrization of 3-manifolds.
		From chapters 1, 2, 3, 4 of Hamilton's Ricci Flow, by Bennett Chow, Peng Lu, Lei Ni, to be published by Science Press, China.
		Hamilton's trace Harnack estimate for the Ricci flow on surfaces and its consequences
	www.claymath.org /programs/summer_school/2005/program.php (535 words)

	Introduction to the Ricci Flow
		The Ricci Flow: An Introduction, by Ben Chow and Dan Knopf.
		Ricci flow on surface (III) : r = 0.
		Ricci flow on surface (IV) : R > 0.
	www.math.cornell.edu /~cao/711fall2006.html (195 words)

	Untitled Document (Site not responding. Last check: 2007-10-29)
		Ricci deformation of the metric on a Riemannian manifold
		On the entropy estimate for the Ricci flow on compact
		The Harnack estimate for the Ricci flow on a surface--
	www.intlpress.com /books/math/ricci-toc.html (88 words)

	Ricci flow - MA607 - University of Warwick - Spring 2007
		Hamilton's original application was to take an arbitrary closed 3-manifold with positive Ricci curvature, and show that the (renormalised) flow deforms it to a spherical space form.
		In the twenty years following its introduction, the Ricci flow was steadily developed, largely by Hamilton and his school, partly with a view to proving Thurston's geometrization conjecture (which includes the Poincaré conjecture).
		Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian, July 25, 2006.
	www.maths.warwick.ac.uk /~topping/RF2007.html (609 words)

	The Manila Times Internet Edition \| OPINION > The mathematician from St. Petersburg
		The equation was adapted from the equations of thermodynamics on the flow of heat.
		In a body with hot and cold spots, heat flows from the hot to the cold regions until the temperature everywhere is the same.
		But applying the Ricci flow equations to the equations of general relativity is equally exciting.
	www.manilatimes.net /national/2006/sept/03/yehey/opinion/20060903opi2.html (724 words)

	Matches for:
		Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds.
		The Ricci flow method is now central to our understanding of the geometry and topology of manifolds.
		The Ricci Flow was nominated for the 2005 Robert W. Hamilton Book Award, which is the highest honor of literary achievement given to published authors at the University of Texas at Austin.
	www.mathaware.org /bookstore?fn=20&arg1=survseries&item=SURV-110 (276 words)

	Long-time behavior of Ricci flow \| Department of Mathematics
		Abstract : Ricci flow is a way to evolve a Riemannian metric on a manifold.
		However, the precise long-time behavior of Ricci flow is not well understood.
		I will outline the Ricci flow strategy to prove the geometrization of three-dimensional manifolds, discuss the open questions concerning long-time behavior and give some results for type-III Ricci flow solutions.
	www.math.osu.edu /node/24230 (97 words)

	Bulletin of the American Mathematical Society
		Chow and S.-C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math.
		Hamilton, The Ricci flow on surfaces, Contemporary Mathematics 71 (1988), 237-261.
		Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom.
	e-math.ams.org /bull/1999-36-01/S0273-0979-99-00773-9/home.html (651 words)

	Science's Breakthrough of the Year 2006, a First for Math, but with Physics Connections
		As it turned out, the equations that defined the Ricci flow were strikingly similar to those that govern the propagation of heat, as embodied in the so-called heat equation.
		This similarity helped develop a theory of the Ricci flow based in part on the machinery that had worked for the heat equation, which physicists and mathematicians had studied since the early 1800s.
		In fact, physicists had discovered the Ricci flow independently from Hamilton in the early 1980s, in the context of quantum theory.
	www.aip.org /isns/reports/2006/021.html (854 words)

	International Press: Books: Collected Papers on Ricci Flow
		Ricci deformation of the metric on a Riemannian manifold, G.
		On the entropy estimate for the Ricci flow on compact 2-orbifolds, B.
		The Harnack estimate for the Ricci flow on a surface--Revisited, R.
	www.intlpress.com /books/RicciFlow.php (381 words)

	CTQM: Ricci flow and geometrization
		The Ricci flow is the geometric evolution equation
		For instance, in 1986 Hamilton proved that on a closed surface, the normalized Ricci flow converges to a metric of constant curvature.
		As an application one obtains the classification of closed 3-manifolds of positive Ricci curvature given by Hamilton in 1982.
	www.ctqm.au.dk /courses/RFG (271 words)

	AMS meeting
		First, I will explain the quasi-second-order fourth order modified Ricci flow for extremal-soliton metrics and will try to prove the convergence of the flow on compact almost homogeneous manifolds with two ends.
		The other one is a dynamical stability and it refers to a convergence of a Ricci flow starting at any metric in a neighbourhood of a considered Ricci flat metric.
		Abstract: The Ricci flow on homogeneous 3-manifolds are studied by J. Isenberg and M. Jackson.
	www.math.ucsb.edu /~yer/ams04.html (1138 words)

	Riemann Surface :Publications :Abstract : Ricci Flow (Site not responding. Last check: 2007-10-29)
		Then Ricci flow is introduced to conformally deform surfaces, such that the solution surfaces have constant Gaussian curvatures.
		The discrete Ricci flow is thoroughly explained, the existence of the solution, the exponential convergence, the variational energy, the Newton¡¯s method are explained.
		Finally, discrete Ricci flow is implemented based a common mesh library.
	www.cs.sunysb.edu /%7Egu/publications/abstracts/Ricci_flow.htm (191 words)

	Outline of GADGET seminar
		A highly biased survey of flow techniques in the search for canonical geometries
		The following outline is only tentative and will evolve in response to the interests of the participants.
		Ricci curvature and what it means for humanity
	www.ma.utexas.edu /~danknopf/Flow.html (38 words)

	Vortrag
		Abstract: The Ricci flow is a geometric evolution equation introduced by Richard Hamilton where one starts with a smooth Riemannian manifold and evolves its metric in the direction of its Ricci tensor.
		In the first part of my talk, I will motivate the Ricci flow equation and explain intuitively the possible behaviour of solutions and the formation of singularities.
		I will also explain Grisha Perelman's new gradient flow approach to the Ricci flow (from 2002) and give some ideas about how this approach can be used to finish Hamilton's program.
	www.math.unizh.ch /index.php?ve_mfs_sem_vor&key1=0&key2=219&key3=448&no_cache=1 (106 words)

	Centre Emile Borel - Ricci curvature and Ricci flow
		Centre Emile Borel - Ricci curvature and Ricci flow
		Ricci flow on manifolds of dimension four and higher
		Introduction to Kähler-Einstein geometry and Kähler-Ricci flow by N. Pali
	www-fourier.ujf-grenoble.fr /~besson/Borel (199 words)

	Clay Mathematics Institute
		Grigory Perelman was awarded a Fields Medal at the Madrid meeting on the International Congress of Mathematicians for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." A number of authors have written detailed expositions of Perelman's work.
		The book on Ricci Flow and Poincaré Conjecture by John Morgan and Gang Tian is available now.
		The 2007 Clay Research Summer School on Homogeneous Flows, Moduli Spaces and Arithmetic was held June 11 - July 6 at the Centro di Ricerca Matematica more....
	www.claymath.org (423 words)

	Lectures on the Ricci Flow - Cambridge University Press
		After describing the basic properties of, and intuition behind the Ricci flow, core elements of the theory are discussed such as consequences of various forms of maximum principle, issues related to existence theory, and basic properties of singularities in the flow.
		A detailed exposition of Perelman's entropy functionals is combined with a description of Cheeger-Gromov-Hamilton compactness of manifolds and flows to show how a 'tangent' flow can be extracted from a singular Ricci flow.
		Finally, all these threads are pulled together to give a modern proof of Hamilton's theorem that a closed three-dimensional manifold which carries a metric of positive Ricci curvature is a spherical space form.
	www.cambridge.org /us/catalogue/catalogue.asp?isbn=0521689473 (197 words)

	WSEAS -- Perelman (Site not responding. Last check: 2007-10-29)
		In recent years Hamilton had been investigating an approach to solve this problem using the Ricci Flow, an equation which evolves and morphs a manifold into a more understandable shape.
		According to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgery, can systematically excise singular regions as they develop, in a controlled way.
		Madrid, "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow".
	worldses.org /perelman (1968 words)

	[No title]
		These flows are characterized by the deformation of geometric objects such as metrics, mappings, and submanifolds by geometric quantities such as curvature and consist of partial differential equations of parabolic type.
		During this past decade the theory of formation of singularities was developed for Ricci flow and mean curvature flow, which has had a large impact on other geometric flows.
		The recent theoretical progress on geometric flows, especially on understanding weak solutions and singularities, together with the recent computational progress on geometric flows makes this an opportune time to hold a workshop which will bring together mathematicians working on the theoretical and numerical aspects of geometric flows.
	www.ipam.ucla.edu /programs/gf2004 (715 words)

	Poincaré, Ricci flow and Super String Theory
		Apparently the proof uses a modified (an extra element) Ricci flow and then the article says that the modification to the Ricci flow pops up in Super String Theory
		Secondly, I believe Perelman, when modifying the Ricci flow equation, added a term to the equation.
		There is a ricci like analogue for the renormalization "group" flow in some qft models....no idea if it comes up in string theory.
	www.physicsforums.com /showthread.php?p=253491 (299 words)

	The Clay Mathematics Institute 2003 Annual Meeting
		In his talk, Richard Hamilton of Columbia University will discuss his work on Ricci flow, a system of differential equations that are somewhat like those that govern heat flow in physics.
		However, the Ricci flow concerns not heat, but geometry, in a manner analogous to Einstein's description of gravity in General Relativity.
		It is therefore all the more remarkable that in 1982, Richard Hamilton used Ricci flow to show that a positively curved space of dimension three evolves to a final state of constant curvature.
	www.eurekalert.org /pub_releases/2003-11/fccc-tcm111203.php (508 words)

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