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The entropy formula for the Ricci flow and its geometric applications Grisha Perelman ∗November 20,2007Introduction 1.The Ricci flow equation,introduced by Richard Hamilton [H 1],is the evolution equation d ∗St.Petersburg branch of Steklov Mathematical Institute,Fontanka 27,St.Petersburg
191011,Russia.Email:perelman@pdmi.ras.ru or perelman@math.sunysb.edu ;I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992,to the SUNY at Stony Brook in the Spring of 1993,and to the UC at Berkeley as a Miller Fellow in 1993-95.I’d like to thank everyone who worked to make tho opportunities available to me.剑桥商务英语官网
1scarra
in dimension four converge,modulo scaling,to metrics of constant positive
curvature.
Without assumptions on curvature the long time behavior of the metric
evolving by Ricciflow may be more complicated.In particular,as t ap-
proaches somefinite time T,the curvatures may become arbitrarily large in some region while staying bounded in its complement.In such a ca,it is
uful to look at the blow up of the solution for t clo to T at a point where curvature is large(the time is scaled with the same factor as the metric ten-
sor).Hamilton[H9]proved a convergence theorem,which implies that a
subquence of such scalings smoothly converges(modulo diffeomorphisms) to a complete solution to the Ricciflow whenever the curvatures of the scaled
metrics are uniformly bounded(on some time interval),and their injectivity radii at the origin are bounded away from zero;moreover,if the size of the
scaled time interval goes to infinity,then the limit solution is ancient,that
is defined on a time interval of the form(−∞,T).In general it may be hard to analyze an arbitrary ancient solution.However,Ivey[I]and Hamilton
[H4]proved that in dimension three,at the points where scalar curvature
onestepatatimeis large,the negative part of the curvature tensor is small compared to the scalar curvature,and therefore the blow-up limits have necessarily nonneg-
ative ctional curvature.On the other hand,Hamilton[H3]discovered a remarkable property of solutions with nonnegative curvature operator in ar-
bitrary dimension,called a differential Harnack inequality,which allows,in
particular,to compare the curvatures of the solution at different points and different times.The results lead Hamilton to certain conjectures on the
游戏伙伴structure of the blow-up limits in dimension three,e[H4,§26];the prent
work confirms them.
The most natural way of forming a singularity infinite time is by pinching
an(almost)round cylindrical neck.In this ca it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries,and
then to continue running the Ricciflow.The exact procedure was described
by Hamilton[H5]in the ca of four-manifolds,satisfying certain curvature assumptions.He also expresd the hope that a similar procedure would
work in the three dimensional ca,without any a priory assumptions,and that afterfinite number of surgeries,the Ricciflow would exist for all time
t→∞,and be nonsingular,in the n that the normalized curvatures ˜Rm(x,t)=tRm(x,t)would stay bounded.The topology of such nonsingular solutions was described by Hamilton[H6]to the extent sufficient to make
sure that no counterexample to the Thurston geometrization conjecture can
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occur among them.Thus,the implementation of Hamilton program would imply the geometrization conjecture for clod three-manifolds.
In this paper we carry out some details of Hamilton program.The more technically complicated arguments,related to the surgery,will be discusd elwhere.We have not been able to confirm Hamilton’s hope that the so-lution that exists for all time t→∞necessarily has bounded normalized curvature;still we are able to show that the region where this does not hold is locally collapd with curvature bounded below;by our earlier(partly unpublished)work this is enough for topological conclusions.
Our prent work has also some applications to the Hamilton-Tian con-jecture concerning K¨a hler-Ricciflow on K¨a hler manifolds with positivefirst Chern class;the will be discusd in a parate paper.
2.The Ricciflow has also been discusd in quantumfield theory,as an ap-proximation to the renorma
lization group(RG)flow for the two-dimensional nonlinearσ-model,e[Gaw,§3]and references therein.While my back-ground in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RGflow.
In this picture,t corresponds to the scale parameter;the larger is t,the larger is the distance scale and the smaller is the energy scale;to compute something on a lower energy scale one has to average the contributions of the degrees of freedom,corresponding to the higher energy scale.In other words,decreasing of t should correspond to looking at our Space through a microscope with higher resolution,where Space is now described not by some(riemannian or any other)metric,but by an hierarchy of riemannian metrics,connected by the Ricciflow equation.Note that we have a paradox here:the regions that appear to be far from each other at larger distance scale may become clo at smaller distance scale;moreover,if we allow Ricci flow through singularities,the regions that are in different connected compo-nents at larger distance scale may become neighboring when viewed through microscope.explicit
Anyway,this connection between the Ricciflow and the RGflow sug-gests that Ricciflow must be gradient-like;the prent work confirms this expectation.
crumble3.The paper is organized as follows.In§1we explain why Ricciflow can be regarded as a gradientflow.In§2,3we prove that Ricciflow,considered as a dynamical system on the space of riemannian metrics modulo diffeomor-phisms and scaling,has no nontrivial periodic orbits.The easy(and known)
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ca of metrics with negative minimum of scalar curvature is treated in§2; the other ca is dealt with in§3,using our main monotonicity formula(3.4) and the Gaussian logarithmic Sobolev inequality,due to L.Gross.In§4we apply our monotonicity formula to prove that for a smooth solution on a finite time interval,the injectivity radius at each point is controlled by the curvatures at nearby points.This result removes the major stumbling block in Hamilton’s approach to geometrization.In§5we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical enmble.In§6we try to interpret the formal expressions,arising in the study of the Ricciflow,as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension.The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricciflow.This for-mula is rigorously proved in§7;it may be more uful than thefirst one in local considerations.In§8it is applied to obtain t
he injectivity radius control under somewhat different assumptions than in§4.In§9we consider one more way to localize the original monotonicity formula,this time using the differential Harnack inequality for the solutions of the conjugate heat equation,in the spirit of Li-Yau and Hamilton.The technique of§9and the logarithmic Sobolev inequality are then ud in§10to show that Ricciflow can not quickly turn an almost euclidean region into a very curved one,no matter what happens far away.The results of ctions1through10require no dimensional or curvature restrictions,and are not immediately related to Hamilton program for geometrization of three manifolds.
The work on details of this program starts in§11,where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits offinite time singularities(they must satisfy a certain noncollaps-ing assumption,which,in the interpretation of§5,corresponds to having bounded entropy).Then in§12we describe the regions of high curvature under the assumption of almost nonnegative curvature,which is guaranteed to hold by the Hamilton and Ivey result,mentioned above.We also prove, under the same assumption,some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball.Finally,in§13we give a brief sketch of the proof of geometrization conjecture.
The subctions marked by*contain historical remarks and references. See also[Cao-C]for a relativel
y recent survey on the Ricciflow.
告示范文4
1Ricciflow as a gradientflow
1.1.Consider the functional F= M(R+|∇f|2)e−f dV for a riemannian metric g ij and a function f on a clod manifold M.Itsfirst variation can be expresd as follows:
δF(v ij,h)= M e−f[−△v+∇i∇j v ij−R ij v ij
−v ij∇i f∇j f+2<∇f,∇h>+(R+|∇f|2)(v/2−h)]
= M e−f[−v ij(R ij+∇i∇j f)+(v/2−h)(2△f−|∇f|2+R)], whereδg ij=v ij,δf=h,v=g ij v ij.Notice that v/2−h vanishes identically iffthe measure dm=e−f dV is keptfixed.Therefore,the symmetric tensor −(R ij+∇i∇j f)is the L2gradient of the functional F m= M(R+|∇f|2)dm, where now f denotes log(dV/dm).Thus given a measure m,we may consider the gradientflow(g ij)t=−2(R ij+∇i∇j f)for F m.For general m thisflow may not exist even for short time;however,when it exists,it is just the Ricciflow,modified by a diffeomorphism.The remarkable fact here is that different choices of m lead to the sameflow,up to a diffeomorphism;that is, the choice of m is analogous to the choice of gauge.
1.2Proposition.Suppo that the gradientflow for F m exists for t∈[0,T]. Then at t=0we have F m≤n
Now we compute蹚浑水
F t≥2n( (R+△f)e−f dV)2=2
