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the ricci flow has been introduced by r. hamilton @xcite with the goal of providing an analytic approach to thurston s geometrization conjecture for three - manifolds @xcite@xmath0 . inspired by the theory of harmonic maps , he considered the geometric evolution equation obtained when one evolves a riemannian metric @xmath1 , on a three - manifold @xmath2 , in the direction of its ricci tensor@xcite @xmath3 , _ i.e. _ @xmath4 in recent years , this geometric flow has gained extreme popularity thanks to the revolutionary breakthroughs of g. perelman @xcite@xmath0 , who , taking the whole subject by storm , has brought to completion hamilton s approach to thurston s conjecture . the prominent themes recurring in hamilton s and perelman s works converge to a proof that the ricci flow , coupled to topological surgery , provides a natural technique for factorizing and uniformizing a three - dimensional riemannian manifold @xmath5 into locally homogeneous geometries . this is a result of vast potential use also in theoretical physics , where the ricci flow often appears in disguise as a natural real - space renormalization group flow . non - linear @xmath6-model theory , describing quantum strings propagating in a background spacetime , affords the standard case study in such a setting @xmath0 . another paradigmatical , perhaps even more direct , application occurs in relativistic cosmology @xcite@xmath0 , ( for a series of recent results see also @xcite and the references cited therein ) . this will be related to the main topic of this talk , and to motivate our interest in it , let us recall that homogeneous and isotropic solutions of einstein s laws of gravity ( the friedman lemaitre  robertson  walker ( flrw ) spacetimes ) do not account for inhomogeneities in the universe . the question whether they do _ on average _ is an issue @xcite that is the subject of considerable debate especially in the recent literature ( see @xcite and follow  up references ; comprehensive lists may be found in @xcite and @xcite ) .    in any case , a member of the family of flrw cosmologies ( the so  called concordance model that is characterized by a dominating cosmological constant in a spatially flat universe model ) provides a successful _ fitting model _ to a large number of observational data , and the generally held view is that the spatial sections of flrw spacetimes indeed describe the _ physical universe _ on a sufficiently large averaging scale . this raises an interesting problem in mathematical cosmology : devise a way to explicitly construct a constant  curvature metric out of a _ scale  dependent _ inhomogeneous distribution of matter and spatial curvature . it is in such a framework that one makes the basic observation that the ricci flow ( [ mflow ] ) and its linearization , provide a natural technique@xcite for deforming , and under suitable conditions smoothing , the geometrical part of scale  dependent cosmological initial data sets . moreover , by taking advantage of some elementary aspects of perelman s results , this technique also provides a natural and unique way for deforming , along the ricci flow , the matter distribution . the expectation is that in this way we can define a deformation of cosmological initial data sets into a one - parameter family of initial data whose time evolution , along the evolutive part of einstein s equations , describe the ricci flow deformation of a cosmological spacetime . to set notation , we emphasize that throughout the paper we shall consider a smooth three - dimensional manifold @xmath2 , which we assume to be closed and without boundary . we let @xmath7 and @xmath8 be the space of smooth functions and of smooth @xmath9tensor fields on @xmath2 , respectively . we shall denote by @xmath10 the group of smooth diffeomorphisms of @xmath2 , and by @xmath11 the space of all smooth riemannian metrics over @xmath12 . the tangent space , @xmath13 , to @xmath11 at @xmath14 can be naturally identified with the space of symmetric bilinear forms @xmath15 over @xmath2 . the hypothesis of smoothness has been made for simplicity . results similar to those described below , can be obtained for initial data sets with finite holder or sobolev differentiability . in such a framework , let us recall that a collection of fields @xmath16 , @xmath17 , @xmath18 , @xmath19 , defined over the three - manifold @xmath2 , characterizes  a set @xmath20 , of physical cosmological initial data for einstein equations if and only if the _ matter fields _ @xmath21 verify the weak energy condition @xmath22 , the dominant energy condition @xmath23 , and their coupling with the geometric fields @xmath24 is such as to satisfy the hamiltonian and divergence constraints ; we adopt the summation convention . the nabla operator denotes covariant derivative with respect to the 3metric . the units are such that @xmath25 . ] : @xmath26 here @xmath27 is the cosmological constant , @xmath28 , and @xmath29  is the scalar curvature of the riemannian metric @xmath1 . if such a set of admissible data is propagated according to the evolutive part of einstein s equations , then the symmetric tensor field @xmath30 can be interpreted as the extrinsic curvature and @xmath31 as the mean curvature of the embedding @xmath32 of @xmath33 in the spacetime @xmath34 resulting from the evolution of @xmath35 , whereas @xmath36 and @xmath37 are , respectively , identified with the mass density and the momentum density of the material self gravitating sources on @xmath33 . the averaging procedure described in @xcite is based on a smooth deformation of the physical initial data @xmath35 into a one - parameter family of initial data sets @xmath38 with @xmath39 being a parameter characterizing the averaging scale . the general idea is to construct the flow ( [ betadata ] ) in such a way as to represent , as @xmath40 increases , a scale - dependent averaging of @xmath41 , and  under suitable hypotheses  reducing it to a constant - curvature initial data set @xmath42 where @xmath43 is a constant curvature metric on @xmath2 , @xmath44 is the ( spatially constant ) trace of the extrinsic curvature ( related to the hubble parameter ) , and @xmath45 is the averaged matter density . under the heading of such a general strategy it is easy to figure out the reasons for an important role played by the ricci flow and its linearization . as we shall recall shortly , they are natural geometrical flows always defining a non - trivial deformation of the metric @xmath1 and of the extrinsic curvature @xmath30 . moreover , when global , they posses remarkable smoothing properties . for instance , if , for @xmath46 , the scalar curvature @xmath47 of @xmath33 is @xmath48 , and if there exist positive constants @xmath49 , @xmath50 , @xmath51 , not depending on @xmath40 , such that @xmath52,and @xmath53,where @xmath54is the trace free part of the ricci tensor , then @xcite the solutions @xmath55 of the ( volume  normalized ) ricci flow and its linearization exist for all @xmath56 , and the pair @xmath57 , uniformly converges , when @xmath58 , to @xmath59 where @xmath43 is a metric with constant positive sectional curvature , and @xmath60 is some vector field on @xmath2 , possibly depending on @xmath61 . the flow @xmath62 , describing a motion by diffeomorphisms over a constant curvature manifold , can be thought of as representing the smoothing of the geometrical part of an initial data set @xmath35 . + it is useful to keep in mind what we can expect and what we can not expect out of such a ricci  flow deformation of cosmological initial data set . let us start by remarking that the family of data ( [ betadata ] ) will correspond to the initial data for physical spacetimes iff the constraints ( [ constraint1 ] ) and ( [ constraints ] ) hold throughout the @xmath40dependent deformation . this is a very strong requirement and , if we have a technically consistent way @xmath63 of deforming the matter distribution and the geometrical data , then the most natural way of implementing the constraints is to use them to define scale dependent backreaction fields @xmath64 , @xmath65 describing the non  linear interaction between matter averaging and geometrical averaging , _ i.e. _ ,    @xmath66\;,\label{hamro}\\ \psi _ { a } ( \beta ) & \doteq & j_{a}(\beta ) -(8\pi \,g)^{-1 } \left[\nabla _ { b}k_{\;\,a}^{b}(\beta ) -\nabla _ { a}k(\beta )    \right]\;. \label{divj}\end{aligned}\ ] ]    to illustrate how this strategy works , let us concentrate , in this talk , on the characterization of the scalar field @xmath64 , providing the backreaction between matter and geometrical averaging . the covector field @xmath67 can , in principle , be controlled by the action of a @xmath40dependent diffeomorphism . however , its analysis requires a subtle interplay with the kinematics of spacetime foliations @xcite@xmath0 , ( _ i.e. _ , how we deal with the lapse function and with the shift vector field in the framework of perelman s approach ) , and will be discussed elsewhere , ( for a pre perelman approach to this issue see @xcite)@xmath0 . + let us start by observing that the matter averaging flow @xmath68 must comply with the preservation of the physical matter content @xmath69 and must be explicitly coupled to the scale of geometrical averaging . in other words , if , for some fixed @xmath70 , we consider that part of the matter distribution @xmath71 which is localized in a given region @xmath72 of size @xmath73 , then we should be able to tell from which localized distribution @xmath74 , at @xmath46 , the selected matter content @xmath75 has evolved . a natural answer to these requirements is provided by perelman s backward localization@xcite of probability measures on ricci evolving manifolds . the idea is to probe the ricci flow with a probability measure whose dynamics can localize the regions of the manifold @xmath76 of geometric interest . this is achieved by considering , along the solution @xmath77 of ( [ mflow ] ) , a @xmath40dependent mapping @xmath78 , in terms of which one constructs on @xmath79 the measure @xmath80 , where @xmath81 is a scale parameter chosen in such a way as to normalize @xmath82 according to the so  called _ perelman s coupling_:@xmath83 . it is easily verified that this is preserved in form along the ricci flow ( [ mflow ] ) , if the mapping @xmath84 and the scale parameter @xmath85 are evolved backward in time @xmath86 according to the coupled flows defined by @xmath87 where @xmath88 is the laplacian with respect to the metric @xmath77 , and @xmath89 , @xmath90 are given ( final ) data . in this connection , note that the equation for @xmath84 is a backward heat equation , and as such the forward evolution @xmath91 is an ill - posed problem . a direct way for circumventing such a difficulty is to interpret ( [ ourp ] ) according to the following two - steps prescription : ( i ) evolve the metric @xmath92 , say up to some @xmath93 , according to the ricci flow , ( if the flow is global we may let @xmath94 ) ; ( ii ) on the ricci evolved riemannian manifold @xmath95 so obtained , select a function @xmath96 and the corresponding scale parameter @xmath97 ,   and evolve them , backward in @xmath40 , according to ( [ ourp ] ) . with these preliminary remarks along the way , let us characterize the various steps involved in constructing the flow ( [ betadata ] ) . ( for ease of exposition , we refer to the standard unnormalized flow ; volume normalization can be enforced by a reparametrization of the deformation parameter ) . +    ( _ geometrical data deformation _ ) given an initial data set @xmath99 for a cosmological spacetime @xmath100 , the ricci flow deformation of its geometrical part @xmath101 is defined by the flow @xmath102 , @xmath103 provided by the ( weakly ) parabolic initial value problem ( the ricci flow , proper ) @xmath104<\infty     \;,$% \end{tabular } \right . \label{fflow}\ ] ] and by its linearization , which , by suitably fixing the action of the diffeomorphism group @xmath10 , takes the form of the parabolic initial value problem @xcite @xmath105 where @xmath106 denotes the lichnerowicz derham laplacian @xmath107 acting on symmetric bilinear forms @xcite@xmath0 . ( _ localization of the deformed data _ ) the geometrical deformation @xmath102 is controlled by the backward localizing flow @xmath108 ,  @xmath109 , defined by the backward heat kernels for the conjugate @xcite parabolic initial value problem associated with the @xmath110dependent laplace - beltrami and lichnerowicz operators @xmath111 . the operator @xmath112 , when acting on ( bi)scalars , reduces to @xmath113 . thus , we can characterize both these kernels in a compact form as solutions of @xmath114 where @xmath115 $ ] , @xmath116 denotes the lichnerowicz derham laplacian with respect to the variable @xmath117 , the heat kernels @xmath118 and @xmath119 are smooth sections of @xmath120 and @xmath121 , respectively , and finally , @xmath122 is the dirac @xmath123tensorial measure , ( @xmath124 ) on @xmath125 . the dirac initial condition is understood in the distributional sense , _ i.e. _ , @xmath126 , for any smooth symmetric bilinear form with compact support @xmath127 , and where the limit is meant in the uniform norm on @xmath128 .    note that heat kernels for generalized laplacians , such as @xmath129 , ( smoothly ) depending on a one  parameter family of metrics @xmath130 , @xmath131 , are briefly dealt with in @xcite@xmath0 . the delicate setting where the parameter dependence is , as in our case , identified with the parabolic time driving the diffusion of the kernel , is discussed in @xcite@xmath0 , ( see appendix a ,  7 for a characterization of the parametrix of the heat kernel in such a case ) , and in @xcite@xmath0 . strictly speaking , in all these works , the analysis is confined to the scalar laplacian , possibly with a potential term , but the theory readily extends to generalized laplacians , always under the assumption that the metric @xmath77 is smooth as @xmath132 . finally , the kernels for @xmath133 and @xmath134 can both be normalized , along the ricci flow , over the round ( collapsing ) 3sphere @xmath135 . let @xmath145 be the ricci flow deformation , on @xmath146 $ ] , of the data @xmath147 as defined above . assume that the underlying ricci flow @xmath148 is of bounded geometry , and let @xmath149 and @xmath118 be the ( backward ) heat kernels ricci - flow conjugated to the laplace  beltrami and to the lichnerowicz derham operator , respectively . then , for all @xmath150 , @xmath151\,d\mu _ { g(x,\eta ) } \ ; , \label{grepres0}\ ] ] @xmath152 @xmath153        where @xmath156 @xmath157 is the parallel transport operator associated with @xmath158 , @xmath159 is the distance function in @xmath160 , and @xmath161 } ^{ab}_{i'k'}(y , x;\eta ) $ ] are smooth sections @xmath162 , ( depending on the geometry of @xmath158 ) , characterizing the asymptotics of the heat kernel @xmath118 . with an obvious adaptation , such an asymptotic behavior also extends to @xmath163 and @xmath164 .    with these results it is rather straightforward to provide useful characterizations of the ( intrinsic ) backreaction field @xmath165 . for illustrative purposes , let us assume that @xmath166 and @xmath167 , then from ( [ hamro ] ) and the above proposition we get that @xmath168\;.\nonumber\end{aligned}\ ] ] if we further assume that the hamiltonian constraint holds at the _ fixed _ observative scale @xmath139 and that @xmath169 $ ] , ( _ i.e. _ , curvature fluctuates around the _ constant curvature background _ @xmath170 ) , then @xmath171 from which it follows that the intrinsic backreaction field @xmath165 is generated by curvature fluctuations around the given background , as expected . m.c . would like to dedicate this paper to t. ruggeri on occasion of his .. th birthday ; he also thanks the organizers for a very stimulating and enjoyable meeting . t.b . acknowledges hospitality at and support from the university of pavia during a working visit . research supported in part by prin grant @xmath172 . i. bakas , _ geometric flows and ( some of ) their physical applications _ , avh conference advances in physics and astrophysics of the 21st century , 6 - 11 september 2005 , varna , bulgaria , arxiv : hep - th/0511057 ( 2005 ) .      t. buchert , _ on average properties of inhomogeneous fluids in general relativity : 2 . perfect fluid cosmologies _ , gen . * 33 * , 1381 ( 2001 ) . t. buchert , _ dark energy from structure : a status report _ ( special issue on dark energy ) , in press ; arxiv:0707.2153 ( 2007 ) . t. buchert and m. carfora , _ regional averaging and scaling in relativistic cosmology _ , class . * 19 * , 6109 - 6145 ( 2002 ) . t. buchert and m. carfora , _ cosmological parameters are dressed _ , phys . .  lett . * 90 * , 31101 - 1 - 4 ( 2003 ) . m. carfora , a. marzuoli , _ model geometries in the space of riemannian structures and hamilton s flow _ quantum grav . 5 , 659 - 693 ( 1988 ) . b. chow , s - c . chu , d. glickenstein , c. guenther , j. isenberg , t. ivey , d. knopf , p. lu , l. ni , _ the ricci flow : techniques and applications : part i : geometric aspects _ , math . surveys and monographs vol . * 135 * , am . ( 2007 ) . ellis and t. buchert , _ the universe seen at different scales _ , phys . . a ( einstein special issue ) * 347 * , 38 ( 2005 ) . d. h. friedan , _ nonlinear models in @xmath173 dimensions _ physics * 163 * , no . 2 , 318419 ( 1985 ) .
ricci flow deformation of cosmological initial data sets in general relativity is a technique for generating families of initial data sets which potentially would allow to interpolate between distinct spacetimes . 
 this idea has been around since the appearance of the ricci flow on the scene , but it has been difficult to turn it into a sound mathematical procedure . in this expository talk we illustrate , how perelman s recent results in ricci flow theory can considerably improve on such a situation . from a physical point of view 
 this analysis can be related to the issue of finding a constant  curvature template spacetime for the inhomogeneous universe , relevant to the interpretation of observational data and , hence , bears relevance to the dark energy and dark matter debates . 
 these techniques provide control on curvature fluctuations ( intrinsic backreaction terms ) in their relation to the averaged matter distribution .