Showing papers on "Ricci flow published in 2003"
Posted Content•
TL;DR: In this article, the Ricci flow with surgeries was constructed, and a lower bound on the volume of maximal horns and the smoothness of solutions was established. But this lower bound was later shown to be unjustified and irrelevant for the other conclusions.
Abstract: This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print: the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.
1,200 citations
Posted Content•
TL;DR: In this article, the Ricci flow with surgery becomes extinct in finite time for any initial riemannian metric on a closed oriented three-manifold, whose prime decomposition contains no aspherical factors.
Abstract: Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal disk argument from 1999 paper by Richard Hamilton, and a regularization of the curve shortening flow, worked out by Altschuler and Grayson.
793 citations
Book•
01 Jan 2003
TL;DR: In this article, the Ricci curvature pinching problem has been studied in the context of Riemannian manifolds, and the authors present several possible approaches to solve it.
Abstract: 0. Vector fields, tensors 1. Tensor Riemannian duality, the connection and the curvature 2. The parallel transport 3. Absolute (Ricci) calculus, commutation formulas 4. Hodge and the Laplacian, Bochners technique 5. Generalizing Gauss-Bonnet, characteristic classes and C. GEOMETRIC MEASURE THEORY AND PSEUDO-HOLOMORPHIC B. HIGHER DIMENSIONS A.THE CASE OF SURFACES IN R3 C. various other bundles 3. Harmonic maps between Riemannian manifolds 4. Low dimensional Riemannian geometry 5. Some generalizations of Riemannian geometry 6. Gromov mm-spaces 7. Submanifolds B. Spinors A. Exterior differential forms (and some others) C. RICCI FLAT KAHLER AND HYPERKAHLER MANIFOLDS 6. Kahlerian manifolds (Kahler metrics) Chapter XI : SOME OTHER IMPORTANT TOPICS 1. Non compact manifolds 2. Bundles over Riemannian manifolds B. QUATERNIONIC-KAHLER MANIFOLDS A. G2 AND Spin(7) HIERRACHY : HOLONOMY GROUPS AND KAHLER MANIFOLDS 1. Definitions and philosophy 2. Examples 3. General structure theorems 4. The classification result 5. The rare cases b. on a given compact manifolds : closures Chapter X : GLOBAL PARALLEL TRANSPORT AND ANOTHER RIEMANNIAN a. collapsing C. THE CASE OF RICCI CURVATURE 12. Compactness, convergence results 13. The set of all Riemannian structures : collapsing B. MORE FINITENESS THEOREMS A. CHEEGERs FINITENESS THEOREM 11. Finiteness results of all Riemannian structures third part : Finiteness, compactness, collapsing and the space D. NEGATIVE VERSUS NONPOSITIVE CURVATURE 10. The negative side : Ricci curvature C. VOLUMES, FUNDAMENTAL GROUP B. QUASI-ISOMETRIES A. INTRODUCTION E. POSSIBLE APPROACHES, LOOKING FOR THE FUTURE 7. Ricci curvature : positive, nonnegative and just below 8. The positive side : scalar curvature 9. The negative side : sectional curvature D. POSITIVITY OF THE CURVATURE OPERATOR C. THE NON-COMPACT CASE B. HOMOLOGY TYPE AND THE FUNDAMENTAL GROUP A. THE KNOWN EXAMPLES 6. The positive side : sectional curvature second part : Curvature of a given sign1. Introduction 2. The positive pinching 3. Pinching around zero 4. The negative pinching 5. Ricci curvature pinching first part : Pinching problems b. hierarchy of curvaturesa. hopfs urge d. the set of constants, ricci flat metrics 18. The Yamabe problem Chapter IX : from curvature to topology 0. Some history and structure of the chapter c. moduli b. uniqueness a. existence b. homogeneous spaces and others 14. Examples from Analysis I : the evolution Ricci flow 15. Examples from Analysis II : the Kahler case 16. The sporadic examples 17. Around existence and uniqueness a. symmetric spaces THIRD PART : EINSTEIN MANIFOLDS 12. Hilberts variational principle and great hopes 13. The examples from the geometric hierachy 10. The case of Min R d/2 when d=4 11. Summing up questions on MinVol and Min(R) d/2 b. the simplicial volume of gromov a. using integral formulas d. cheeger-rong examples 9. Some cases where MinVol > 0 , Min Rd/2 > 0 c. nilmanifolds and the converse : almost flat manifolds b. wallachs type examples a. s1 fibrations and more examples MinDiam = 0 MinVol, MinDiam 5. Definitions 6. The case of surfaces 7. Generalities, compactness, finiteness and equivalence 8.Cases where MinVol = Min R d/2 = 0 and SECOND PART : WHICH METRIC IS THE LESS CURVED : Min R d/2 , FIRST PART: PURE GEOMETRIC FUNCTIONALS 1. Systolic quotients 2. Counting periodic geodesics 3. The embolic volume 4. Diameter/Injectivity riemannian metric on a given compact manifold ? 0. Introduction and a possible scheme of attack c. the structure on a given Sd and KPn 19. Inverse problems II : conjugacy of geodesics flows Chapter VIII : the search for distinguished metrics : what is the best b. bott and samelson theorems a. definitions and the need to be careful are closed 14. The case of negative curvature 15. The case of nonpositive curvature 16. Entropies on various space forms 17. From Osserman to Lohkamp 18. Inverse problems I : manifolds all of whose geodesics b. the various notions of
661 citations
TL;DR: In this article, the Bakry-Emery tensor is shown to be non-decreasing under a Riemannian manifold with fiber transport preserving measures up to constants.
Abstract: The Bakry-Emery tensor gives an analog of the Ricci tensor for a
Riemannian manifold with a smooth measure. We show that some of
the topological consequences of having a positive or nonnegative
Ricci tensor are also valid for the Bakry-Emery tensor. We show
that the Bakry-Emery tensor is nondecreasing under a Riemannian
submersion whose fiber transport preserves measures up to constants.
We give some
relations between the Bakry-Emery tensor and measured
Gromov-Hausdorff limits.
332 citations
TL;DR: In this paper, it was shown that Ricci flow converges exponentially fast to Thurston's circle packing on surfaces, and a new proof of the existence of the circle packing theorem is obtained.
Abstract: We show that the analogue of Hamilton’s Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston’s circle packing on surfaces. As a consequence, a new proof of Thurston’s existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.
325 citations
TL;DR: In this article, the authors constructed new families of Kahler-Ricci solitons on complex line bundles over ℂℙn−1, n ≥ 2, and exhibited a noncompact Ricci flow that shrinks smoothly and self-similarly for t 0.
Abstract: We construct new families of Kahler-Ricci solitons on complex line bundles over ℂℙn−1, n ≥ 2. Among these are examples whose initial or final condition is equal to a metric cone ℂn/ℤk. We exhibit a noncompact Ricci flow that shrinks smoothly and self-similarly for t 0; this evolution is smooth in space-time except at a single point, at which there is a blowdown of a ℂℙn−1. We also construct certain shrinking solitons with orbifold point singularities.
300 citations
TL;DR: In this article, it was shown that the induced Ricci tensor of a totally umbilical light-like submanifold is symmetric if and only if its screen distribution is integrable.
Abstract: This paper provides new results on a class of totally umbilical lightlike submanifolds in semi-Riemannian manifolds of constant curvature. We prove that the induced Ricci tensor of any such submanifold is symmetric if and only if its screen distribution is integrable.
103 citations
TL;DR: A conformal invariant in dimension four is 1 2 Q (AR R2+3 IRicI2) as discussed by the authors, where R denotes the scalar curvature and Ric the Ricci tensor.
Abstract: An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, e.g. a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser [20] proved that for every positive function f on S2 satisfying f (x) = f(-x) for all x E S2 there exists a conformal metric on S2 whose Gauss curvature is equal to f. A natural conformal invariant in dimension four is 1 2 Q (AR R2+ 3 IRicI2), where R denotes the scalar curvature and Ric the Ricci tensor. This formula can also be written in the form
83 citations
Posted Content•
TL;DR: In this paper, the extinction time of the Ricci flow on the 3-sphere has been studied and lower bounds for the area of a min-max surface of a 3-manifold have been shown.
Abstract: In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question of Grisha Perelman asked to the first author at a dinner in New York City on April 25th of 2003. His question was ``what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In particular does the flow become extinct in finite time?'' He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min--max argument where the minimal of all maximal slices of sweep-outs is a minimal surface; see, for instance, [CD]. The idea is then to look at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold and as we will see for certain 3-manifolds (those, like the 3-sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see [Pe1]) answering his original question about finite extinction time. However, even after the appearance of his paper, then we still think that our slightly different approach may be of interest. In part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke and thus our approach is in some respects technically easier.
79 citations
Posted Content•
TL;DR: In this paper, it was shown that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.
Abstract: Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.
78 citations
Posted Content•
Journal Article•
TL;DR: In this paper, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained in a 3D trans-Sasakian manifold in cases of the manifold being η-Einstein or satisfying R (X, Y) · S = 0.
Abstract: In a 3-dimensional trans-Sasakian manifold, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained. In particular, expressions for Ricci tensor are obtained in a 3-dimensional trans-Sasakian manifold in cases of the manifold being η-Einstein or satisfying R (X, Y) · S = 0.
Book•
01 Jan 2003
TL;DR: The Ricci flow is a hot topic at the forefront of mathematics research as discussed by the authors, and a selection of papers on the Riemannian Ricci Flow is intended both for the graduate student or researcher unfamiliar with the Ricci Flows and for geometers already familiar to the Flows.
Abstract: The Ricci flow is a hot topic at the forefront of mathematics research. This selection of papers on the Riemannian Ricci flow is intended both for the graduate student or researcher unfamiliar with the Ricci flow and for geometers already familiar to the Ricci flow.
TL;DR: In this paper, the Ricci flow is shown to converge in the Gromov-Hausdor sense to a space which is not a manifold but only a metric space.
Abstract: Consider a sequence of pointed n{dimensional complete Riemannian manifolds f(Mi;gi (t );Oi )g such that t2 [0;T] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n{dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov{Hausdor sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.
TL;DR: In this article, a sharp linear trace Li-Yau-Hamilton inequality for Kahler-Ricci flow is proved and applied to the Liouville properties of the plurisubharmonic functions on complete Kahler manifolds with bounded nonnegative holomorphic bisectional curvature.
Abstract: In this paper, a sharp linear trace Li-Yau-Hamilton inequality for Kahler-Ricci flow is proved. The new inequality extends the previous trace Harnack inequality obtained by H.-D. Cao. We also establish sharp gradient estimates for the positive solution of the time-dependent heat equation for some cases. Finally, we apply this new linear trace Li-Yau-Hamilton inequality to study the Liouville properties of the plurisubharmonic functions on complete Kahler manifolds with bounded nonnegative holomorphic bisectional curvature.
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TL;DR: In this paper, it was shown that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect, and this was generalized to show that in manifolds with positive curvature, in the integral sense, two minimal surfaces must be close to each other.
Abstract: In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci cur- vature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.
TL;DR: In this paper, the Ricci tensor of a slant submanifold in a complex space form is estimated in terms of the main extrinsic invariant, namely the squared mean curvature.
Abstract: B.-Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. The Lagrangian version of this inequality was proved by the same author.
ewline In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold $M$ in a complex space form $\widetilde M(4c)$, in terms of the main extrinsic invariant, namely the squared mean curvature. If, in particular, $M$ is a Kaehlerian slant submanifold which satisfies the equality case identically, then it is minimal.
TL;DR: In this paper, a new method for the determination of Ricci Collineations (RC) and Matter Collineation (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form with the metric, is presented.
Abstract: A new method is presented for the determination of Ricci Collineations (RC) and Matter Collineations (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form with the metric. This method reduces the problem of finding the RCs and the MCs to that of determining the KVs whereas at the same time uses already known results on the motions of the metric. We employ this method to determine all hypersurface homogeneous locally rotationally symmetric spacetimes, which admit proper RCs and MCs. We also give the corresponding collineation vectors.
Posted Content•
TL;DR: In contrast to Chen and Tian as mentioned in this paper, we do not rely on the exsitence of the Harnack inequality for the Ricci flow on a compact Kahler manifold with positive bisectional curvature.
Abstract: We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact Kahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of Kahler-Einstein metrics on $M^n$, but instead on the first author's Harnack inequality for the Kahler-Ricc flow, and a very recent local injectivity radius estimate of Perelman for the Ricci flow.
TL;DR: Cao et al. as discussed by the authors showed that the curvature of the Ricci flow on a compact Kahler manifold with positive bisectional curvature can be estimated based on the Harnack inequality.
TL;DR: The conformal Ricci flow equation as mentioned in this paper is a variation of the classical Ricci equation that modifies the unit volume constraint of that equation to a scalar curvature constraint, which is analogous to the incompressible Navier-Stokes equations of fluid mechanics.
Abstract: We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equations are analogous to the incompressible Navier-Stokes equations of fluid mechanics inasmuch as a conformal pressure arises as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points are Einstein metrics with a negative Einstein constant and the conformal pressue is shown to be zero at an equilibrium point and strictly positive otherwise. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. That the constraint force does not lose derivatives is exactly analogous to the fact that the real physical pressure force that occurs in the Navier-Stokes equations is a bounded function of the velocity. Using a nonlinear Trotter product formula, existence and uniqueness of solutions to the conformal Ricci flow equations is proven. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.
TL;DR: In this paper, a complete classification of cylindrically symmetric static Lorentzian manifolds according to their Ricci collineations (RCs) is provided, and the Lie algebras of RCs for the non-degenerate Ricci tensor have dimensions 3 to 10, excluding 8 and 9.
Abstract: A complete classification of cylindrically symmetric static Lorentzian manifolds according to their Ricci collineations (RCs) is provided. The Lie algebras of RCs for the non-degenerate Ricci tensor have dimensions 3 to 10, excluding 8 and 9. For the degenerate tensor the algebra is mostly but not always infinite dimensional; there are cases of 10-, 5-, 4- and 3-dimensional algebras. The RCs are compared with the Killing vectors (KVs) and homothetic motions (HMs). The (non-linear) constraints corresponding to the Lie algebras are solved to construct examples which include some exact solutions admitting proper RCs. Their physical interpretation is given. The classification of plane symmetric static spacetimes emerges as a special case of this classification when the cylinder is unfolded.
TL;DR: In this article, a modified Ricci flow analysis of the Thurston conjecture is presented, in which the topology of the geometry is encoded in the parameters of an underlying field theory.
Abstract: We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can construct modified Ricci flow equations in which the topology of the geometry is encoded in the parameters of an underlying field theory.
TL;DR: In this article, conformally flat, semi-Riemannian manifolds with curvature tensors and Ricci operators were classified. But the Ricci operator has pure imaginary eigenvalues.
Abstract: We classify the conformally flat, semi-Riemannian manifolds satisfying $R(X,Y) \cdot Q = 0$, where $R$ and $Q$ are the curvature tensor and the Ricci operator, respectively. As the cases which do not occur in the Riemannian manifolds, the Ricci operator $Q$ has pure imaginary eigenvalues or it satisfies $Q^2 = 0$.
Posted Content•
TL;DR: In this article, the formation of singularities in Ricci flows was studied using numerical techniques and the authors found critical behavior at the threshold of singularity formation in the case of S 2 neck pinching.
Abstract: We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S 3 with varying amounts of S 2 neck pinching, we find critical behavior at the threshold of singularity formation
TL;DR: In this paper, the vanishing of the first Betti number on compact Hermitian-Weyl manifold admits a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry.
Abstract: We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian–Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that every such manifold has first Betti number b 1 =1 and Hodge numbers h p ,0 =0 for p >0, h 0,1 =1, h 0, q =0 for q >1.
TL;DR: In this article, it was shown that the Riemann compatibility theorem still holds for C 1-metric tensors in a distributional sense and that immersion is induced by immersion.
Abstract: A classical theorem in differential geometry asserts that if a C2-metric tensor satisfies the Riemann compatibility conditions, then it is induced by an immersion. We prove that this theorem still holds true for C1-metric tensors satisfying the Riemann compatibility conditions in a distributional sense.
TL;DR: Camci et al. as discussed by the authors derived general expressions for the components of the Ricci collineation vector and related constraints for all spacetime manifolds admitting symmetries larger than so(3).
Abstract: General expressions for the components of the Ricci collineation vector are derived and the related constraints are obtained. These constraints are then solved to obtain Ricci collineations and the related constraints on the Ricci tensor components for all spacetime manifolds (degenerate or non-degenerate, diagonal or non-diagonal) admitting symmetries larger than so(3) and already known results are recovered. A complete solution is achieved for the spacetime manifolds admitting so(3) as the maximal symmetry group with non-degenerate and non diagonal Ricci tensor components. It is interesting to point out that there appear cases with finite number of Ricci collineations although the Ricci tensor is degenerate and also the cases with infinitely many Ricci collineations even in the case of non-degenerate Ricci tensor. Interestingly, it is found that the spacetime manifolds with so(3) as maximal symmetry group may admit two extra proper Ricci collineations, although they do not admit a G5 as the maximal symmetry group. Examples are provided which show and clarify some comments made by Camci et al. [Camci, U., and Branes, A. (2002). Class. Quantum Grav.19, 393–404]. Theorems are proved which correct the earlier claims made in [Carot, J., Nunez, L. A., and Percoco, U. (1997). Gen. Relativ. Gravit.29, 1223–1237; Contreras, G., Nunez, L. A., and Percolo, U. (2000). Gen. Relativ. Gravit.32, 285–294].
TL;DR: In this article, the distance to halfway points of minimal geodesics in terms of the distantce to end points on some types of Riemannian manifolds was established.
Abstract: The authors establish some uniform estimates for the distance to halfway points of minimal geodesics in terms of the distantce to end points on some types of Riemannian manifolds, and then prove some theorems about the finite generation of fundamental group of Riemannian manifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.