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1 Oberwolfach Preprints OWP Gerhar Huisken Geometric Flows an 3-Manifols Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN

2 Oberwolfach Preprints (OWP) Starting in 2007, the MFO publishes a preprint series which mainly contains research results relate to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs- Programme (RiP) an the Oberwolfach-Leibniz-Fellows (OWLF), but this can also inclue an Oberwolfach Lecture, for example. A preprint can have a size from pages, an the MFO will publish it on its website as well as by har copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free har copies (DIN A4, black an white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only nees the right to publish it on its website as a ocumentation of the research work one at the MFO, which you are accepting by sening us your file. In case of interest, please sen a pf file of your preprint by to or respectively. The file shoul be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO. There are no requirements for the format of the preprint, except that the introuction shoul contain a short appreciation an that the paper size (respectively format) shoul be DIN A4, "letter" or "article". On the front page of the har copies, which contains the logo of the MFO, title an authors, we shall a a running number (20XX - XX). We corially invite the researchers within the RiP or OWLF programme to make use of this offer an woul like to thank you in avance for your cooperation. Imprint: Mathematisches Forschungsinstitut Oberwolfach ggmbh (MFO) Schwarzwalstrasse Oberwolfach-Walke Germany Tel Fax URL The Oberwolfach Preprints (OWP, ISSN ) are publishe by the MFO. Copyright of the content is hol by the authors.

3 Geometric Flows an 3-Manifols Gerhar Huisken Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut Am Mühlenberg 1, Golm, Germany The current article arose from a lecture 1 given by the author in October 2005 on the work of R. Hamilton an G. Perelman on Ricci-flow an explains central analytical ingreients in geometric parabolic evolution equations that allow the application of these flows to geometric problems incluing the Uniformisation Theorem an the proof of the Poincare conjecture. Parabolic geometric evolution equations of secon orer are nonlinear extensions of the orinary heat equation to a geometric setting, so we begin by remining the reaer of the linear heat equation an its properties. We will then introuce key ieas in the simpler equations of curve shortening an 2- Ricci-flow before iscussing aspects of three-imensional Ricci-flow. 1 The Heat Equation A function u : IR n [0, T ] IR is a solution of the heat equation if n t u = u = D i D i u (1.1) hols everywhere on [0, T ]. The heat equation is the graient flow of the Dirichlet energy E(u) = 1 2 i IR n Du 2 µ (1.2) with respect to the L 2 -norm an has the funamental property of smoothing out all erivatives of a solution even for non-smooth initial ata in a precise quantitative way: For example, a uniform boun on the absolute value of u on some time interval [0, t) implies a corresponing boun sup D k u(, t) C k sup u (1.3) t k/2 IR n [0,t) 1 Oberwolfach Lecture elivere in connection with the general meeting of the Gesellschaft für mathematische Forschung e.v. (GMF) in Oberwolfach on October 16th,

4 for each higher erivative D k u after some waiting time. The scaling behavior of the equation arises from the fact that with u for each positive λ also the function u λ (x, t) = u(λx, λ 2 t) (1.4) is a solution of the heat equation. With respect to this scaling the heat-kernel ρ ( x, t) = 1 (4πt) exp ( x 2 ) n/2 4t (1.5) is a selfsimilar solution of the heat equation an thus connects the geometry of IR n with important analytical properties of heat-flow such as the Harnack inequality: For any positive solution u > 0 of the heat equation on IR n (0, T ) one has the ifferential inequality u Du 2 + n u 0, (1.6) u 2t with equality being vali on the heat-kernel. This inequality is equivalent to t log u D log u 2 n 2t (1.7) an implies by integration a sharp Harnack inequality for the values of u. General ifferential Harnack inequalities in the context of smooth Riemannian manifols (M, g) were establishe in the work of Li an Yau, compare [9]. In the Eucliean case the inequality reas u(x 2, t 2 ) u(x 1, t 1 ) ( ( t ) 1 n/2 exp x 2 x 1 2 ) (1.8) t 2 4(t 2 t 1 ) for all x 1, x 2 IR an t 2 > t 1 > 0 since the expression x 2 x 1 2 4(t 2 t 1 ) (1.9) minimizes the action 1 γ 2 τ (1.10) 4 γ among curves γ connecting the events (x 1, t 1 ), (x 2, t 2 ). It turns out that sharp inequalities of this type that hol with equality on selfsimilar solutions are one key to the unerstaning of the behavior of the Ricci-flow an its singularities. 2 Harmonic Map Heat-flow When consiering more general maps u : (M m, g) (N n, h) between two Riemannian manifols one stuies again the L 2 - graient flow of the Dirichlet energy E(u) = 1 2 M m u 2 g,h µ g, (2.1) 2

5 where the energy ensity u 2 is now compute with respect to the metrics g, h. The resulting parabolic system of equations tu = g,hu (2.2) is linear in the secon erivatives of u but has a quaratic nonlinearity in u ue to the epenence of the target metric on the map u. The stationary points of this system are harmonic maps an it was the first triumph of geometric evolution equations when Eells an Sampson were able to show that in the case of negatively curve target manifols the flow has a long time solution that converges to a harmonic map in the same homotopy class, see [1]. This striking result motivate the search for other evolution equations that woul be able to eform a given geometric object into some canonical representative of its class. On the other han the harmonic map heat-flow also provie the first examples of singularitie s in solutions of a geometric evolution equation when positive curvature in the target manifol force so calle bubbling phenomena. 3 Mean Curvature Flow For F 0 : M n IR n+1 a smooth immersion of an n imensional hypersurface in Eucliean space, n 1, the evolution of M 0 = F 0 (M) by mean curvature flow is the one parameter family of smooth immersions F : M n [0, T [ IR n+1 satisfying F t (p, t) = H(p, t)ν(p, t), p Mn, t 0, (3.1) F (, 0) = F 0. (3.2) Here H(p, t) an ν(p, t) are the mean curvature an the outer normal respectively at the point F (p, t) of the surface M n t = F (, t)(m n ). The signs are chosen such that Hν = H is the mean curvature vector an the mean curvature of a convex surface is positive. In case n = 1 this flow is calle the curve shortening flow. It is the graient flow of n-imensional area with respect to the L 2 -norm an has similar scaling an smoothing properties as the heat equation an the harmonic map heat-flow. However, since now the righthan sie of the equation, H(p, t)ν(p, t) = t F (p, t), (3.3) is compute from the Laplace-Beltrami operator with respect to the evolving inuce metric on the hypersurface, the system of equations epens on first erivatives of the solution an is only quasilinear. It leas to a system of reaction-iffusion equations for the evolving secon funamental form of the solution surface, t hi j = h i j + A 2 h i j. (3.4) 3

6 The nonlinear term can cause blowup of the curvature, an example is a shrinking sphere with M n t = S n R(t) (0) an R(t) = R 2 0 2nt. Here the curvature H 2 blows up with the rate (T t) 1 at the singular time T. It is a remarkable fact that espite the singularity the iffusion part of the equation is strong enough to enforce a selfsimilar structure in the formation of the singularity. The following uniformisation theorem for curves in the plane proven by Grayson [2] following work of Gage an Hamilton is a highly nontrivial one-imensional emonstration of how a geometric evolution equation can straighten out a given geometry: Theorem 3.1 (M. Grayson) If F 0 : S 1 IR 2 is an embee initial curve, then the solution of the curve shortening flow (3.1) remains embee an becomes convex after some finite time. It then converges smoothly to a point while its shape approaches a selfsimilar shrinking circle. One successful strategy for the proof of this result that we woul like to briefly sketch consists in classifying all possible singularities of the curve shortening flow in the plane together with arguments that rule out all such possibilities accept the known shrinking circle. Main ingreient for this approach is a monotonicity formula involving surface area with a heat-kernel as a weighting function: Let u(x, t) = 2(T t)ρ(x, T t) be the backwar heat-kernel aapte to the n-imensional hypersurface, then u µ = H + ν log u 2 u µ, (3.5) t M n M n compare [7]. Since the zeros of the RHS are exactly the selfsimilar shrinking solutions of mean curvature flow, one can euce with parabolic rescaling that all singularities of the flow with the natural blowup rate 1/ 2(T t) for the secon funamental form are asymptotically selfsimilar - an among these the shrinking circle is the only embee possibility. The proof of Grayson s result is then complete if one can show that there is no singularity with a higher blowup rate of curvature than observe in the case of the shrinking circle. By a rescaling argument again this can be one if all convex translating solutions of mean curvature flow can be rule out as the profile of the singularity. The key ingreient in this last step is a Harnack inequality for the curvature of convex curves moving by the curve shortening flow, that is fully analogous to the ifferential Harnack inequality (1.6) in the linear heat equation: κ ss κ s 2 κ + 1 κ 0. (3.6) 2t It can be euce that the grim reaper curve y(x, t) = log cos(x) + t is the only convex translating solution of the flow an this singularity profile can then be rule out using the global embeeness assumption for the initial curve in a quantitative way. In summary, it is not necessary to control the shape of the evolving curves at intermeiate times, the classification of all relevant singularities is sufficient. An this classification is achieve with the help of a sharp Harnack inequality an sharp integral estimates that 4

7 characterize selfsimilar solutions of the flow. We will fin exactly the same ingreients when stuying the approach to the classical uniformisation theorem an the Poincare conjecture by Ricci-flow. 4 Ricci-Flow Given a Riemannian manifol (M n, g 0 ) in 1982 Richar Hamilton propose to solve the evolution equation [3] g = 2Rc(g) (4.1) t with the given metric g 0 as initial atum. Here Rc(g(t)) is the Ricci curvature of the evolving metric an the manifol is assume to be compact without bounary. The equation is a quasilinear parabolic system of equations which is in many ways analogous to mean curvature flow. E.g. the evolution equation for the curvature resulting from Ricci-flow is again a system of reaction-iffusion equations that have the Laplace-Beltrami operator as their leaing part in all components of the system: Riem = Riem + Qua(Riem). (4.2) t The algebraic properties of the quaratic reaction term etermine the interplay of the various curvature quantities uring the flow. In three imensions the Riemann curvature tensor can be fully expresse in terms of the Ricci tensor, allowing a concentration of attention on the behavior of the three eigenvalues of this symmetric tensor fiel uring the evolution. In [3] Hamilton was able to fully unerstan the 3-imensional case with positive Ricci curvature: Theorem 4.1 If the initial Riemannian metric g 0 on a close 3-manifol has positive Ricci curvature then the solution of Ricci flow contracts smoothly to zero volume in finite time an appropriate rescalings of the metric converge to a smooth metric of constant sectional curvature. In particular, any 3-manifol carrying a metric of positive Ricci curvature is iffeomorphic to a spherical space-form. In three imensions the algebraic properties of the reaction terms in (4.2) strongly favor metrics of constant positive sectional curvature while the secon Bianchi ientities allow control of the graient of the scalar curvature through the graient of the tracefree part of the Ricci tensor. It was then possible to prove the theorem with the maximum principle being the essential tool. On two imensional spheres the maximum principle seems not to be enough to control the size of the curvature an an appeal to integral estimates an a Harnack estimate becomes an alternative avenue just like in the curve shortening of embee curves. Hamilton [4] prove that a Li-Yau type Harnack inequality hols for the scalar curvature R in 2- Ricci flow that is analogous to the orinary heat equation, we only state the scalar version R DR 2 R + n R 0. (4.3) 2t 5

8 Using this inequality it is possible in a fashion analogous to the case of curve shortening to prove that the only possible singularities of 2-spheres are either asymptotic to shrinking selfsimilar spheres or to a cigar -type cappe cyliner that satisfies Ricci-flow by moving uner iffeomorphisms along a raial graient vectorfiel on IR 2. The metric of such a translating soliton solution is explicitly given by an satisfies the corresponing soliton equation s 2 = x2 + y x 2 + y 2 (4.4) t g ij = (L X g) ij = 2D i D j f = 2R ij, (4.5) where X = Df is the graient vector fiel riving the iffeomorphism. Again in parallel to the case of curve shortening this last moel for a singularity can then be rule out for example by an isoperimetric inequality satisfie by 2- Ricci-flow. Gathering all cases an using an extra argument ue to Chow it follows from the work of Hamilton that the Ricci flow has the best possible behavior on all 2- surfaces: Theorem 4.2 For any initial metric on a two-imensional Riemann surface the solution of Ricci flow converges to a metric of constant Gauss curvature in the same conformal class (when appropriately rescale). We note that in the spherical case the shrinking surface has to be scale up while in the case of negative Gauss curvature the Ricci flow is expaning the surface such that it has to be scale own; in the case of the torus the volume stays constant an no rescaling is necessary. As a final comment we note that the eternal selfsimilar solutions of curve shortening flow ( grim reaper ) an Ricci-flow respectively ( cigar ) that appeare in our analysis of singularities are of inepenent interest in the theory of Renormalisationgroup flows where they are known as the hairpin solution of curve shortening flow an the 2- Riemannian black hole respectively. 5 Ricci-Flow singularities in 3 imensions If the assumption of positive Ricci curvature is roppe in imension n = 3 new singularities become possible, in particular when a long thin neck of type S 2 [a, b] connecting two larger pieces of the three-manifol begins to pinch off uner the influence of the large sectional curvature present in the small S 2. Such neckpinch singularities will inevitably happen for general initial ata an cannot be avoie. The great iea riving the work of Hamilton an Perelman in three imensions is that this is essentially the only singularity that the Ricci flow can evelop in finite time an that this singular behavior of the flow can in fact be turne to avantage: Accoring to the geometrisation conjecture of Thurston every close three-manifol amits a ecomposition into irreucible pieces of eight ifferent types along spherical an toroial necks. The 6

9 expectation is that stanar neckpinch singularities at finitely many space-time instances of Ricci flow will happen at just the right places to effect the esire ecomposition along spherical necks automatically an that the pinching of toroial necks only happens in infinite Ricci flow time in a controllable way. One hopes to construct a stanar surgery proceure for thin necks which replaces a piece of a spherical neck close to some cyliner S 2 [a, b] by two positively curve spherical caps while keeping track of all curvature quantities -this proceure shoul then be carefully use each time a neckpinch forms. In particular, on a simply connecte close 3-manifol it is expecte that after finitely many spherical neckpinch singularities which cut up the 3-manifol into isjoint pieces there will be only finitely many singularities asymptotic to selfsimilar shrinking 3-spheres analogous to the 2- case: If this were true, we coul retrace our flow backwars an conclue that the original surface was a finite connecte sum of stanar 3-spheres, hence a 3-sphere, thus proving the Poincare conjecture. There are several major analytical obstacles to this program: There will be egenerate spherical neckpinches where some small 3-imensional bubble oesn t pinch off properly an instea a 3-imensional hemisphere gets squashe into a long thin horn eveloping rapily towars a cusp. Again, this behavior is expecte an cannot be avoie. A further ifficulty lies in the nee to emonstrate that these singularities are almost exactly axisymmetric in a precise quantitative way: Only then it will be possible to evise a etaile quantitative surgery algorithm for both cutting off the horns an cutting out the spherical necks. By using first maximum principle estimates on the reaction-iffusion system for the Ricci curvature establishing that all singularities in 3- are asymptotically non-negatively curve (Hamilton-Ivey estimates) an then proving a higher imensional version of the Li- Yau type Harnack inequality for the Ricci tensor Hamilton gave a preliminary classification of finite time singularities of 3- Ricci flow in [5]: The classification inclues the selfsimilar shrinking 3-sphere, the expecte selfsimilar spherical neckpinch singularity S 2 IR an the translating Ricci solitons of strictly positive curvature moelling the tip of the horn in a egenerate spherical neckpinch. Unfortunately there is one more possible singularity moel on the list, namely the cigar type translating 2- soliton from (4.4) cross IR. The presence of such a singularity coul inicate a egenerating cross-section of a pinching neck an woul make the propose surgery an continuation of the flow impossible. A further complication lay in the fact that Hamilton s classification only applie in regions where the curvature of the manifol is of comparable size to the maximum curvature in the region - making it inapplicable in certain situations. Just like in curve shortening or in 2- Ricci flow a new estimate was neee to rule out the unesirable cigar singularity. Perelman in his breakthrough contributions in [10] iscovere new estimates for weighte volume istributions on the evolving three-manifol that enable him to rule out the cigar type solitons since they have very little volume in large regions of small curvature, so calle collapse regions, which are not present in the other singularity moels. One of the crucial new concepts evelope by Perelman for this purpose is the concept of a new action on curves γ : [τ 1, τ 2 ] (M 3, g(t)) in Ricci flow space-time, given by L(γ) := γ τ(r(γ(τ)) + γ(τ) 2 )τ (5.1) 7

10 that leas to the concept of a L-shortest curve in a natural way. Fixing a point p M 3 an τ 1 = 0 we let L(q, τ) be the L-length of the L-shortest curve connecting p an q an enote by l(q, τ) = 1 2 L(q, τ) the reuce istance. Then Perelman proves τ Theorem 5.1 The reuce volume V (τ) = τ n 2 exp ( l(q, τ))q (5.2) M is monotonically increasing in τ if the metric satisfies g τ ij = 2R ij. This result following from a careful analysis of the variational behavior of the L geoesics shoul be seen as relate in spirit to the monotonicity formulae for mean curvature flow an harmonic map heat-flow iscusse earlier in this article since the integran resembles a nonlinear backwar heat-kernel aapte to the space-time geometry of the Ricci flow. Also compare the L integral with the action appearing in the Li-Yau Harnack inequality (1.10). The theorem just explaine can be seen to exclue collapsing behavior on finite time intervals, thus ruling out the unesirable cigar -type singularity in 3- Ricci flow at finite times. This is the starting point for a precise quantitative escription of the remaining singularities involving spherical neckpinches an for encorporating the quantitative surgery proceure on necks an horns evelope by Hamilton in [6]. Perelman sketches in [11] how to set up an algorithm from smooth Ricci flow an intermittent surgeries that maintains all the a priori estimates controlling the flow an keeps the total number of surgeries finite on finite time intervals. Finally, in [12] Perelman outlines that for a subclass of 3-manifols containing all possible counterexamples of the Poincare conjecture the Ricci flow must stop at a finite time when the volume tens to zero. Simply connecteness an the classification of singularities escribe above ensures that there are only finitely many spheres left at this stage of the proceure which is the esire outcome implying the Poincare conjecture. Since the time of the lecture in Oberwolfach three careful manuscripts have appeare that give etails of the above results, by Kleiner-Lott [13], by Cao-Zhu [14] an Morgan- Tian [15]. The first two manuscripts inclue etails also for the whole geometrisation conjecture while the last manuscript follows Perelman s arguments in [12] on finite time extinction proviing a shorter route to the Poincare conjecture. Hamilton has announce an algorithm base on his previous work an Perelman s non-collapsing estimate that follows the strategy of [6]. Finally, in [8] Huisken an Sinestrari constructe a mean curvature flow with surgery for two-convex hypersurfaces that is inspire by Hamilton s surgery approach in [6]. References [1] J. Eells Jr. an J.H. Sampson,Harmonic Mappings of Riemannian Manifols, American Journal of Math. 86 (1964), [2] M.A. Grayson, Shortening embee curves, Annals of Mathematics, 129 (1989),

11 [3] R.S. Hamilton, Three-manifols with positive Ricci curvature, J. Differential Geom. 17 (1982), [4] R.S. Hamilton, The Ricci flow on surfaces, Mathematics an General Relativity, Contemp. Math. 71 (1986), AMS, J. Isenberg E., [5] R.S. Hamilton, Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1995) 7 136, International Press, Boston. [6] R.S. Hamilton, Four manifols with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), [7] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), [8] G. Huisken an C. Sinestrari, Mean curvature flow with surgeries for two-convex surfaces, preprint (2006), 87pp. [9] P. Li an S.T. Yau, On the parabolic kernel of the Schroeinger operator, Acta Math. 156 (1986), [10] G. Perelman, The entropy formula for the Ricci flow an its geometric applications, preprint (2002). [11] G. Perelman, Ricci flow with surgery on three-manifols, preprint (2003). [12] G. Perelman, Finite time extinction for the solutions to the Ricci flow on certain three-manifols, preprint (2003). [13] B. Kleiner an J. Lott, Notes on Perelman s papers, preprint, arxiv:math.dg/ [14] H.-D. Cao an X.-P. Zhu, A Complete Proof of the Poincar an Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow, Asian J. of Math. 10 (2006), [15] J.W. Morgan an G. Tian, Ricci Flow an the Poincare conjecture, Arxiv preprint math.dg/

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