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New Geometric Flows National Science Review REVIEW 5: 534–541, 2018 doi: 10.1093/nsr/nww053 Advance access publication 12 September 2016 MATHEMATICS New geometric flows Gang Tian1,2 ABSTRACT Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 This is an expository paper on geometric flows. I will start with a brief tour on Ricci flow, thenIwill discuss two geometric flows introduced in my joint work with J. Streets and some recent workson them. Keywords: Ricci flow, metrics, AMMP, Kahler,¨ Hermitian, symplectic RICCI FLOW AND ITS APPLICATION First, we recall that a metric g is Einstein if Ric(g) = λ g, where λ is a constant, often nor- We will assume that M is a closed manifold of dimen- malized to be −(n − 1), 0 or n − 1. Note sion n and g is fixed. The Ricci flow was introduced 0 that Ric(g) = (R ) denotes the Ricci curvature by Hamilton in [1]: ij of g and measures the deviation of volume form from the Euclidean one. In dimension 2, an Ein- ∂g stein metric is simply a metric of constant Gauss ij =−2R , g (0) = a given metric. ∂t ij curvature. (1) Now we assume that M is a closed 3-manifold. It Hamilton and later DeTurck proved that for any ini- is known that any Einstein metric on a 3-manifold tial metric there is a unique solution g(t)onM × [0, is of constant sectional curvature. Therefore, it fol- T)forsomeT > 0. Hamilton also established an an- lows from the classical uniformization theorem that alytic theory for Ricci flow. It provides a way of con- if M admits an Einstein metric, then its universal cov- structing Einstein metrics as one has done for 2D or ering is of the form either round sphere S3 or R3 or 3D manifolds. the hyperbolic 3-ball H3. Thus, if we can construct an If M is a 2-manifold, it is proved in [2,3] that given Einstein metric, then we have a similar picture for 3- 1School of any initial g0 on M, (1) has a global solution g(t) after manifolds as we have for surfaces. However, not ev- Mathematical normalization and g(t) converges to a metric g∞ on ery 3-manifold admits an Einstein metric. One can Sciences, Peking 1 M. One can show that g∞ is of constant Gauss curva- easily construct such examples, such as × S for University, Beijing any surface of genus > 1. This is because its funda- 100871, China and ture. The proof is trivial if M has non-positive Euler 2 number and contained in [4]ifM has positive Eu- mental group is the product of a surface group with Department of Z Mathematics, ler number. Thus, combined with the classical uni- which is neither an Abelian group nor the funda- Princeton University, formization theorem on surfaces, we can use Ricci mental group of any hyperbolic compact 3-manifold Princeton, NJ 08544, flow to prove the following. The universal covering (cf. [5]). USA of M is isomorphic to either the 2D round sphere S2 It was known [6] that any closed 3-manifold can or the Euclidean plane R2 or the hyperbolic disk D. be decomposed along embedded 2-spheres into ir- E-mail: Of course, this conclusion can be also obtained by reducible 3-manifolds; moreover, such a decompo- [email protected] the elliptic method or the Riemann mapping theo- sition is essentially unique. Thurston’s Geometriza- rem. However, the method of Ricci flow has much tion Conjecture claims (cf. [5]) that any irreducible Received 18 July more advantages since it can be generalized to higher 3-manifold can be decomposed along incompress- 2016; Revised 15 dimensions. ible tori into finitely many complete Einstein 3- August 2016; A celebrated example is Perelman’s resolution for manifolds plus some Graph manifolds. The famous Accepted 15 August Thurston’s Geometrization Conjecture, which im- Poincare´ Conjecture is a special case of this Ge- 2016 plies the famous PoincareConjecture.´ ometrization Conjecture. C The Author(s) 2016. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. All rights reserved. For permissions,please e-mail: [email protected] REVIEW Tian 535 The Geometrization Conjecture was solved by If M is simply connected or its fundamental group Perelman (cf. [7,8]) using the Ricci flow. is not too big, Perelman showed us in [9]whyMt has If the Ricci flow has a solution g(t), then we can to become empty for t sufficiently large, i.e. the Ricci choose a scaling λ(t) > 0 and a reparametrization flow with surgery becomes extinct in finite time. In t = t(s) with λ(0) = 1 and t(0) = 0suchthat particular, the Poincare´ Conjecture follows. One can g˜(s ) = λ(s )g (t(s )) has fixed volume and satisfies find detailed proof of this finite extinction in[10] the normalized Ricci flow: and [11]. To solve Thurston’s Geometrization Con- ∂g˜ R˜(s ) ij =−2 R˜ − g˜ , (2) jecture, one needs to study asymptotic behavior ∂s ij 3 ij of the above Ricci flow with surgery (in 2012, Bamler found and fixed a gap in section 6.4 of Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 where R˜ij denotes the Ricci curvature of g˜(s ), and [9] that had been missed) and prove a result R˜(s ) denotes the average of the scalar curvature of on collapsing 3-manifolds with curvature locally g˜(s ). If the normalized Ricci flow (2) has a global bounded from below and geodesically convex solution g˜(s )foralls ≥ 0 and g˜(s ) converges to a boundary. smooth metric g∞ as s goes to ∞, then its limiting Perelman did not prove that there are only metric g∞ is an Einstein metric, so the universal cov- finitely many surgeries. It was accomplished by Bam- ering of M is standard. The Geometrization Conjec- ler in [12]. Moreover, he proved that after finitely ture follows. many surgeries, the normalized Ricci flow g(t)has In 1982, Hamilton proved that for any 3- bounded curvature and consequently, converges to manifold with a metric g0 with positive Ricci curva- limiting spaces (possibly collapsed) that fall into one ture, (2) with initial metric g0 has a global solution of the eight geometries classified by Thurston. This that converges smoothly to a metric of constant pos- limit should be unique, but a complete proof is still itive curvature. It follows that M is a quotient of S3 missing though it seems to be doable. Thus, if we by a finite group. could further prove that finite-time surgery is canon- However, in general, (1) or (2) develops singu- ical, then one would have a refined version of the larity at finite time, so one has to introduce amore Geometrization Conjecture: given (M, g0), there is general evolution process called ‘Ricci flow with a unique Ricci flow with surgery (Mt, g(t)) with ini- surgery’. This evolution process is still given by(Mt, tial value (M, g0) such that either Mt becomes ex- g(t)) satisfying: (i) there is a discrete set of times t0 tinct when t is sufficiently large or after normaliza- = 0 < t1 < ... < ti < ... without accumulation point tion, (Mt, g(t)) converges to a unique limit in the such that in each (ti, ti+1), Mt is a fixed smooth mani- eight geometries. fold and g(t) is the usual Ricci flow on Mt; (ii) at each For higher dimensions, we do not know much ti for i ≥ 1, the manifolds and metrics undergo topo- about singularity formation of Ricci flow unless the logical and metric discontinuities (surgeries), in par- underlying manifolds have certain structures. If the ticular, the topology type of Mt for T < ti changes initial metric g0 has either positive curvature op- to that of Mt for t > ti as t moves across ti.Inthis erator or positive bisectional curvature or positive definition, we did not specify how g(t) moves across isotropic curvature, so does g(t) along the Ricci flow. ti. We expect there is a canonical way for g(t)togo In particular, Brendle and Schoen solved a long- through the surgery time. This is still an open prob- standing problem: if a closed manifold admits a met- lem and has applications though it is not needed in ric with curvature 1/4-pinched, then it is diffeomor- Perelman’s resolution of the Geometrization Con- phic to a sphere. jecture. If the initial metric g0 is Kahler,¨ so does g(t) One of Perelman’s crucial advances is that one along the Ricci flow, thus we can restrict the Ricci needs to do surgeries only along two spheres; con- flow to Kahler¨ metrics and call it Kahler-Ricci¨ flow. sequently, he understood completely the change in The analytic minimal model program, abbreviated topology. More precisely, he proved: ‘There is a Ricci as AMMP (it is also referred as the Song-Tian pro- flow with surgery (Mt, g(t)) defined for all t ∈ [0, gram), is to use the Kahler-Ricci¨ flow to deform ∞) with any given initial 3-manifold (M, g0). The any Kahler¨ metric towards a canonical model metric, topological change in the 3-manifold as one crosses possibly through finitely many finite-time surgeries. a surgery time is a connected sum decomposition This gives a promising approach to classify all closed together with removal of connected components, Kahler¨ manifolds birationally. Let us briefly describe each of which is diffeomorphic to one of S2 × S1, the AMMP.
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