In this Note we introduce and study dynamical systems related to the Ricci
operator on the space of Kahler metrics as discretizations of certain geometric
flows. We pose a conjecture on their convergence towards canonical Kahler
metrics and study the case where the first Chern class is negative, zero or
positive. This construction has several applications in Kahler geometry, among
them an answer to a question of Nadel and a construction of multiplier ideal
sheaves.Comment: v2: shortened introduction. v3: corrected some typos. v4: shortened
  to fit in C. R. Acad. Sci. Pari

Rubinstein, Yanir A.

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 345 (2007) 445–448http://france.elsevier.com/direct/CRASS1/Differential GeometryThe Ricci iteration and its applicationsYanir A. Rubinstein ∗Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAReceived 22 June 2007; accepted after revision 27 September 2007Presented by Jean-Michel BismutAbstractIn this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics as discretiza-tions of certain geometric flows. We pose a conjecture on their convergence towards canonical Kähler metrics and study the casewhere the first Chern class is negative, zero or positive. This construction has several applications in Kähler geometry, among theman answer to a question of Nadel and a construction of multiplier ideal sheaves. To cite this article: Y.A. Rubinstein, C. R. Acad.Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméL’itération de Ricci et ses applications. Dans cette Note nous introduisons et étudions des systèmes dynamiques reliées àl’opérateur de Ricci sur l’espace des métriques kählériennes comme discrétisations des certains flots géométriques. Nous posonsune conjecture concernant leurs convergence vers des métriques kählériennes canoniques and nous étudions le cas où la premièreclasse de Chern est négative, zéro ou positive. Cette construction a plusieurs applications en géométrie kählérienne, parmi elles uneréponse à une question de Nadel et une construction des faisceaux d’idéaux multiplicateurs. Pour citer cet article : Y.A. Rubinstein,C. R. Acad. Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionOur main purpose in this Note is to describe a new method for the construction of canonical Kähler metrics viathe discretization of certain geometric flows. The idea is to turn a geometric flow into a set of difference equations.Complete proofs will appear elsewhere [12].The search for a canonical metric representative of a fixed Kähler class has been at the heart of Kähler geometrysince its birth and has a long history starting from Kähler and continuing, among many others, with the work of Calabi,Aubin, Yau, Tian and Donaldson. For general extremal metrics a general existence theory is not presently availablealthough the so-called Yau–Tian–Donaldson conjecture suggests that it should be related with notions of stability inalgebraic geometry.* Present address: Department of Mathematics, Princeton University, Princeton, NJ 08544, USA.E-mail addresses: yanir@member.ams.org, yanir@math.princeton.edu.1631-073X/$ – see front matter © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.09.020446 Y.A. Rubinstein / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 445–448One of the main tools in the existence theory of Kähler–Einstein metrics and Kähler–Ricci solitons has been theRicci flow introduced by Hamilton [6]. Cao has shown that Yau’s continuity method proof may be phrased in termsof the convergence of the Ricci flow [3]. Much work has gone into understanding the Ricci flow on Fano manifoldsand recently Perelman and Tian and Zhu proved that the analogous convergence result holds in this case [14]. Theidea that there might be another way of approaching canonical metrics, in the form of a discrete iterative dynamicalsystem, was suggested by Nadel [8] and is the main motivation for our work.2. The Ricci iterationLet (M, J,ω) be a connected compact closed Kähler manifold of complex dimension n and let Ω denote a Kählerclass. Let ω = −∂̄ ◦ ∂̄ − ∂̄ ◦ ∂̄ denote the Laplacian with respect to ω. Let Hω denote the Hodge projection operatorfrom the space of closed forms onto the kernel of ω . Let V = Ωn([M]). Denote by DΩ the space of all closed(1,1)-forms cohomologous to Ω , by HΩ the subspace of Kähler forms and by H+Ω the subspace of Kähler formswhose Ricci curvature is positive. We introduce the following dynamical system on HΩ which is our main object ofstudy in this Note:Definition 2.1. Given a Kähler form ω ∈ HΩ define the (time one) Ricci iteration1 to be the sequence of forms{ωk}k0, satisfying the following equations for each k ∈ N for which a solution existsωk+1 = ωk + Hk+1 Ricωk+1 − Ricωk+1, k + 1 ∈ N, ω0 = ω.This system of equations may be viewed as a discrete version of the flow ∂ω(t)∂t= −Ricω(t) + Ht Ricω(t). Thisflow, first studied by Guan [5], can in turn be considered as a Kähler version of Hamilton’s Ricci flow. Our work ismotivated by the following conjecture (an analogue may also be posed for the flow):Conjecture 2.1. Let (M, J) be a compact closed Kähler manifold, and assume that there exists a constant scalarcurvature Kähler metric in HΩ . Then for any ω ∈ HΩ the Ricci iteration exists for all k ∈ N and converges in anappropriate sense to a constant scalar curvature metric.Our motivation for posing this conjecture comes from the following theorem:Theorem 2.2. Let (M, J) be a compact closed Kähler manifold admitting a Kähler–Einstein metric. Let Ω be a Kählerclass such that μΩ = c1 with μ ∈ {0,±1}. Then for any ω ∈HΩ the Ricci iteration exists for all k ∈ N and convergesin the sense of Cheeger–Gromov to a Kähler–Einstein metric.Remark 1. Note that the above conjecture may also be posed for solitons of the above flow, i.e., solutions of LXω =Ricω − Hω Ricω for a holomorphic vector field X, and that a result analogous to Theorem 2.2 then holds for thesemetrics using the same methods and discretizing the flow twisted by the one-parameter subgroup of automorphismscorresponding to X.Sketch of proof. First we prove that the iteration exists for each k ∈ N. Since Hω Ricω = μω, this amounts to solvingω1 = ω0 + μω1 − Ricω1. Let ω1 = ωϕ1 := ω +√−1 ∂∂̄ϕ1. This can be written as a complex Monge–Ampère equa-tion: ωnϕ1 = ωnefω−(μ−1)ϕ1 ,∫Mωnefω−(μ−1)ϕ1 = V , where √−1 ∂∂̄fω = Ricω − ω,∫Mefωωn = V . The existence ofsolutions to such equations is known when μ  0 by the work of Aubin [1] and Yau [15], and when μ = 1 by the workof Yau. Hence the iteration exists for each k ∈ N.We divide the discussion into three cases, according to the sign of the first Chern class.Assume first that c1 < 0 and let Ω = −c1. For each k write ωk = ωψk with ψk =∑kl=1 ϕl . We have the followingsystem of Monge–Ampère equations: ωnψk = ωnefω+ψk+ϕk , k ∈ N. One readily sees that an inductive argument using1 Most of the results hold also for discretizations corresponding to other time intervals.Y.A. Rubinstein / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 445–448 447the maximum principle implies that ‖ψk − ψk−1‖C0  C2−k . This uniform bound implies higher order a priori esti-mates, by elliptic regularity theory. Therefore the sequence converges exponentially fast to a smooth function that wedenote by ψ∞.Consider the Chen–Tian functionals Ek [4]. We now observe the following monotonicity result. Its proof can bededuced from some of our previous results [11].Lemma 2.3. Along the iteration E0 is monotonically decreasing whenever ω0 ∈ HΩ . When μ = 1 the same is truefor E1, and if ω0 ∈H+c1 , also for Ek, k  2.Coming back to the proof of the theorem, note that unless ω0 is itself Kähler–Einstein, the functional E0 is strictlydecreasing along the iteration. In particular since ω∞ is a fixed point of the iteration it must be Kähler–Einstein. Thecase μ = 0 is similar and so we omit the details.Finally, we turn to the case μ = 1 and assume for simplicity that there are no holomorphic vector fields. In the caseμ = 1 the corresponding iteration takes a very special form.Definition 2.4. Define the inverse Ricci operator Ric(−1) :Dc1 → Hc1 by letting Ric(−1) ω := ωϕ with ωϕ the uniqueKähler form in Hc1 (given by the Calabi–Yau Theorem [15]) satisfying Ricωϕ = ω. Similarly denote higher orderiterates of this operator by Ric(−l), l ∈ Z, with Ric(0) := Id.We then see that the dynamical system for μ = 1 is nothing but the evolution of iterates of the inverse Riccioperator, ωk = Ric(−k) ω0. For this case we are solving the system of equations ωnψk = ωnefω−ψk−1 , k ∈ N. Let Gkbe a Green function for −k := −ωϕk satisfying∫MGk(·,y)ωnψk (y) = 0. Set Ak = − infM×M Gk . Let I (ω,ωϕ) =V −1∫Mϕ(ωn − ωnϕ). Application of the Green formula gives |ψk|  n(A0 + Ak) + I (ω0,ωψk ). Since E0 is properon Hc1 in the sense of Tian [13], we conclude that I (ω,ωψk ) is uniformly bounded. The crucial technical ingredientis now a uniform upper bound on the diameter. Its derivation hinges on properties of the energy functionals, thedefinition of the iteration, and finally on an argument due to Perelman adapted to this “discrete” situation. Now,combining the diameter estimate with the Green function estimate of Bando and Mabuchi [2], we conclude that|ψk|  C. By monotonicity of the energy functionals we conclude that a subsequence can be chosen, converging to aKähler–Einstein metric. This completes the outline of the proof of Theorem 2.2.There are a number of applications of these constructions to several well-known objects of study in Kähler geome-try, among them canonical metrics, energy functionals, the Moser–Trudinger–Onofri inequality, balanced metrics andthe structure of the space of Kähler metrics. We point out two of the most obvious ones and describe others elsewhere.Also, these constructions can be generalized in some interesting directions [12].The first application is an answer to a question raised by Nadel [8]: Given ω ∈ Hc1 define a sequence of metricsω,Ricω,Ric(Ricω), . . . , as long as positivity is preserved; what are the periodic orbits of this dynamical system?The cases k = 2,3 in the following theorem are due to Nadel:Theorem 2.5. Let (M,J,ω) be a Fano manifold and assume that Ric(k)(ω) = ω for some k ∈ Z. Then ω is Kähler–Einstein.This is an immediate corollary of Lemma 2.3.The second application is the construction of multiplier ideal sheaves [7] when a Kähler–Einstein metric does notexist. It may be seen as a discrete counterpart to a recent result of Phong–Šešum–Sturm [9].Theorem 2.6. Let (M, J) be a Fano manifold that does not admit a Kähler–Einstein metric and let γ > 1. One mayextract a subsequence {ψkj } such that limj→∞ ψkj = ψ∞ exists in L1loc(M) and defines a nontrivial Nadel-typemultiplier ideal sheaf defined for each open set U ⊆ M by local sections {h ∈ OM(U): |h|2e−γψ∞ ∈ L1loc(M)}.AcknowledgementsI would like to express my deep gratitude to my teacher, Gang Tian. I thank X.-X. Chen, S. Donaldson,V. Guillemin, J. Morgan, T. Mrowka, I. Singer, J. Song and S. Zelditch for their interest in this work and for their448 Y.A. Rubinstein / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 445–448warm encouragement. Part of this research was carried out at Peking University in Summer 2005 where the first ver-sion [10] of [12] was written and I thank that institution for its hospitality. This material is based upon work supportedunder a National Science Foundation Graduate Research Fellowship.References[1] T. Aubin, Équations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. 102 (1978) 63–95.[2] S. Bando, T. Mabuchi, Uniqueness of Kähler–Einstein metrics modulo connected group actions, in: Algebraic Geometry, Sendai, 1985,Kinokuniya, 1987, pp. 11–40.[3] H.-D. Cao, Deformations of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985) 359–372.[4] X.-X. Chen, G. Tian, Ricci flow on Kähler–Einstein surfaces, Invent. Math. 147 (2002) 487–544.[5] D.Z.-D. Guan, Extremal-solitons and C∞ convergence of the modified Calabi flow on certain CP 1 bundles, preprint, December 22, 2006.[6] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255–306.[7] A.M. Nadel, Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature, Ann. Math. 132 (1990) 549–596.[8] A.M. Nadel, On the absence of periodic points for the Ricci curvature operator acting on the space of Kähler metrics, in: Modern Methods inComplex Analysis, Princeton University Press, 1995, pp. 277–281.[9] D.H. Phong, N. Šešum, J. Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, preprint, arxiv: math.DG/0611794.[10] Y.A. Rubinstein, On iteration of the Ricci operator on the space of Kähler metrics, I, manuscript, August 14, 2005, unpublished.[11] Y.A. Rubinstein, On energy functionals, Kähler–Einstein metrics and the Moser–Trudinger–Onofri neighborhood, preprint, arxiv:math.DG/0612440.[12] Y.A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics I, preprint, 2007;II, preprint, in preparation.[13] G. Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1–37.[14] G. Tian, X.-H. Zhu, Convergence of Kähler–Ricci flow, J. Amer. Math. Soc. 20 (2007) 675–699.[15] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampére equation, I, Comm. Pure Appl. Math. 31(1978) 339–411.

The Ricci iteration and its applications

Yanir A. Rubinstein

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The Ricci iteration and its applications

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