## Space of Ricci flows. II. Part B: Weak compactness of the flows.(English)Zbl 1479.53103

Denote by $$\mathcal{L M}: = \{(M^n, g(t), J, L, h(t)), \: t \in (-T, T) \subset {\mathbb R} \}$$ a polarized Kähler Ricci flow solution on a Kähler manifold $$(M^n, g =g(0), J)$$ with a pluri-anti-canonical line bundle $$L = K_{M^n} ^{-\nu}$$, i.e., $$g(t)$$ is a Kähler Ricci flow and $$h(t)$$ is a family of smooth metrics on $$L$$ whose curvature is $$\omega(t)$$. We write $$\omega (t) = \omega (0) + \sqrt{ -1} \partial \bar \partial \varphi$$. Then $$\dot \varphi$$ is the Ricci potential, i.e., $$\sqrt{-1} \partial \bar \partial\dot \varphi = - \mathrm{Ric} +\lambda g$$. Denote by $$\mathcal {K} (n, A)$$ the collection of all polarized Kähler Ricci flows on Kähler manifolds $$M$$ of dimension $$n$$ satisfying the following estimates for all $$t \in (-T, T)$$ $T \ge 2,$ $C_ S(M) + \frac{1}{\mathrm{Vol}(M)} + | \dot \varphi|_{ C^1 (M) } + | R - n \lambda| _{ C^0 (M) } \le A.$ Here $$C_S(M)$$ is the Sobolev constant of $$M$$, $$R$$ is the scalar curvature of $$M$$ and $\lambda = \frac{c_1 (M) } {c_1 (L)}.$
The main result of the paper under review (Theorems 1.5, 4.40) concerns the convergence of sequences $$\{\mathcal {L M} _i \in \mathcal {K} (n, A)$$, $$x_i \in M_i\}$$ of polarized Kähler-Ricci flows with base point in the pointed $$\hat C^\infty$$-Cheeger-Gromov topology, i.e., the pointed Gromov-Hausdorff topology plus the pointed $$C^\infty$$-Cheeger-Gromov topology on the regular part of the limiting underlying (possibly singular) analytic normal variety $$\bar M$$, whose structure is analyzed in the paper under the review. As a consequence, the authors prove the weak compactness of anti-canonical Kähler-Ricci flow solutions on a Fano manifold, which can be used for an alternative proof of the stability conjecture proved by Chen-Donaldson-Sun. Another important consequence of their results is the confirmation of the partial $$C^0$$-conjecture of Tian.
The paper under review is the continuation of the authors study on the Ricci flow in [X. Chen and B. Wang, Commun. Pure Appl. Math. 65, No. 10, 1399–1457 (2012; Zbl 1252.53076); J. Eur. Math. Soc. (JEMS) 14, No. 6, 2001–2038 (2012; Zbl 1257.53094); Forum Math. Sigma 5, Paper No. e32, 103 p. (2017; Zbl 1385.53033)].

### MSC:

 53E20 Ricci flows 53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows) 58D27 Moduli problems for differential geometric structures 32Q20 Kähler-Einstein manifolds

### Citations:

Zbl 1252.53076; Zbl 1385.53033; Zbl 1257.53094
Full Text:

### References:

 [1] M. Anderson,Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102(2):429-445, 1990. MR1074481, Zbl 0711.53038. · Zbl 0711.53038 [2] S. Bando, T. Mabuchi,Uniqueness of Einstein K¨ahler metrics modulo connected group actions, In: Oda, T. (ed.) Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure. Math., vol. 10, Amsterdam: North-Holland and Tokyo: Kinokuniya 1987. MR0946233, Zbl 0641.53065. · Zbl 0641.53065 [3] E. Calabi,Improper affine hyperspheres of convex type and a generalization of a theorem by K. J¨orgens, Michigan Math. J., 5:105-126, 1958. MR0106487, Zbl 0113.30104. · Zbl 0113.30104 [4] H. Cao, N. Sesum,A compactness result for K¨ahler Ricci solitons, Adv. Math., 211(2):794-818, 2007. MR2323545, Zbl 1127.53055. · Zbl 1127.53055 [5] H. Cao, X.P. Zhu,A complete proof of the Poincar“e and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math., 10(2):165-492, 2006. MR2233789, Zbl 1200.53057. Erratum to “A Complete Proof of the Poincar´e and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci Flow”, Asian Journal of Math., 10:663-664, 2006. MR2282358, Zbl 1200.53058. [6] X. Cao, R. Hamilton,Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal., 19(4):989-1000, 2009. MR2570311, Zbl 1183.53059. · Zbl 1183.53059 [7] X. Cao, Q.S. Zhang,The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math., 228(5):2891-2919, 2011. MR2838064, Zbl 1238.53038. · Zbl 1238.53038 [8] J. Cheeger,Degeneration of Riemannian metrics under Ricci curvature bounds, Publications of the Scuola Normale Superiore, Edizioni della Normale, October 1, 2001. MR2006642, Zbl 1055.53024. · Zbl 1055.53024 [9] J. Cheeger,Integral Bounds on curvature, elliptic estimates and rectifiability of singular sets, GAFA, 13:20-72, 2003. MR1978491, Zbl 1086.53051. · Zbl 1086.53051 [10] J. Cheeger, T.H. Colding,Lower bounds on Ricci curvature and the almost rigidity of warped products, Annals. Math., 144(1):189-237, 1996. MR1405949, Zbl 0865.53037. · Zbl 0865.53037 [11] J. Cheeger, T.H. Colding,On the structure of spaces with Ricci curvature bounded below.I, J. Differential Geometry, 45:406-480, 1997. MR1484888, Zbl 0902.53034. · Zbl 0902.53034 [12] J. Cheeger, T.H. Colding, G. Tian,On the Singularities of Spaces with Bounded Ricci Curvature, GAFA, Geom. Funct. Anal., 12:873-914, 2002. MR1937830, Zbl 1030.53046. · Zbl 1030.53046 [13] J. Cheeger, M. Gromov, M. Taylor,Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom., 17:15-53, 1982. MR0658471, Zbl 0493.53035. · Zbl 0493.53035 [14] J. Cheeger, A. Naber,Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math., 191(2):321-339, 2013. MR3010378, Zbl 1268.53053. · Zbl 1268.53053 [15] T.H. Colding,Shape of manifolds with positive Ricci curvature, Invent. Math., 124:175-191, 1996. MR1369414, Zbl 9871.53027. · Zbl 0871.53027 [16] X.X. Chen,Space of K¨ahler metrics(IV)—On the lower bound of the K-energy, arXiv:0809.4081. [17] X.X. Chen, S. Donaldson,Volume estimates for K¨ahler-Einstein metrics: the three dimensional case, J. Differential Geom., 93(2):175-189, 2013. MR3024304, Zbl 1279.32020. · Zbl 1279.32020 [18] X.X. Chen, S. Donaldson,Volume estimates for K¨ahler-Einstein metrics and rigidity of complex structures, J. Differential Geom., 93(2):191-201, 2013. MR3024305, Zbl 1281.32019. · Zbl 1281.32019 [19] X.X. Chen, S. Donaldson and S. Sun,K¨ahler-Einstein metrics and Stability, IMRN, 2014(8):2119-2125. MR3194014, Zbl 1331.32011. · Zbl 1331.32011 [20] X.X. Chen, S. Donaldson and S. Sun,K¨ahler-Einstein metrics on Fano manifolds, I: Approximation of metrics with cone singularities, JAMS, 28(1):183-197. MR3264766, Zbl 1312.53096. · Zbl 1312.53096 [21] X.X. Chen, S. Donaldson and S. Sun,K¨ahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than2π, JAMS, 28(1):199-234. MR3264767, Zbl 1312.53097. · Zbl 1312.53097 [22] X.X. Chen, S. Donaldson and S. Sun,K¨ahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches2πand completion of the main proof, JAMS, 28(1):235-278. MR3264768, Zbl 1311.53059. · Zbl 1311.53059 [23] X.X. Chen, S. Sun,Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K¨ahler metrics, Ann. of Math. (2), 180(2):407-454, 2014. MR3224716, Zbl 1307.53058. · Zbl 1307.53058 [24] X.X. Chen, S. Sun, B. Wang,K¨ahler Ricci flow, K¨ahler Einstein metric, and K-stability, Geom. Topol., 22(6):3145-3173, 2018. MR3858762, Zbl 06945124. · Zbl 1404.53058 [25] X.X. Chen, B. Wang,K¨ahler Ricci flow on Fano Surfaces(I), Math. Z., 270(1- 2):577-587, 2012. MR2875850, Zbl 1237.53066. · Zbl 1237.53066 [26] X.X. Chen, B. Wang,Remarks on K¨ahler Ricci flow, J. Geom. Anal., 20(2):335- 353, 2010. MR2579513, Zbl 1185.53075. · Zbl 1185.53075 [27] X.X. Chen, B. Wang,Space of Ricci flows (I), Comm. Pure Appl. Math., 65(10):1399-1457, 2012. MR2957704, Zbl 1252.53076. · Zbl 1252.53076 [28] X.X. Chen, B. Wang,Space of Ricci flows (II), arXiv:1405.6797. [29] X.X. Chen, B. Wang,Space of Ricci flows (II)—Part A: compactness of the moduli of model spaces, Forum of Mathematics, Sigma(2017), vol. 5. MR3739253, Zbl 06824442. [30] X.X. Chen, B. Wang,K¨ahler Ricci flow on Fano manifolds(I), J. Eur. Math. Soc., 14(6):2001-2038, 2012. MR2984594, Zbl 1257.53094. · Zbl 1257.53094 [31] X.X. Chen, B. Wang,On the conditions to extend Ricci flow (III), IMRN, 2013(10):2349-2367. MR3061942, Zbl 1317.53082. · Zbl 1317.53082 [32] B. Chow, P. Lu, L. Ni,Hamilton’s Ricci Flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006. MR2274812, Zbl 1118.53001. · Zbl 1118.53001 [33] T.C. Collins, G. Sz´ekelyhidi,The twisted K¨ahler Ricci flow, J. Reine Angew. Math., 716:179-205, 2016. MR3518375, Zbl 1357.53076. · Zbl 1357.53076 [34] O. Debarre,Higher-dimensional algebraic geometry, Universitext, SpringerVerlag, New York, 2001. MR1841091, Zbl 0978.14001. · Zbl 0978.14001 [35] W.Y. Ding, G. Tian,K¨ahler-Einstein metrics and the generalized Futaki invariants, Invent. Math., 110(2):315-335, 1992. MR1185586, Zbl 0779.53044. · Zbl 0779.53044 [36] S.K. Donaldson,Scalar curvature and stability of toric varieties, J. Diff. Geom., 62(2):289-349, 2002. MR1988506, Zbl 1074.53059. · Zbl 1074.53059 [37] S.K. Donaldson, S. Sun,Gromov-Hausdorff limits of K¨ahler manifolds and algebraic geometry, Acta Math., 213(1):63-106, 2014. MR3261011, Zbl 1318.53037. · Zbl 1318.53037 [38] R.S. Hamilton,Formation of singularities in the Ricci flow, Surveys in Diff. Geom., 2:7-136, 1995. MR1375255, Zbl 0867.53030. · Zbl 0867.53030 [39] R.S. Hamilton,A compactness property for solutions of the Ricci flow, Amer. J. Math., 117(3):545-572, 1995. MR1333936, Zbl 0840.53029. · Zbl 0840.53029 [40] W.S. Jiang,Bergman Kernel along the K¨ahler Ricci flow and Tian’s conjecture, J. Reine Angew. Math., 717:195-226, 2016. MR3530538, Zbl 1345.53069. · Zbl 1345.53069 [41] B. Kleiner, J. Lott,Notes on Perelman’s papers, Geometry and Topology, 12(5):2587-2855, 2008. MR2460872, Zbl 1204.53033. · Zbl 1204.53033 [42] B. Kotschwar,A local version of Bando’s theorem on the real-analyticity of solutions to the Ricci flow, Bull. Lond. Math. Soc., 45(1):153-158, 2013. MR3033963, Zbl 1259.53065. · Zbl 1259.53065 [43] Z. Lu,On the lower order terms of the asymptotic expansion of Tian-Yau- Zelditch, Amer. J. Math., 122(2):235-273, 2000. MR1749048, Zbl 0972.53042. · Zbl 0972.53042 [44] R.J. McCann, P.M. Topping,Ricci flow, entropy and optimal transportation, Amer. J. Math., 132(3):711-730, 2010. MR2666905, Zbl 1203.53065. · Zbl 1203.53065 [45] J. Morgan, G. Tian,Ricci flow and the Poincar´e conjecture, Clay mathematics monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR2334563, Zbl 1179.57045. · Zbl 1179.57045 [46] S. Paul,Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics, Ann. of Math. (2), 175(1):255-296, 2012. MR2874643, Zbl 1243.14038. · Zbl 1243.14038 [47] S. Paul,A Numerical Criterion for K-Energy maps of Algebraic Manifolds, arXiv:1210.0924. [48] S. Paul,Stable Pairs and Coercive Estimates for The Mabuchi Functional, arXiv:1308.4377. [49] G. Perelman,The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159. · Zbl 1130.53001 [50] V.P. Petersen, G.F. Wei,Relative volume comparison with integral curvature bounds, GAFA, 7(6):1031-1045, 1997. MR1487753, Zbl 0910.53029. · Zbl 0910.53029 [51] D.H. Phong, J. Sturm,On stability and the convergence of the K¨ahler-Ricci flow, J. Differential Geom., 72(1):149-168, 2006. MR2215459, Zbl 1125.53048. · Zbl 1125.53048 [52] A.V. Pogorelov,The Minkowski multidimensional problem. Translated from the Russian by Vladimir Oliker and Introduction by Louis Nirenberg, Scripta Series in Mathematics. Washington, D.C., Winston, 1978. MR0478079, Zbl 0387.53023. · Zbl 0387.53023 [53] D. Riebesehl, F. Schulz,A priori estimates and a Liouville theorem for complex Monge-Amp‘ere equations, Math. Z., 186(1):57-66, 1984. MR0735051, Zbl 0566.32013. · Zbl 0566.32013 [54] N. Sesum,Convergence of a K¨ahler Ricci flow, Math. Res. Lett., 12(5-6):623- 632, 2005. MR2189226, Zbl 1087.53063. · Zbl 1087.53063 [55] N. Sesum, G. Tian,Bounding scalar curvature and diameter along the K¨ahler Ricci flow (after Perelman) and some applications, J. Inst. Math. Jussieu, 7(3):575-587, 2008. MR2427424, Zbl 1147.53056. [56] J. Song, G. Tian,The K¨ahler-Ricci flow through singularities, Invent. Math., 207(2):519-595, 2017. MR3595934, Zbl 06685346. · Zbl 1440.53116 [57] J. Song, B. Weinkove,Contracting exceptional divisors by the K¨ahler-Ricci flow, Duke Math. J., 162(2):367-415, 2013. MR3018957, Zbl 1266.53063. · Zbl 1266.53063 [58] J. Song, B. Weinkove,Lecture notes on the K¨ahler Ricci flow, arXiv:1212.3653. [59] S. Sun, Y.Q. Wang,On the K¨ahler Ricci flow near a K¨ahler Einstein metric, J. Reine Angew. Math., 699:143-158, 2015. MR3305923, Zbl 1314.53123. · Zbl 1314.53123 [60] G. Sz´ekelyhidi,The K¨ahler-Ricci flow and K-polystability, Amer. J. Math., 132(4):1077-1090, 2010. MR2663648, Zbl 1206.53075. · Zbl 1206.53075 [61] G. Sz´ekelyhidi,The partialC0-estimate along the continuity method, JAMS., 29(2):537-560, 2016. MR3454382, Zbl 1335.53098. · Zbl 1335.53098 [62] G. Tian,On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., 101(1):101-172. MR1055713, Zbl 0716.32019. · Zbl 0716.32019 [63] G. Tian,On a set of polarized K¨ahler metrics on algebraic manifolds, J. Differential Geom., 32(1):99-130, 1990. MR1064867, Zbl 0706.53036. · Zbl 0706.53036 [64] G. Tian,K¨ahler-Einstein metrics on algebraic manifolds, Proc. of Int. Congress of Math., Kyoto, 1990, Vol. I, 587-598(1991). MR1159246, Zbl 0747.53038. · Zbl 0747.53038 [65] G. Tian,K¨ahler-Einstein metrics with positive scalar curvature, Invent. Math., 130(1):1-37, 1997. MR1471884, Zbl 0892.53027. · Zbl 0892.53027 [66] G. Tian,Existence of Einstein metrics on Fano manifolds, Metric and differential geometry: The Jeff Cheeger Anniversary volume, X. Dai and X. Rong, edt., Prog. Math., 297:119-159, 2012. MR3220441, Zbl 1250.53044. [67] G. Tian, B. Wang,On the structure of almost Einstein manifolds, JAMS, 28(4):1169-1209, 2015. MR3369910, Zbl 1320.53052. · Zbl 1320.53052 [68] G. Tian, X.H. Zhu,Convergence of K¨ahler Ricci flow, JAMS., 20(3):675-699, 2007. MR2291916, Zbl 1185.53078. · Zbl 1185.53078 [69] G. Tian, X.H. Zhu,Convergence of the K¨ahler Ricci flow on Fano manifolds, J. Reine Angew. Math., 678:223-245, 2013. MR3056108, Zbl1276.14061. · Zbl 1276.14061 [70] G. Tian, Z.L. Zhang,Regularity of K¨ahler Ricci flows on Fano manifolds, Acta Math., 216(1):127-176, 2016. MR3508220, Zbl 1356.53067. · Zbl 1356.53067 [71] V. Tosatti,K¨ahler Ricci flow on stable Fano manifolds, J. Reine Angew. Math., 640:67-84, 2010. MR2629688, Zbl 1189.53069. · Zbl 1189.53069 [72] B. Wang,Ricci flow on orbifold, arXiv:1003.0151. [73] B. Wang,On the Conditions to Extend Ricci Flow(II), IMRN, 2012(14):3192- 3223. MR2946223, Zbl 1251.53040. · Zbl 1251.53040 [74] S.T. Yau,Open problems in geometry, Differential geometry: partial differential equations on manifolds, Los Angeles, CA, 1990, 1-28, Proc. Symp. Pure · Zbl 0801.53001 [75] R.G. Ye,The logarithmic Sobolev inequality along the Ricci flow: The case λ0(g0) = 0, Commun. Math. Stat., 2(3-4):363-368, 2014. MR3326237, Zbl 1316.53080. · Zbl 1316.53080 [76] Q.S. Zhang,A uniform Sobolev inequality under Ricci flow, IMRN, 2007(17), article ID rnm056. MR2354801, Zbl 1141.53064. · Zbl 1141.53064 [77] Q.S. Zhang,Some gradient estimates for the heat equation on domains and for an equation by Perelman, IMRN, 2006(15), article ID 92314. MR2250008, Zbl 1123.35006. · Zbl 1123.35006 [78] Q.S. Zhang,Bounds on volume growth of geodesic balls under Ricci flow, Math. Res. Lett., 19(1):245-253, 2012. MR2923189, Zbl 1272.53056. · Zbl 1272.53056
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