Regularity Scales and Convergence of the Calabi Flow

The Journal of Geometric Analysis - Tập 28 - Trang 2050-2101 - 2017
Haozhao Li1, Bing Wang2, Kai Zheng3
1Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
2Department of Mathematics, University of Wisconsin-Madison, Madison, USA
3Mathematics Institute, University of Warwick, Coventry, UK

Tóm tắt

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kähler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.

Tài liệu tham khảo

Anderson, M.T.: Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102, 429–445 (1990) Anderson, M.T., Cheeger, J.: \(C^{\alpha }\)-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differ. Geom. 35(2), 265–281 (1992) Berman, R.: A thermodynamical formalism for Monge-Ampre equations, Moser-Trudinger inequalities and Kähler Einstein metrics. Adv. Math. 248, 1254–1297 (2013) Berman, R., Berndtsson, B.: Convexity of the \(K\)-energy on the space of Kähler metrics and uniqueness of extremal metrics. arXiv:1405.0401 Calabi, E.: Extremal Kähler metrics, Seminar on Differential Geometry. Ann. of Math. Stud., vol. 102, pp. 259–290. Princeton University Press, Princeton (1982) Calabi, E.: Extremal Kähler Metrics. II. Differential Geometry and Complex Analysis. Springer, Berlin (1985) Calabi, E., Chen, X.X.: The space of Kähler metrics.II. J. Differ. Geom. 61(2), 173–193 (2002) Calabi, E., Hartman, P.: On the smoothness of isometries. Duke Math. J. 37, 741–750 (1970) Cao, H.D.: Deformation of Kähler metrics to Kähler Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985) Chau, A., Tam, L.F.: On the complex structure of Kähler manifolds with nonnegative curvature. J. Differ. Geom. 73(3), 491–530 (2006) Chen, X.X.: Space of Kähler metrics. J. Differ. Geom. 56, 189–234 (2000) Chen, X.X.: Calabi flow in Riemann surfaces revisited: a new point of view. Int. Math. Res. Not. 6, 275–297 (2001) Chen, X.X.: Private communication Chen, X.X.: Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance. Invent. Math. 175(3), 453–503 (2009) Chen, X.X., He, W.Y.: On the Calabi flow. Am. J. Math. 130(2), 539–570 (2008) Chen, X.X., He, W.Y.: The Calabi flow on toric Fano surfaces. Math. Res. Lett. 17(2), 231–241 (2010) Chen, X.X., He, W.Y.: The Calabi flow on Kähler surfaces with bounded Sobolev constant (I). Math. Ann. 354(1), 227–261 (2012) Chen, X.X., Lebrun, C., Weber, B.: On conformally Kähler Einstein manifolds. J. Am. Math. Soc. 21(4), 1137–1168 (2008) Chen, X.X., Li, L., Paun, M.: Approximation of weak geodesics and subharmonicity of Mabuchi energy. arXiv:1409.7896 Chen, X.X., Sun, S.: Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics. Ann. Math. 180, 407–454 (2014) Chen, X.X., Tian, G.: Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci., No. 107, pp. 1–107 (2008) Chen, X.X., Wang, B.: On the conditions to extend Ricci flow (III). Int. Math. Res. Not., No. 10, pp. 2349–2367 (2013) Chen, X.X., Wang, B.: Space of Ricci flows (I). Commun. Pure Appl. Math. 65(10), 1399–1457 (2012) Chen, X.X., Wang, B.: Space of Ricci flows (II). arXiv: 1405.6797 Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence, RI (2006) Chruciel, P.T.: Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation. Commun. Math. Phys. 137, 289–313 (1991) Collins, T.C., Székelyhidi, G.: The twisted Kähler Ricci flow. arXiv: 1207.5441 Donaldson, S.K.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Northen California Symplectic Geometry Seminar. Amer. Math. Soc. Transl. Ser. No. 2 vol. 196, pp. 13–33. Am. Math. Soc., Providence, RI (1999) Donaldson, S.K.: Stability, birational transformations, and the Kähler-Einstein problem. Surveys in Differential Geometry, vol. 17. International Press (2012) Donaldson, S.K.: Conjectures in Kähler geometry. Strings and Geometry, Clay Math. Proc., vol. 3, pp. 71–78. Amer. Math. Soc., Providence, RI (2004) Donaldson, S.K.: Private communication Fine, J.: Calabi flow and projective embeddings. With an appendix by Kefeng Liu and Xiaonan Ma. J. Differ. Geom. 84(3), 489–523 (2010) Fujiki, A.: Moduli space of polarized algebraic manifolds and Kähler metrics. [trans. of Sugaku 42(1990), no. 3, 231-243; MR 1073369], Sugaku Exposition 5, 173–191 (1992) Glickenstein, D.: Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates. Geom. Topol. 7, 487–510 (2003) Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982) He, W.Y.: On the convergence of the Calabi flow. arXiv:1303.3056 Huang, H.N., Feng, R.J.: The Gbloal Existence and Convergence of the Calabi flow on \({\mathbb{C}}^{n}/ {\mathbb{Z}}_{n+i}{\mathbb{Z}}_{N}\) Huang, H.N., Zheng, K.: Stability of the Calabi flow near an extremal metric. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(1), 167–175 (2012) Li, H.Z., Zheng, K.: Kähler non-collapsing, eigenvalues and the Calabi flow. J. Funct. Anal. 267(5), 1593–1636 (2014) Mabuchi, T.: Some symplectic geometry on compact Kähler manifolds. I. Osaka J. Math. 24(2), 227–252 (1987) Semmes, S.: Complex monge-ampere and symplectic manifolds. Am. J. Math. 114, 495–550 (1992) Sesum, N.: Curvature tensor under the Ricci flow. Am. J. Math. 127(6), 1315–1324 (2005) Sesum, N., Tian, G.: Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman) and some applications. J. Inst. Math. Jussieu 7(3), 575–587 (2008) Streets, J.: Long time existence of minimizing movement solutions of Calabi flow. Adv. Math. 259, 688–729 (2014) Streets, J.: The consistency and convergence of \(K\)-energy minimizing movements. arXiv:1301.3948 (to appear) Streets, J.: The long time behavior of fourth order curvature flows. Calc. Var. Partial Differ. Equ. 46(1–2), 39–54 (2013) Struwe, M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(2), 247–274 (2002) Székelyhidi, G.: The Calabi functional on a ruled surface. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 837–856 (2009) Tian, G., Yau, S.T.: Complete Kähler manifolds with zero Ricci curvature. I. J. Am. Math. Soc. 3(3), 579–609 (1990) Tian, G., Zhang, Z.L.: Regularity of Kähler Ricci flows on Fano manifolds. arXiv:1310.5897 Tian, G., Zhu, X.H.: Convergence of Kähler Ricci flow. J. Am. Math. Soc. 20, 675–699 (2007) Tian, G., Zhu, X.H.: Convergence of Kähler-Ricci flow on Fano manifolds. J. Reine Angew. Math. 678, 223–245 (2013) Wang, B.: On the conditions to extend Ricci flow (II). Int. Math. Res. Not. 14, 3192–3223 (2012)