Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

The partial $ C^0$-estimate along the continuity method


Author: Gábor Székelyhidi
Journal: J. Amer. Math. Soc. 29 (2016), 537-560
MSC (2010): Primary 53C55; Secondary 53C23
DOI: https://doi.org/10.1090/jams/833
Published electronically: April 29, 2015
MathSciNet review: 3454382
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the partial $ C^0$-estimate holds for metrics along Aubin's continuity method for finding Kähler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on the analogous problem for conical Kähler-Einstein metrics.


References [Enhancements On Off] (What's this?)

  • [1] Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429-445. MR 1074481 (92c:53024), https://doi.org/10.1007/BF01233434
  • [2] Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63-95 (French, with English summary). MR 494932 (81d:53047)
  • [3] Thierry Aubin, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d'une inégalité, J. Funct. Anal. 57 (1984), no. 2, 143-153 (French). MR 749521 (85k:58084), https://doi.org/10.1016/0022-1236(84)90093-4
  • [4] Zbigniew Błocki, On the regularity of the complex Monge-Ampère operator, Complex geometric analysis in Pohang (1997), Contemp. Math., vol. 222, Amer. Math. Soc., Providence, RI, 1999, pp. 181-189. MR 1653050 (99m:32018), https://doi.org/10.1090/conm/222/03161
  • [5] G. Carron, Some old and new results about rigidity of critical metric, available at arXiv:1012.0685.
  • [6] Jeff Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds, Lezioni Fermiane. [Fermi Lectures], Scuola Normale Superiore, Pisa, 2001. MR 2006642 (2004j:53049)
  • [7] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406-480. MR 1484888 (98k:53044)
  • [8] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom. 54 (2000), no. 1, 13-35. MR 1815410 (2003a:53043)
  • [9] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), no. 1, 37-74. MR 1815411 (2003a:53044)
  • [10] J. Cheeger, T. H. Colding, and G. Tian, On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal. 12 (2002), no. 5, 873-914. MR 1937830 (2003m:53053), https://doi.org/10.1007/PL00012649
  • [11] Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics and stability, Int. Math. Res. Not. IMRN 8 (2014), 2119-2125. MR 3194014
  • [12] Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $ 2\pi $, J. Amer. Math. Soc. 28 (2015), no. 1, 199-234. MR 3264767, https://doi.org/10.1090/S0894-0347-2014-00800-6
  • [13] Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $ 2\pi $ and completion of the main proof, J. Amer. Math. Soc. 28 (2015), no. 1, 235-278. MR 3264768, https://doi.org/10.1090/S0894-0347-2014-00801-8
  • [14] Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), no. 3, 477-501. MR 1454700 (98d:53050), https://doi.org/10.2307/2951841
  • [15] Jean-Pierre Demailly and János Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), no. 4, 525-556 (English, with English and French summaries). MR 1852009 (2002e:32032), https://doi.org/10.1016/S0012-9593(01)01069-2
  • [16] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289-349. MR 1988506 (2005c:32028)
  • [17] S. K. Donaldson, Discussion of the Kähler-Einstein problem (2009). http://www2.imperial.ac.uk/~skdona/KENOTES.PDF.
  • [18] Simon Donaldson and Song Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math. 213 (2014), no. 1, 63-106. MR 3261011, https://doi.org/10.1007/s11511-014-0116-3
  • [19] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725 (80b:14001)
  • [20] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu, and S. Semmes; Translated from the French by Sean Michael Bates. MR 2307192 (2007k:53049)
  • [21] C. Denson Hill and Michael Taylor, The complex Frobenius theorem for rough involutive structures, Trans. Amer. Math. Soc. 359 (2007), no. 1, 293-322 (electronic). MR 2247892 (2007f:32033), https://doi.org/10.1090/S0002-9947-06-04067-0
  • [22] Zhiqin Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235-273. MR 1749048 (2002d:32034)
  • [23] Albert Nijenhuis and William B. Woolf, Some integration problems in almost-complex and complex manifolds., Ann. of Math. (2) 77 (1963), 424-489. MR 0149505 (26 #6992)
  • [24] S. T. Paul, Stable pairs and coercive estimates for the Mabuchi functional, available at arXiv:1308.4377.
  • [25] Sean Timothy Paul, Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics, Ann. of Math. (2) 175 (2012), no. 1, 255-296. MR 2874643 (2012m:32029), https://doi.org/10.4007/annals.2012.175.1.7
  • [26] Sean Timothy Paul and Gang Tian, CM stability and the generalized Futaki invariant II, Astérisque 328 (2009), 339-354 (2010) (English, with English and French summaries). MR 2674882 (2012a:32027)
  • [27] D. H. Phong, Julius Ross, and Jacob Sturm, Deligne pairings and the Knudsen-Mumford expansion, J. Differential Geom. 78 (2008), no. 3, 475-496. MR 2396251 (2008k:32072)
  • [28] D. H. Phong, J. Song, and J. Sturm, Degenerations of Kähler-Ricci solitons on Fano manifolds, available at arXiv:1211.5849.
  • [29] D. H. Phong, Jian Song, Jacob Sturm, and Ben Weinkove, The Moser-Trudinger inequality on Kähler-Einstein manifolds, Amer. J. Math. 130 (2008), no. 4, 1067-1085. MR 2427008 (2009e:32027), https://doi.org/10.1353/ajm.0.0013
  • [30] Wei-Dong Ruan, Canonical coordinates and Bergmann [Bergman] metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589-631. MR 1638878 (2000a:32050)
  • [31] Wei-Dong Ruan, On the convergence and collapsing of Kähler metrics, J. Differential Geom. 52 (1999), no. 1, 1-40. MR 1743466 (2001e:53042)
  • [32] Richard Schoen and Karen Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982), no. 2, 307-335. MR 664498 (84b:58037a)
  • [33] Yum Tong Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53-156. MR 0352516 (50 #5003)
  • [34] Yum Tong Siu and Shing Tung Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 363-380. MR 645748 (83g:32027)
  • [35] G. Tian, K-stability and Kähler-Einstein metrics, available at arXiv:1211.4669.
  • [36] G. Tian, Kähler-Einstein metrics on algebraic manifolds, Proc. of Int. Congress of Math., Mathematical Society of Japan, Tokyo, 1990, pp. 587-598.
  • [37] G. Tian, Stability of pairs, available at arXiv:1310.5544.
  • [38] Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99-130. MR 1064867 (91j:32031)
  • [39] G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101-172. MR 1055713 (91d:32042), https://doi.org/10.1007/BF01231499
  • [40] Gang Tian, Compactness theorems for Kähler-Einstein manifolds of dimension $ 3$ and up, J. Differential Geom. 35 (1992), no. 3, 535-558. MR 1163448 (93g:53066)
  • [41] Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 137 (1997), no. 3, 1-37.
  • [42] Gang Tian, Canonical metrics in Kähler geometry, Birkhäuser Verlag, Basel, 2000. Lectures in Mathematics ETH Zürich.
  • [43] Gang Tian, Extremal metrics and geometric stability, Houston J. Math. 28 (2002), no. 2, 411-432. Special issue for S. S. Chern. MR 1898198 (2003i:53062)
  • [44] Gang Tian, Existence of Einstein metrics on Fano manifolds, Metric and differential geometry, Progr. Math., vol. 297, Birkhäuser/Springer, Basel, 2012, pp. 119-159. MR 3220441, https://doi.org/10.1007/978-3-0348-0257-4_5
  • [45] G. Tian and Z. Zhang, Regularity of Kähler-Ricci flows on Fano manifolds, available at arXiv:1310.5897.
  • [46] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. MR 480350 (81d:53045), https://doi.org/10.1002/cpa.3160310304
  • [47] Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1-28. MR 1216573 (94k:53001)
  • [48] Steve Zelditch, Szegő kernels and a theorem of Tian, Int. Math. Res. Not. IMRN 6 (1998), 317-331. MR 1616718 (99g:32055), https://doi.org/10.1155/S107379289800021X

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 53C55, 53C23

Retrieve articles in all journals with MSC (2010): 53C55, 53C23


Additional Information

Gábor Székelyhidi
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: gszekely@nd.edu

DOI: https://doi.org/10.1090/jams/833
Received by editor(s): November 5, 2013
Received by editor(s) in revised form: June 26, 2014, and March 18, 2015
Published electronically: April 29, 2015
Additional Notes: The author was supported in part by NSF grant DMS-1306298.
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society