Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

Abstract

We study the uniformization conjecture of Yau by using the Gromov-Hausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if $M^n$ is a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, then $M$ is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kähler manifolds with nonnegative bisectional curvature.

Note: To view the article, click on the URL link for the DOI number.

  • [[CC1]] Go to document J. Cheeger and T. H. Colding, "Lower bounds on Ricci curvature and the almost rigidity of warped products," Ann. of Math., vol. 144, iss. 1, pp. 189-237, 1996.
    @article{[CC1], mrkey = {1405949},
      author = {Cheeger, Jeff and Colding, Tobias H.},
      title = {Lower bounds on {R}icci curvature and the almost rigidity of warped products},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {144},
      year = {1996},
      number = {1},
      pages = {189--237},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53C21 (53C20 53C23)},
      mrnumber = {1405949},
      mrreviewer = {Joseph E. Borzellino},
      doi = {10.2307/2118589},
      zblnumber = {0865.53037},
      }
  • [[CC2]] Go to document J. Cheeger and T. H. Colding, "On the structure of spaces with Ricci curvature bounded below. I," J. Differential Geom., vol. 46, iss. 3, pp. 406-480, 1997.
    @article{[CC2], mrkey = {1484888},
      author = {Cheeger, Jeff and Colding, Tobias H.},
      title = {On the structure of spaces with {R}icci curvature bounded below. {I}},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {46},
      year = {1997},
      number = {3},
      pages = {406--480},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C21 (53C20)},
      mrnumber = {1484888},
      mrreviewer = {William P. Minicozzi, II},
      url = {http://projecteuclid.org/euclid.jdg/1214459974},
      zblnumber = {0902.53034},
      }
  • [[CC3]] Go to document J. Cheeger and T. H. Colding, "On the structure of spaces with Ricci curvature bounded below. II," J. Differential Geom., vol. 54, iss. 1, pp. 13-35, 2000.
    @article{[CC3], mrkey = {1815410},
      author = {Cheeger, Jeff and Colding, Tobias H.},
      title = {On the structure of spaces with {R}icci curvature bounded below. {II}},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {54},
      year = {2000},
      number = {1},
      pages = {13--35},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C21 (49Q15 53C20 53C23)},
      mrnumber = {1815410},
      mrreviewer = {Man Chun Leung},
      url = {http://projecteuclid.org/euclid.jdg/1214342145},
      zblnumber = {1027.53042},
      }
  • [[CC4]] Go to document J. Cheeger and T. H. Colding, "On the structure of spaces with Ricci curvature bounded below. III," J. Differential Geom., vol. 54, iss. 1, pp. 37-74, 2000.
    @article{[CC4], mrkey = {1815411},
      author = {Cheeger, Jeff and Colding, Tobias H.},
      title = {On the structure of spaces with {R}icci curvature bounded below. {III}},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {54},
      year = {2000},
      number = {1},
      pages = {37--74},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C21 (49Q15 53C20 53C23)},
      mrnumber = {1815411},
      mrreviewer = {Man Chun Leung},
      url = {http://projecteuclid.org/euclid.jdg/1214342146},
      zblnumber = {1027.53043},
      }
  • [[CCT]] Go to document J. Cheeger, T. H. Colding, and G. Tian, "On the singularities of spaces with bounded Ricci curvature," Geom. Funct. Anal., vol. 12, iss. 5, pp. 873-914, 2002.
    @article{[CCT], mrkey = {1937830},
      author = {Cheeger, Jeff and Colding, Tobias H. and Tian, G.},
      title = {On the singularities of spaces with bounded {R}icci curvature},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {12},
      year = {2002},
      number = {5},
      pages = {873--914},
      issn = {1016-443X},
      coden = {GFANFB},
      mrclass = {53C21 (53C20)},
      mrnumber = {1937830},
      mrreviewer = {Zhongmin Shen},
      doi = {10.1007/PL00012649},
      zblnumber = {1030.53046},
      }
  • [[CFLZ]] Go to document B. Chen, X. Fu, L. Yin, and X. Zhu, "Sharp dimension estimates of holomorphic functions and rigidity," Trans. Amer. Math. Soc., vol. 358, iss. 4, pp. 1435-1454, 2006.
    @article{[CFLZ], mrkey = {2186981},
      author = {Chen, Bing-Long and Fu, Xiao-Yong and Yin, Le and Zhu, Xi-Ping},
      title = {Sharp dimension estimates of holomorphic functions and rigidity},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {358},
      year = {2006},
      number = {4},
      pages = {1435--1454},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {32Q30 (32Q10 32Q15)},
      mrnumber = {2186981},
      mrreviewer = {Peng Lu},
      doi = {10.1090/S0002-9947-05-04105-X},
      zblnumber = {1159.32303},
      }
  • [[CT]] Go to document A. Chau and L. Tam, "On the complex structure of Kähler manifolds with nonnegative curvature," J. Differential Geom., vol. 73, iss. 3, pp. 491-530, 2006.
    @article{[CT], mrkey = {2228320},
      author = {Chau, Albert and Tam, Luen-Fai},
      title = {On the complex structure of {K}ähler manifolds with nonnegative curvature},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {73},
      year = {2006},
      number = {3},
      pages = {491--530},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C44 (32Q15 53C21)},
      mrnumber = {2228320},
      mrreviewer = {Zhou Zhang},
      url = {http://projecteuclid.org/euclid.jdg/1146169936},
      zblnumber = {1161.53351},
      }
  • [[CTZ]] Go to document B. Chen, S. Tang, and X. Zhu, "A uniformization theorem for complete non-compact Kähler surfaces with positive bisectional curvature," J. Differential Geom., vol. 67, iss. 3, pp. 519-570, 2004.
    @article{[CTZ], mrkey = {2153028},
      author = {Chen, Bing-Long and Tang, Siu-Hung and Zhu, Xi-Ping},
      title = {A uniformization theorem for complete non-compact {K}ähler surfaces with positive bisectional curvature},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {67},
      year = {2004},
      number = {3},
      pages = {519--570},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C44 (53C21)},
      mrnumber = {2153028},
      mrreviewer = {Peng Lu},
      url = {http://projecteuclid.org/euclid.jdg/1102091357},
      zblnumber = {1100.32009},
      }
  • [[CY]] Go to document S. Y. Cheng and S. T. Yau, "Differential equations on Riemannian manifolds and their geometric applications," Comm. Pure Appl. Math., vol. 28, iss. 3, pp. 333-354, 1975.
    @article{[CY], mrkey = {0385749},
      author = {Cheng, S. Y. and Yau, S. T.},
      title = {Differential equations on {R}iemannian manifolds and their geometric applications},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {28},
      year = {1975},
      number = {3},
      pages = {333--354},
      issn = {0010-3640},
      mrclass = {53C20 (58G99)},
      mrnumber = {0385749},
      mrreviewer = {Lung Ock Chung},
      doi = {10.1002/cpa.3160280303},
      zblnumber = {0312.53031},
      }
  • [[CZ]] Go to document B. Chen and X. Zhu, "On complete noncompact Kähler manifolds with positive bisectional curvature," Math. Ann., vol. 327, iss. 1, pp. 1-23, 2003.
    @article{[CZ], mrkey = {2005119},
      author = {Chen, Bing-Long and Zhu, Xi-Ping},
      title = {On complete noncompact {K}ähler manifolds with positive bisectional curvature},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {327},
      year = {2003},
      number = {1},
      pages = {1--23},
      issn = {0025-5831},
      coden = {MAANA},
      merciless = {53C21 (32Q10 53C55)},
      mrnumber = {2005119},
      mrreviewer = {Man Chun Leung},
      doi = {10.1007/s00208-003-0385-1},
      zblnumber = {1034.32015},
      }
  • [[D]] J. P. Demailly, Analytic Methods in Algebraic Geometry, Somerville, MA: International Press, 2012, vol. 1.
    @book{[D],
      author = {Demailly, J. P.},
      title = {Analytic Methods in Algebraic Geometry},
      series = {Surveys of Modern Mathematics},
      publisher = {International Press},
      address = {Somerville, MA},
      volume = {1},
      pages = {viii+231},
      year = {2012},
      isbn = {978-1-57146-234-3},
      mrclass = {32-02 (14C30 14F18 32J25 32Q15 32U40)},
      mrnumber = {2978333},
      zblnumber = {1271.14001},
      }
  • [[DS]] Go to document S. Donaldson and S. Sun, "Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry," Acta Math., vol. 213, iss. 1, pp. 63-106, 2014.
    @article{[DS], mrkey = {3261011},
      author = {Donaldson, Simon and Sun, Song},
      title = {Gromov-{H}ausdorff limits of {K}ähler manifolds and algebraic geometry},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {213},
      year = {2014},
      number = {1},
      pages = {63--106},
      issn = {0001-5962},
      mrclass = {53C55 (32Q20 53C23)},
      mrnumber = {3261011},
      mrreviewer = {Valentino Tosatti},
      doi = {10.1007/s11511-014-0116-3},
      zblnumber = {1318.53037},
      }
  • [[G]] M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Boston: Birkhäuser, 1999.
    @book{[G], mrkey = {1699320},
      author = {Gromov, Misha},
      title = {Metric Structures for {R}iemannian and non-{R}iemannian Spaces},
      series = {Progr. Math.},
      number = {152},
      publisher = {Birkhäuser},
      address = {Boston},
      year = {1999},
      pages = {xx+585},
      isbn = {0-8176-3898-9},
      mrclass = {53C23 (53-02)},
      mrnumber = {1699320},
      mrreviewer = {Igor Belegradek},
      zblnumber = {0953.53002},
      }
  • [[GPR]] Go to document H. Grauert, T. Peternell, and R. Remmert, Several Complex Variables. VII, New York: Springer-Verlag, 1994, vol. 74.
    @book{[GPR],
      author = {Grauert, H. and Peternell, Th. and Remmert, R.},
      mrkey = {1326617},
      title = {Several Complex Variables. {VII}},
      series = {Encycl. Math. Sci.},
      volume = {74},
      publisher = {Springer-Verlag},
      year = {1994},
      pages = {vi+369},
      isbn = {3-540-56259-1},
      mrclass = {32-02 (32-06 32Cxx 32Dxx 32Fxx 32Jxx 32Lxx)},
      mrnumber = {1326617},
      mrreviewer = {J. S. Joel},
      doi = {10.1007/978-3-662-09873-8},
      address = {New York},
      zblnumber = {0793.00010},
      }
  • [[H]] Go to document R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
    @book{[H],
      author = {Hartshorne, R.},
      title = {Algebraic Geometry},
      series = {Graduate Texts in Math},
      volume = {52},
      year = {1977},
      pages = {xvi+496},
      publisher = {Springer-Verlag},
      address = {New York},
      mrclass = {14-01},
      zblnumber = {0531.14001},
      mrnumber = {0463157},
      doi = {10.1007/978-1-4757-3849-0},
      }
  • [[Ho]] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Third ed., Amsterdam: North-Holland Publishing Co., 1990, vol. 7.
    @book{[Ho], mrkey = {1045639},
      author = {H{ö}rmander, Lars},
      title = {An Introduction to Complex Analysis in Several Variables},
      series = {North-Holland Math. Library},
      volume = {7},
      edition = {Third},
      publisher = {North-Holland Publishing Co.},
      address = {Amsterdam},
      year = {1990},
      pages = {xii+254},
      isbn = {0-444-88446-7},
      mrclass = {32-01 (35N15)},
      mrnumber = {1045639},
      zblnumber = {0685.32001},
      }
  • [[HT]] C. T. Huang and L. -F. Tam, Kähler-Ricci flow with unbounded curvature, 2015.
    @misc{[HT],
      author = {Huang, C. T. and Tam, L.-F.},
      title = {Kähler-{R}icci flow with unbounded curvature},
      arxiv = {1506.00322},
      year = {2015},
      }
  • [[LT2]] Go to document P. Li and L. Tam, "Complete surfaces with finite total curvature," J. Differential Geom., vol. 33, iss. 1, pp. 139-168, 1991.
    @article{[LT2], mrkey = {1085138},
      author = {Li, Peter and Tam, Luen-Fai},
      title = {Complete surfaces with finite total curvature},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {33},
      year = {1991},
      number = {1},
      pages = {139--168},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C21 (53C22 53C23 53C45)},
      mrnumber = {1085138},
      mrreviewer = {Chun-Li Shen},
      url = {http://projecteuclid.org/euclid.jdg/1214446033},
      zblnumber = {0749.53025},
      }
  • [[LS]] Go to document P. Li and R. Schoen, "$L^p$ and mean value properties of subharmonic functions on Riemannian manifolds," Acta Math., vol. 153, iss. 3-4, pp. 279-301, 1984.
    @article{[LS], mrkey = {0766266},
      author = {Li, Peter and Schoen, Richard},
      title = {{$L\sp p$} and mean value properties of subharmonic functions on {R}iemannian manifolds},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {153},
      year = {1984},
      number = {3-4},
      pages = {279--301},
      issn = {0001-5962},
      coden = {ACMAA8},
      mrclass = {58G20 (31C12)},
      mrnumber = {0766266},
      mrreviewer = {Helmut Kaul},
      doi = {10.1007/BF02392380},
      zblnumber = {0556.31005},
      }
  • [[L1]] G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, 2013.
    @misc{[L1],
      author = {Liu, G.},
      title = {Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds},
      arxiv = {1308.0710},
      year = {2013},
      note={to appear in {\em Duke Math. J.}},
      }
  • [[L2]] Go to document G. Liu, "On the volume growth of Kähler manifolds with nonnegative bisectional curvature," J. Differential Geom., vol. 102, iss. 3, pp. 485-500, 2016.
    @article{[L2], mrkey={3466805},
      author = {Liu, G.},
      TITLE = {On the volume growth of {K}ähler manifolds with nonnegative bisectional curvature},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {102},
      YEAR = {2016},
      NUMBER = {3},
      PAGES = {485--500},
      ISSN = {0022-040X},
      MRCLASS = {53C20 (32Q15 53C55)},
      MRNUMBER = {3466805},
      ZBLNUMBER = {06571777},
      URL = {http://projecteuclid.org/euclid.jdg/1456754016},
      }
  • [[LY]] Go to document P. Li and S. Yau, "On the parabolic kernel of the Schrödinger operator," Acta Math., vol. 156, iss. 3-4, pp. 153-201, 1986.
    @article{[LY], mrkey = {0834612},
      author = {Li, Peter and Yau, Shing-Tung},
      title = {On the parabolic kernel of the {S}chrödinger operator},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {156},
      year = {1986},
      number = {3-4},
      pages = {153--201},
      issn = {0001-5962},
      coden = {ACMAA8},
      mrclass = {58G11 (35J10)},
      mrnumber = {0834612},
      mrreviewer = {Harold Donnelly},
      doi = {10.1007/BF02399203},
      zblnumber = {0611.58045},
      }
  • [[Mo]] Go to document N. Mok, "An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties," Bull. Soc. Math. France, vol. 112, iss. 2, pp. 197-250, 1984.
    @article{[Mo], mrkey = {0788968},
      author = {Mok, Ngaiming},
      title = {An embedding theorem of complete {K}ähler manifolds of positive bisectional curvature onto affine algebraic varieties},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {112},
      year = {1984},
      number = {2},
      pages = {197--250},
      issn = {0037-9484},
      coden = {BSMFAA},
      mrclass = {53C55 (32C10)},
      mrnumber = {0788968},
      mrreviewer = {Hung-Hsi Wu},
      url = {http://www.numdam.org/item?id=BSMF_1984__112__197_0},
      zblnumber = {0536.53062},
      }
  • [[MSY]] Go to document N. Mok, Y. T. Siu, and S. T. Yau, "The Poincaré-Lelong equation on complete Kähler manifolds," Compositio Math., vol. 44, iss. 1-3, pp. 183-218, 1981.
    @article{[MSY], mrkey = {0662462},
      author = {Mok, Ngaiming and Siu, Yum Tong and Yau, Shing Tung},
      title = {The {P}oincaré-{L}elong equation on complete {K}ähler manifolds},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {44},
      year = {1981},
      number = {1-3},
      pages = {183--218},
      issn = {0010-437X},
      coden = {CMPMAF},
      mrclass = {32C10 (32F99 35J99 53C55)},
      mrnumber = {0662462},
      mrreviewer = {Eric Bedford},
      url = {http://www.numdam.org/item?id=CM_1981__44_1-3_183_0},
      zblnumber = {0531.32007},
      }
  • [[Na]] Go to document M. Nagata, "Invariants of a group in an affine ring," J. Math. Kyoto Univ., vol. 3, pp. 369-377, 1963/1964.
    @article{[Na], mrkey = {0179268},
      author = {Nagata, Masayoshi},
      title = {Invariants of a group in an affine ring},
      journal = {J. Math. Kyoto Univ.},
      fjournal = {Journal of Mathematics of Kyoto University},
      volume = {3},
      year = {1963/1964},
      pages = {369--377},
      issn = {0023-608X},
      mrclass = {13.93 (14.05)},
      mrnumber = {0179268},
      mrreviewer = {G. E. Bredon},
      zblnumber = {0146.04501},
      URL = {http://projecteuclid.org/euclid.kjm/1250524787},
     }
  • [[N1]] Go to document L. Ni, "A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature," J. Amer. Math. Soc., vol. 17, iss. 4, pp. 909-946, 2004.
    @article{[N1], mrkey = {2083471},
      author = {Ni, Lei},
      title = {A monotonicity formula on complete {K}ähler manifolds with nonnegative bisectional curvature},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {17},
      year = {2004},
      number = {4},
      pages = {909--946},
      issn = {0894-0347},
      mrclass = {32Q30 (32Q10 32Q15)},
      mrnumber = {2083471},
      doi = {10.1090/S0894-0347-04-00465-5},
      zblnumber = {1071.58020},
      }
  • [[N2]] Go to document L. Ni, "Ancient solutions to Kähler-Ricci flow," Math. Res. Lett., vol. 12, iss. 5-6, pp. 633-653, 2005.
    @article{[N2], mrkey = {2189227},
      author = {Ni, Lei},
      title = {Ancient solutions to {K}ähler-{R}icci flow},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {12},
      year = {2005},
      number = {5-6},
      pages = {633--653},
      issn = {1073-2780},
      mrclass = {53C44 (53C55)},
      mrnumber = {2189227},
      mrreviewer = {John Urbas},
      doi = {10.4310/MRL.2005.v12.n5.a3},
      zblnumber = {1087.53061},
      }
  • [[NT1]] Go to document L. Ni and L. Tam, "Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature," J. Differential Geom., vol. 64, iss. 3, pp. 457-524, 2003.
    @article{[NT1], mrkey = {2032112},
      author = {Ni, Lei and Tam, Luen-Fai},
      title = {Plurisubharmonic functions and the structure of complete {K}ähler manifolds with nonnegative curvature},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {64},
      year = {2003},
      number = {3},
      pages = {457--524},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {32Q30 (32Q15 32U10 53C55)},
      mrnumber = {2032112},
      mrreviewer = {Fangyang Zheng},
      url = {http://projecteuclid.org/euclid.jdg/1090427001},
      zblnumber = {1088.32013},
      }
  • [[NT2]] Go to document L. Ni and L. Tam, "Poincaré-Lelong equation via the Hodge-Laplace heat equation," Compos. Math., vol. 149, iss. 11, pp. 1856-1870, 2013.
    @article{[NT2], mrkey = {3133296},
      author = {Ni, Lei and Tam, Luen-Fai},
      title = {Poincaré-{L}elong equation via the {H}odge-{L}aplace heat equation},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {149},
      year = {2013},
      number = {11},
      pages = {1856--1870},
      issn = {0010-437X},
      mrclass = {53C55 (32Q15 32U40 35B40 35R01)},
      mrnumber = {3133296},
      mrreviewer = {Mohammed El A{ï}di},
      doi = {10.1112/S0010437X12000322},
      zblnumber = {1286.53048},
      }
  • [[Shi1]] Go to document W. Shi, Ricci Deformation of the Metric on Complete Noncompact Kahler Manifolds, Ann Arbor, MI: ProQuest LLC, 1990.
    @book{[Shi1], mrkey = {2638857},
      author = {Shi, Wan-Xiong},
      title = {Ricci Deformation of the Metric on Complete Noncompact {K}ahler Manifolds},
      note = {Ph.D thesis, Harvard University},
      publisher = {ProQuest LLC},
      address = {Ann Arbor, MI},
      year = {1990},
      pages = {203},
      mrclass = {Thesis},
      mrnumber = {2638857},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9035552},
      }
  • [[Shi2]] Go to document W. Shi, "Ricci flow and the uniformization on complete noncompact Kähler manifolds," J. Differential Geom., vol. 45, iss. 1, pp. 94-220, 1997.
    @article{[Shi2], mrkey = {1443333},
      author = {Shi, Wan-Xiong},
      title = {Ricci flow and the uniformization on complete noncompact {K}ähler manifolds},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {45},
      year = {1997},
      number = {1},
      pages = {94--220},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53C55 (53C21 58E11)},
      mrnumber = {1443333},
      mrreviewer = {Knut Smoczyk},
      url = {http://projecteuclid.org/euclid.jdg/1214459756},
      zblnumber = {0954.53043},
      }
  • [[Ti5]] Go to document G. Tian, "Partial $C^0$-estimate for Kähler-Einstein metrics," Commun. Math. Stat., vol. 1, iss. 2, pp. 105-113, 2013.
    @article{[Ti5], mrkey = {3197855},
      author = {Tian, Gang},
      title = {Partial {$C\sp 0$}-estimate for {K}ähler-{E}instein metrics},
      journal = {Commun. Math. Stat.},
      fjournal = {Communications in Mathematics and Statistics},
      volume = {1},
      year = {2013},
      number = {2},
      pages = {105--113},
      issn = {2194-6701},
      mrclass = {32Q20 (32Q10 53C23 53C55)},
      mrnumber = {3197855},
      mrreviewer = {Slimane Benelkourchi},
      doi = {10.1007/s40304-013-0011-9},
      zblnumber = {1280.32014},
      }
  • [[Y1]] S. Yau, "Open problems in geometry," in Differential Geometry: Partial Differential Equations on Manifolds, Providence, RI: Amer. Math. Soc., 1993, vol. 54, pp. 1-28.
    @incollection{[Y1], mrkey = {1216573},
      author = {Yau, Shing-Tung},
      title = {Open problems in geometry},
      booktitle = {Differential Geometry: Partial Differential Equations on Manifolds},
      venue = {{L}os {A}ngeles, {CA},
      1990},
      series = {Proc. Sympos. Pure Math.},
      volume = {54},
      pages = {1--28},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1993},
      mrclass = {53-02},
      mrnumber = {1216573},
      zblnumber = {0801.53001},
      }
  • [[Y2]] Go to document S. Yau, "Nonlinear analysis in geometry," Enseign. Math., vol. 33, iss. 1-2, pp. 109-158, 1987.
    @article{[Y2], mrkey = {0896385},
      author = {Yau, Shing-Tung},
      title = {Nonlinear analysis in geometry},
      journal = {Enseign. Math.},
      fjournal = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
      volume = {33},
      year = {1987},
      number = {1-2},
      pages = {109--158},
      issn = {0013-8584},
      coden = {ENMAAR},
      mrclass = {58-02 (58G30)},
      mrnumber = {0896385},
      zblnumber = {0631.53002},
      URL = {http://www.e-periodica.ch/digbib/view?pid=ens-001:1987:33#242},
      }

Authors

Gang Liu

Northwestern University, Evanston, IL