$L^2$ curvature bounds on manifolds with bounded Ricci curvature

Abstract

Consider a Riemannian manifold with bounded Ricci curvature $|\mathrm{Ric}|\leq n-1$ and the noncollapsing lower volume bound $\mathrm{Vol}(B_1(p))>\mathrm{v}>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $⨏_{B_1(p)}|\mathrm{Rm}|^2(x)\, dx < C(n,\mathrm{v})$,which proves the $L^2$ conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if $(M^n_j,d_j,p_j) \longrightarrow (X,d,p)$ is a $\mathrm{GH}$-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set $\mathcal{S}(X)$ is $n-4$ rectifiable with the uniform Hausdorff measure estimates $H^{n-4}(\mathcal{S}(X)\cap B_1) < C(n,\mathrm{v})$ which, in particular, proves the $n-4$-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for $n-4$ a.e. $x\in \mathcal{S}(X)$, the tangent cone of $X$ at $x$ is unique and isometric to $\mathbb{R}^{n-4}\times C(S^3/\Gamma_x)$ for some $\Gamma_x\subseteq O(4)$ that acts freely away from the origin.

  • [NaVa_Rect_harmonicmap] Go to document A. Naber and D. Valtorta, "Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps," Ann. of Math. (2), vol. 185, iss. 1, pp. 131-227, 2017.
    @ARTICLE{NaVa_Rect_harmonicmap,
      author = {Naber, Aaron and Valtorta, Daniele},
      title = {Rectifiable-{R}eifenberg and the regularity of stationary and minimizing harmonic maps},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {185},
      year = {2017},
      number = {1},
      pages = {131--227},
      issn = {0003-486X},
      mrclass = {58E20 (53C43)},
      mrnumber = {3583353},
      mrreviewer = {Andreas Gastel},
      doi = {10.4007/annals.2017.185.1.3},
      url = {https://doi.org/10.4007/annals.2017.185.1.3},
      zblnumber = {1393.58009},
      }
  • [A89] Go to document M. T. Anderson, "Ricci curvature bounds and Einstein metrics on compact manifolds," J. Amer. Math. Soc., vol. 2, iss. 3, pp. 455-490, 1989.
    @ARTICLE{A89,
      author = {Anderson, Michael T.},
      title = {Ricci curvature bounds and {E}instein metrics on compact manifolds},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {2},
      year = {1989},
      number = {3},
      pages = {455--490},
      issn = {0894-0347},
      mrclass = {53C20 (53C25 58D17 58G30)},
      mrnumber = {0999661},
      mrreviewer = {Maung Min-Oo},
      doi = {10.2307/1990939},
      url = {https://doi.org/10.2307/1990939},
      zblnumber = {0694.53045},
      }
  • [Anderson_Einstein] Go to document M. T. Anderson, "Convergence and rigidity of manifolds under Ricci curvature bounds," Invent. Math., vol. 102, iss. 2, pp. 429-445, 1990.
    @ARTICLE{Anderson_Einstein,
      author = {Anderson, Michael T.},
      title = {Convergence and rigidity of manifolds under {R}icci curvature bounds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {102},
      year = {1990},
      number = {2},
      pages = {429--445},
      issn = {0020-9910},
      mrclass = {53C23 (53C21 58D27)},
      mrnumber = {1074481},
      mrreviewer = {Gudlaugur Thorbergsson},
      doi = {10.1007/BF01233434},
      url = {https://doi.org/10.1007/BF01233434},
      zblnumber = {0711.53038},
      }
  • [Anderson_Hausdorff] Go to document M. T. Anderson, "Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem," Duke Math. J., vol. 68, iss. 1, pp. 67-82, 1992.
    @ARTICLE{Anderson_Hausdorff,
      author = {Anderson, Michael T.},
      title = {Hausdorff perturbations of {R}icci-flat manifolds and the splitting theorem},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {68},
      year = {1992},
      number = {1},
      pages = {67--82},
      issn = {0012-7094},
      mrclass = {53C21 (53C20)},
      mrnumber = {1185818},
      doi = {10.1215/S0012-7094-92-06803-7},
      url = {https://doi.org/10.1215/S0012-7094-92-06803-7},
      zblnumber = {0767.53029},
      }
  • [Anderson_ICM94] Go to document M. T. Anderson, "Einstein metrics and metrics with bounds on Ricci curvature," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 1995, pp. 443-452.
    @INPROCEEDINGS{Anderson_ICM94,
      author = {Anderson, Michael T.},
      title = {Einstein metrics and metrics with bounds on {R}icci curvature},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2 ({Z}ürich, 1994)},
      pages = {443--452},
      publisher = {Birkhäuser, Basel},
      year = {1995},
      mrclass = {53C21 (53C23 53C25)},
      mrnumber = {1403944},
      mrreviewer = {Zhongmin Shen},
      doi = {10.1007/978-3-0348-9078-6_37},
      url = {https://doi.org/10.1007/978-3-0348-9078-6_37},
      zblnumber = {0840.53036},
      }
  • [BKN89] Go to document S. Bando, A. Kasue, and H. Nakajima, "On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth," Invent. Math., vol. 97, iss. 2, pp. 313-349, 1989.
    @ARTICLE{BKN89,
      author = {Bando, Shigetoshi and Kasue, Atsushi and Nakajima, Hiraku},
      title = {On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {97},
      year = {1989},
      number = {2},
      pages = {313--349},
      issn = {0020-9910},
      mrclass = {53C20 (53C25)},
      mrnumber = {1001844},
      mrreviewer = {Thomas H. Otway},
      doi = {10.1007/BF01389045},
      url = {https://doi.org/10.1007/BF01389045},
      zblnumber = {0682.53045},
      }
  • [Cheeger01] J. Cheeger, Degeneration of Riemannian Metrics Under Ricci Curvature Bounds, Scuola Normale Superiore, Pisa, 2001.
    @BOOK{Cheeger01,
      author = {Cheeger, Jeff},
      title = {Degeneration of {R}iemannian Metrics Under {R}icci Curvature Bounds},
      series = {Lezioni Fermiane. [Fermi Lectures]},
      publisher = {Scuola Normale Superiore, Pisa},
      year = {2001},
      pages = {ii+77},
      mrclass = {53C21 (53C20 53C23)},
      mrnumber = {2006642},
      mrreviewer = {Vitali Kapovitch},
      zblnumber = {1055.53024},
      }
  • [Cheeger] Go to document J. Cheeger, "Integral bounds on curvature elliptic estimates and rectifiability of singular sets," Geom. Funct. Anal., vol. 13, iss. 1, pp. 20-72, 2003.
    @ARTICLE{Cheeger,
      author = {Cheeger, Jeff},
      title = {Integral bounds on curvature elliptic estimates and rectifiability of singular sets},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {13},
      year = {2003},
      number = {1},
      pages = {20--72},
      issn = {1016-443X},
      mrclass = {53C21 (49Q99 53C20)},
      mrnumber = {1978491},
      mrreviewer = {Silvano Delladio},
      doi = {10.1007/s000390300001},
      url = {https://doi.org/10.1007/s000390300001},
      zblnumber = {1086.53051},
      }
  • [ChC1] Go to document J. Cheeger and T. H. Colding, "Lower bounds on Ricci curvature and the almost rigidity of warped products," Ann. of Math. (2), vol. 144, iss. 1, pp. 189-237, 1996.
    @ARTICLE{ChC1,
      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. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {144},
      year = {1996},
      number = {1},
      pages = {189--237},
      issn = {0003-486X},
      mrclass = {53C21 (53C20 53C23)},
      mrnumber = {1405949},
      mrreviewer = {Joseph E. Borzellino},
      doi = {10.2307/2118589},
      url = {https://doi.org/10.2307/2118589},
      zblnumber = {0865.53037},
      }
  • [ChC2] 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{ChC2,
      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},
      mrclass = {53C21 (53C20)},
      mrnumber = {1484888},
      mrreviewer = {William P. Minicozzi, II},
      doi = {10.4310/jdg/1214459974},
      url = {https://doi.org/10.4310/jdg/1214459974},
      zblnumber = {0902.53034},
      }
  • [CCTi_eps_reg] 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{CCTi_eps_reg,
      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},
      mrclass = {53C21 (53C20)},
      mrnumber = {1937830},
      mrreviewer = {Zhongmin Shen},
      doi = {10.1007/PL00012649},
      url = {https://doi.org/10.1007/PL00012649},
      zblnumber = {1030.53046},
      }
  • [CheegerNaber_Ricci] Go to document J. Cheeger and A. Naber, "Lower bounds on Ricci curvature and quantitative behavior of singular sets," Invent. Math., vol. 191, iss. 2, pp. 321-339, 2013.
    @ARTICLE{CheegerNaber_Ricci,
      author = {Cheeger, Jeff and Naber, Aaron},
      title = {Lower bounds on {R}icci curvature and quantitative behavior of singular sets},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {191},
      year = {2013},
      number = {2},
      pages = {321--339},
      issn = {0020-9910},
      mrclass = {53C21 (32Q20 53C23 53C25)},
      mrnumber = {3010378},
      mrreviewer = {Leonid V. Kovalev},
      doi = {10.1007/s00222-012-0394-3},
      url = {https://doi.org/10.1007/s00222-012-0394-3},
      zblnumber = {1268.53053},
      }
  • [CheegerNaber_Codimensionfour] Go to document J. Cheeger and A. Naber, "Regularity of Einstein manifolds and the codimension 4 conjecture," Ann. of Math. (2), vol. 182, iss. 3, pp. 1093-1165, 2015.
    @ARTICLE{CheegerNaber_Codimensionfour,
      author = {Cheeger, Jeff and Naber, Aaron},
      title = {Regularity of {E}instein manifolds and the codimension 4 conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {182},
      year = {2015},
      number = {3},
      pages = {1093--1165},
      issn = {0003-486X},
      mrclass = {53C25 (53C23)},
      mrnumber = {3418535},
      mrreviewer = {Luis Guijarro},
      doi = {10.4007/annals.2015.182.3.5},
      url = {https://doi.org/10.4007/annals.2015.182.3.5},
      zblnumber = {1335.53057},
      }
  • [CheegerTian05] Go to document J. Cheeger and G. Tian, "Anti-self-duality of curvature and degeneration of metrics with special holonomy," Comm. Math. Phys., vol. 255, iss. 2, pp. 391-417, 2005.
    @ARTICLE{CheegerTian05,
      author = {Cheeger, Jeff and Tian, Gang},
      title = {Anti-self-duality of curvature and degeneration of metrics with special holonomy},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {255},
      year = {2005},
      number = {2},
      pages = {391--417},
      issn = {0010-3616},
      mrclass = {53C23 (53C29)},
      mrnumber = {2129951},
      mrreviewer = {Lorenz J. Schwachhöfer},
      doi = {10.1007/s00220-004-1279-0},
      url = {https://doi.org/10.1007/s00220-004-1279-0},
      zblnumber = {1081.53038},
      }
  • [Chen_Donaldson14] Go to document X. -X. Chen and S. K. Donaldson, "Integral bounds on curvature and Gromov-Hausdorff limits," J. Topol., vol. 7, iss. 2, pp. 543-556, 2014.
    @ARTICLE{Chen_Donaldson14,
      author = {Chen, X.-X. and Donaldson, S. K.},
      title = {Integral bounds on curvature and {G}romov-{H}ausdorff limits},
      journal = {J. Topol.},
      fjournal = {Journal of Topology},
      volume = {7},
      year = {2014},
      number = {2},
      pages = {543--556},
      issn = {1753-8416},
      mrclass = {53C23 (53C20 53C21)},
      mrnumber = {3217630},
      mrreviewer = {Yu Ding},
      doi = {10.1112/jtopol/jtt037},
      url = {https://doi.org/10.1112/jtopol/jtt037},
      zblnumber = {1308.53057},
      }
  • [ColdingMinicozzi_tangentcone] Go to document T. H. Colding and W. P. Minicozzi II, "On uniqueness of tangent cones for Einstein manifolds," Invent. Math., vol. 196, iss. 3, pp. 515-588, 2014.
    @ARTICLE{ColdingMinicozzi_tangentcone,
      author = {Colding, Tobias Holck and Minicozzi, II, William P.},
      title = {On uniqueness of tangent cones for {E}instein manifolds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {196},
      year = {2014},
      number = {3},
      pages = {515--588},
      issn = {0020-9910},
      mrclass = {53C25 (53C21 53C23)},
      mrnumber = {3211041},
      mrreviewer = {Megan M. Kerr},
      doi = {10.1007/s00222-013-0474-z},
      url = {https://doi.org/10.1007/s00222-013-0474-z},
      zblnumber = {1302.53048},
      }
  • [Fed] H. Federer, Geometric Measure Theory, Springer-Verlag New York Inc., New York, 1969, vol. 153.
    @BOOK{Fed,
      author = {Federer, Herbert},
      title = {Geometric Measure Theory},
      series = {Grundlehren math. Wiss.},
      volume = {153},
      publisher = {Springer-Verlag New York Inc., New York},
      year = {1969},
      pages = {xiv+676},
      mrclass = {28.80 (26.00)},
      mrnumber = {0257325},
      mrreviewer = {J. E. Brothers},
      zblnumber = {0176.00801},
      }
  • [FOT_Dirichlet] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, extended ed., Walter de Gruyter & Co., Berlin, 2011, vol. 19.
    @BOOK{FOT_Dirichlet,
      author = {Fukushima, Masatoshi and Oshima, Yoichi and Takeda, Masayoshi},
      title = {Dirichlet Forms and Symmetric {M}arkov Processes},
      series = {de Gruyter Stud. Math.},
      volume = {19},
      edition = {extended},
      publisher = {Walter de Gruyter \& Co., Berlin},
      year = {2011},
      pages = {x+489},
      isbn = {978-3-11-021808-4},
      mrclass = {60J25 (28A12 31C45 60F10 60J40 60J45 60J55)},
      mrnumber = {2778606},
      zblnumber = {1227.31001},
      }
  • [Hamilton_gradient] Go to document R. S. Hamilton, "A matrix Harnack estimate for the heat equation," Comm. Anal. Geom., vol. 1, iss. 1, pp. 113-126, 1993.
    @ARTICLE{Hamilton_gradient,
      author = {Hamilton, Richard S.},
      title = {A matrix {H}arnack estimate for the heat equation},
      journal = {Comm. Anal. Geom.},
      fjournal = {Communications in Analysis and Geometry},
      volume = {1},
      year = {1993},
      number = {1},
      pages = {113--126},
      issn = {1019-8385},
      mrclass = {58G11 (35K05)},
      mrnumber = {1230276},
      mrreviewer = {Wei Yue Ding},
      doi = {10.4310/CAG.1993.v1.n1.a6},
      url = {https://doi.org/10.4310/CAG.1993.v1.n1.a6},
      zblnumber = {0799.53048},
      }
  • [Kot_hamilton_gradient] Go to document B. L. Kotschwar, "Hamilton’s gradient estimate for the heat kernel on complete manifolds," Proc. Amer. Math. Soc., vol. 135, iss. 9, pp. 3013-3019, 2007.
    @ARTICLE{Kot_hamilton_gradient,
      author = {Kotschwar, Brett L.},
      title = {Hamilton's gradient estimate for the heat kernel on complete manifolds},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the American Mathematical Society},
      volume = {135},
      year = {2007},
      number = {9},
      pages = {3013--3019},
      issn = {0002-9939},
      mrclass = {58J35},
      mrnumber = {2317980},
      mrreviewer = {Peng Lu},
      doi = {10.1090/S0002-9939-07-08837-5},
      url = {https://doi.org/10.1090/S0002-9939-07-08837-5},
      zblnumber = {1127.58021},
      }
  • [LiYau_heatkernel86] 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{LiYau_heatkernel86,
      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},
      mrclass = {58G11 (35J10)},
      mrnumber = {0834612},
      mrreviewer = {Harold Donnelly},
      doi = {10.1007/BF02399203},
      url = {https://doi.org/10.1007/BF02399203},
      zblnumber = {0611.58045},
      }
  • [Na_14ICM] A. Naber, "The geometry of Ricci curvature," in Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, 2014, pp. 911-937.
    @INPROCEEDINGS{Na_14ICM,
      author = {Naber, Aaron},
      title = {The geometry of {R}icci curvature},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians---{S}eoul 2014. {V}ol. {II}},
      pages = {911--937},
      publisher = {Kyung Moon Sa, Seoul},
      year = {2014},
      mrclass = {53C21},
      mrnumber = {3728645},
      mrreviewer = {Shouhei Honda},
      zblnumber = {1376.53003},
      }
  • [NaVa_CriticalSets] Go to document A. Naber and D. Valtorta, "Volume estimates on the critical sets of solutions to elliptic PDEs," Comm. Pure Appl. Math., vol. 70, iss. 10, pp. 1835-1897, 2017.
    @ARTICLE{NaVa_CriticalSets,
      author = {Naber, Aaron and Valtorta, Daniele},
      title = {Volume estimates on the critical sets of solutions to elliptic {PDE}s},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {70},
      year = {2017},
      number = {10},
      pages = {1835--1897},
      issn = {0010-3640},
      mrclass = {35J15 (35A20 35R01)},
      mrnumber = {3688031},
      mrreviewer = {Xinhua Ji},
      doi = {10.1002/cpa.21708},
      url = {https://doi.org/10.1002/cpa.21708},
      zblnumber = {1376.35021},
      }
  • [Petersen_RiemannianGeometry] Go to document P. Petersen, Riemannian Geometry, Third ed., Springer, Cham, 2016, vol. 171.
    @BOOK{Petersen_RiemannianGeometry,
      author = {Petersen, Peter},
      title = {Riemannian Geometry},
      series = {Grad. Texts in Math.},
      volume = {171},
      edition = {Third},
      publisher = {Springer, Cham},
      year = {2016},
      pages = {xviii+499},
      isbn = {978-3-319-26652-7; 978-3-319-26654-1},
      mrclass = {53-01 (53C20 53C21 53C23)},
      mrnumber = {3469435},
      doi = {10.1007/978-3-319-26654-1},
      url = {https://doi.org/10.1007/978-3-319-26654-1},
      zblnumber = {1417.53001},
      }
  • [Reifenberg] Go to document E. R. Reifenberg, "Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type," Acta Math., vol. 104, pp. 1-92, 1960.
    @ARTICLE{Reifenberg,
      author = {Reifenberg, E. R.},
      title = {Solution of the {P}lateau {P}roblem for {$m$}-dimensional surfaces of varying topological type},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {104},
      year = {1960},
      pages = {1--92},
      issn = {0001-5962},
      mrclass = {49.00},
      mrnumber = {0114145},
      mrreviewer = {Wendell H. Fleming},
      doi = {10.1007/BF02547186},
      url = {https://doi.org/10.1007/BF02547186},
      zblnumber = {0099.08503},
      }
  • [SY_Redbook] R. Schoen and S. -T. Yau, Lectures on Differential Geometry, International Press, Cambridge, MA, 1994.
    @BOOK{SY_Redbook,
      author = {Schoen, R. and Yau, S.-T.},
      title = {Lectures on Differential Geometry},
      series = {Conference Proceedings and Lecture Notes in Geometry and Topology, I},
      note = {lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, translated from the Chinese by Ding and S. Y. Cheng, with a preface translated from the Chinese by Kaising Tso},
      publisher = {International Press, Cambridge, MA},
      year = {1994},
      pages = {v+235},
      isbn = {1-57146-012-8},
      mrclass = {53-01 (53-02 53C21 58G30)},
      mrnumber = {1333601},
      mrreviewer = {Man Chun Leung},
      zblnumber = {0830.53001},
      }
  • [SoZha] Go to document P. Souplet and Q. S. Zhang, "Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds," Bull. London Math. Soc., vol. 38, iss. 6, pp. 1045-1053, 2006.
    @ARTICLE{SoZha,
      author = {Souplet, Philippe and Zhang, Qi S.},
      title = {Sharp gradient estimate and {Y}au's {L}iouville theorem for the heat equation on noncompact manifolds},
      journal = {Bull. London Math. Soc.},
      fjournal = {The Bulletin of the London Mathematical Society},
      volume = {38},
      year = {2006},
      number = {6},
      pages = {1045--1053},
      issn = {0024-6093},
      mrclass = {35K05 (35B40 58J35)},
      mrnumber = {2285258},
      doi = {10.1112/S0024609306018947},
      url = {https://doi.org/10.1112/S0024609306018947},
      zblnumber = {1109.58025},
      }
  • [T90] Go to document G. Tian, "On Calabi’s conjecture for complex surfaces with positive first Chern class," Invent. Math., vol. 101, iss. 1, pp. 101-172, 1990.
    @ARTICLE{T90,
      author = {Tian, G.},
      title = {On {C}alabi's conjecture for complex surfaces with positive first {C}hern class},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {101},
      year = {1990},
      number = {1},
      pages = {101--172},
      issn = {0020-9910},
      mrclass = {32L07 (32F07 53C25 53C55)},
      mrnumber = {1055713},
      mrreviewer = {M. Kalka},
      doi = {10.1007/BF01231499},
      url = {https://doi.org/10.1007/BF01231499},
      zblnumber = {0716.32019},
      }
  • [Topping] Go to document P. Topping, Lectures on the Ricci Flow, Cambridge University Press, Cambridge, 2006, vol. 325.
    @BOOK{Topping,
      author = {Topping, Peter},
      title = {Lectures on the {R}icci Flow},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {325},
      publisher = {Cambridge University Press, Cambridge},
      year = {2006},
      pages = {x+113},
      isbn = {978-0-521-68947-2; 0-521-68947-3},
      mrclass = {53C44},
      mrnumber = {2265040},
      mrreviewer = {Peng Lu},
      doi = {10.1017/CBO9780511721465},
      url = {https://doi.org/10.1017/CBO9780511721465},
      zblnumber = {1105.58013},
      }

Authors

Wenshuai Jiang

School of Mathematical Sciences, Zhejiang University, Hangzhou, China

Aaron Naber

Department of Mathematics, Northwestern University, Evanston, IL, USA