Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Abstract

We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Hölder continuous way along the geodesic. We give examples that show that the Hölder exponent, along with essentially all the other consequences that follow from this estimate, are sharp.
Among the applications is that the regular set is convex for any noncollapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

  • [AbGl] Go to document U. Abresch and D. Gromoll, "On complete manifolds with nonnegative Ricci curvature," J. Amer. Math. Soc., vol. 3, iss. 2, pp. 355-374, 1990.
    @article {AbGl, MRKEY = {1030656},
      AUTHOR = {Abresch, Uwe and Gromoll, Detlef},
      TITLE = {On complete manifolds with nonnegative {R}icci curvature},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {3},
      YEAR = {1990},
      NUMBER = {2},
      PAGES = {355--374},
      ISSN = {0894-0347},
      MRCLASS = {53C21},
      MRNUMBER = {1030656},
      MRREVIEWER = {Ji-Ping Sha},
      DOI = {10.2307/1990957},
      ZBLNUMBER = {0704.53032},
      }
  • [A1] 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 {A1, MRKEY = {1074481},
      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},
      CODEN = {INVMBH},
      MRCLASS = {53C23 (53C21 58D27)},
      MRNUMBER = {1074481},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      DOI = {10.1007/BF01233434},
      ZBLNUMBER = {0711.53038},
      }
  • [A2] M. T. Anderson, "Einstein metrics and metrics with bounds on Ricci curvature," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 443-452.
    @inproceedings {A2, MRKEY = {1403944},
      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},
      VENUE={{Z}ürich, 1994},
      PAGES = {443--452},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1995},
      MRCLASS = {53C21 (53C23 53C25)},
      MRNUMBER = {1403944},
      MRREVIEWER = {Zhongmin Shen},
      ZBLNUMBER = {0840.53036},
      }
  • [BGP] Go to document Y. Burago, M. Gromov, and G. Perelman, "A. D. Aleksandrov spaces with curvatures bounded below," Russian Math. Surveys, vol. 47, iss. 2(284), pp. 1-58, 1992.
    @article {BGP, MRKEY = {1185284},
      AUTHOR = {Burago, Yu. and Gromov, M. and Perelman, G.},
      TITLE = {A. {D}. {A}leksandrov spaces with curvatures bounded below},
      JOURNAL = {Russian Math. Surveys},
      VOLUME = {47},
      YEAR = {1992},
      NUMBER = {2(284)},
      PAGES = {1--58},
      ISSN = {0042-1316},
      MRCLASS = {53C21 (53C23)},
      MRNUMBER = {1185284},
      MRREVIEWER = {Tadeusz Januszkiewicz},
      DOI = {10.1070/RM1992v047n02ABEH000877},
      }
  • [BKN] 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 {BKN, MRKEY = {1001844},
      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},
      CODEN = {INVMBH},
      MRCLASS = {53C20 (53C25)},
      MRNUMBER = {1001844},
      MRREVIEWER = {Thomas H. Otway},
      DOI = {10.1007/BF01389045},
      ZBLNUMBER = {0682.53045},
      }
  • [Ca] Go to document E. Calabi, "On Ricci curvature and geodesics," Duke Math. J., vol. 34, pp. 667-676, 1967.
    @article {Ca, MRKEY = {0216429},
      AUTHOR = {Calabi, Eugenio},
      TITLE = {On {R}icci curvature and geodesics},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {34},
      YEAR = {1967},
      PAGES = {667-676},
      ISSN = {0012-7094},
      MRCLASS = {53.72},
      MRNUMBER = {0216429},
      MRREVIEWER = {M. do Carmo},
      ZBLNUMBER = {0153.51501},
      DOI = {10.1215/S0012-7094-67-03469-2},
     }
  • [Ch] J. Cheeger, Degeneration of Riemannian Metrics under Ricci Curvature Bounds, Pisa: Scuola Normale Superiore, 2001.
    @book {Ch, MRKEY = {2006642},
      AUTHOR = {Cheeger, Jeff},
      TITLE = {Degeneration of {R}iemannian Metrics under {R}icci Curvature Bounds},
      SERIES = {Lezioni Fermiane. [Fermi Lectures]},
      PUBLISHER = {Scuola Normale Superiore},
      ADDRESS = {Pisa},
      YEAR = {2001},
      PAGES = {ii+77},
      MRCLASS = {53C21 (53C20 53C23)},
      MRNUMBER = {2006642},
      MRREVIEWER = {Vitali Kapovitch},
      ZBLNUMBER = {1055.53024},
      }
  • [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., vol. 144, iss. 1, pp. 189-237, 1996.
    @article {ChC1, 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},
      }
  • [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, 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},
      }
  • [ChC3] 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 {ChC3, 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},
      }
  • [ChC4] 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 {ChC4, 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},
      }
  • [C1] Go to document T. H. Colding, "Shape of manifolds with positive Ricci curvature," Invent. Math., vol. 124, iss. 1-3, pp. 175-191, 1996.
    @article {C1, MRKEY = {1369414},
      KEY={Col96a},
      AUTHOR = {Colding, Tobias H.},
      TITLE = {Shape of manifolds with positive {R}icci curvature},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {124},
      YEAR = {1996},
      NUMBER = {1-3},
      PAGES = {175--191},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {53C23 (53C21)},
      MRNUMBER = {1369414},
      MRREVIEWER = {Man Chun Leung},
      DOI = {10.1007/s002220050049},
      ZBLNUMBER = {0871.53027},
      }
  • [C2] Go to document T. H. Colding, "Large manifolds with positive Ricci curvature," Invent. Math., vol. 124, iss. 1-3, pp. 193-214, 1996.
    @article {C2, MRKEY = {1369415},
      KEY={Col96b},
      AUTHOR = {Colding, Tobias H.},
      TITLE = {Large manifolds with positive {R}icci curvature},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {124},
      YEAR = {1996},
      NUMBER = {1-3},
      PAGES = {193--214},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {53C23 (53C21)},
      MRNUMBER = {1369415},
      MRREVIEWER = {Man Chun Leung},
      DOI = {10.1007/s002220050050},
      ZBLNUMBER = {0871.53028},
      }
  • [C3] Go to document T. H. Colding, "Ricci curvature and volume convergence," Ann. of Math., vol. 145, iss. 3, pp. 477-501, 1997.
    @article {C3, MRKEY = {1454700},
      AUTHOR = {Colding, Tobias H.},
      TITLE = {Ricci curvature and volume convergence},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {145},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {477--501},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {53C21 (53C23)},
      MRNUMBER = {1454700},
      MRREVIEWER = {Zhongmin Shen},
      DOI = {10.2307/2951841},
      ZBLNUMBER = {0879.53030},
      }
  • [C4] T. H. Colding, "Spaces with Ricci curvature bounds," in Proceedings of the International Congress of Mathematicians, Vol. II, 1998, pp. 299-308.
    @inproceedings {C4, MRKEY = {1648080},
      AUTHOR = {Colding, Tobias H.},
      TITLE = {Spaces with {R}icci curvature bounds},
      BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {II}},
      VENUE={{B}erlin, 1998},
      SERIES = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      YEAR = {1998},
      VOLUME = {Extra Vol. II},
      PAGES = {299--308},
      ISSN = {1431-0635},
      MRCLASS = {53C23 (53C21)},
      MRNUMBER = {1648080},
      MRREVIEWER = {William P. Minicozzi, II},
      ZBLNUMBER = {0955.53020},
      }
  • [CN1] T. H. Colding and A. Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications.
    @misc{CN1,
      author = {Colding, Tobias H. and Naber, A.},
      TITLE = {Characterization of tangent cones of noncollapsed limits with lower {R}icci bounds and applications},
      ARXIV={1108.3244},
     }
  • [CN2] T. H. Colding and A. Naber, Lower Ricci curvature, branching and the bi-Lipschitz structure of uniform Reifenberg spaces.
    @misc{CN2,
      author = {Colding, Tobias H. and Naber, A.},
      TITLE = {Lower {R}icci curvature, branching and the bi-{L}ipschitz structure of uniform {R}eifenberg spaces},
      ARXIV={1111.2184},
      NOTE={in preparation},
     }
  • [EH] Go to document T. Eguchi and A. J. Hanson, "Asymptotically flat self-dual solutions to Euclidean gravity," Phys. Lett. B, vol. 74, iss. 1, pp. 82-106, 1978.
    @article {EH,
      author = {Eguchi, Tohru and Hanson, Andrew J.},
      TITLE = {Asymptotically flat self-dual solutions to Euclidean gravity},
      JOURNAL = {Phys. Lett. B},
      FJOURNAL = {Phys. Lett. B},
      VOLUME = {74},
      YEAR = {1978},
      NUMBER = {1},
      PAGES = {82--106},
      ISSN = {0003-4916},
      CODEN = {ADNYA6},
      MRCLASS = {83C20 (57R25)},
      DOI = {10.1016/0370-2693(78)90566-X},
      }
  • [F1] Go to document K. Fukaya, "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator," Invent. Math., vol. 87, iss. 3, pp. 517-547, 1987.
    @article {F1, MRKEY = {0874035},
      AUTHOR = {Fukaya, Kenji},
      TITLE = {Collapsing of {R}iemannian manifolds and eigenvalues of {L}aplace operator},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {87},
      YEAR = {1987},
      NUMBER = {3},
      PAGES = {517--547},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {58G25 (53C20 58C40)},
      MRNUMBER = {0874035},
      MRREVIEWER = {Atsushi Katsuda},
      DOI = {10.1007/BF01389241},
      ZBLNUMBER = {0589.58034},
      }
  • [F2] Go to document K. Fukaya, "Metric Riemannian geometry," in Handbook of Differential Geometry. Vol. II, Elsevier/North-Holland, Amsterdam, 2006, pp. 189-313.
    @incollection {F2, MRKEY = {2194670},
      AUTHOR = {Fukaya, Kenji},
      TITLE = {Metric {R}iemannian geometry},
      BOOKTITLE = {Handbook of Differential Geometry. {V}ol. {II}},
      PAGES = {189--313},
      PUBLISHER = {Elsevier/North-Holland, Amsterdam},
      YEAR = {2006},
      MRCLASS = {53C23 (53C20)},
      MRNUMBER = {2194670},
      DOI = {10.1016/S1874-5741(06)80007-5},
      ZBLNUMBER = {1163.53021},
      }
  • [FY] Go to document K. Fukaya and T. Yamaguchi, "Isometry groups of singular spaces," Math. Z., vol. 216, iss. 1, pp. 31-44, 1994.
    @article {FY, MRKEY = {1273464},
      AUTHOR = {Fukaya, Kenji and Yamaguchi, Takao},
      TITLE = {Isometry groups of singular spaces},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {216},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {31--44},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {53C23 (53C20)},
      MRNUMBER = {1273464},
      MRREVIEWER = {Zhongmin Shen},
      DOI = {10.1007/BF02572307},
      ZBLNUMBER = {0797.53033},
      }
  • [Ga] S. Gallot, "Volumes, courbure de Ricci et convergence des variétés (d’après T. H. Colding et Cheeger-Colding)," in Séminaire Bourbaki. Vol. 1997/98, , 1998, vol. 252, p. exp. no. 835, 3, 7-32.
    @incollection {Ga, MRKEY = {1685585},
      AUTHOR = {Gallot, Sylvestre},
      TITLE = {Volumes, courbure de {R}icci et convergence des variétés (d'après {T}. {H}. {C}olding et {C}heeger-{C}olding)},
      BOOKTITLE = {S{é}minaire Bourbaki. Vol. 1997/98},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {252},
      YEAR = {1998},
      PAGES = {Exp. No. 835, 3, 7--32},
      ISSN = {0303-1179},
      MRCLASS = {53C23 (53C20 53C21)},
      MRNUMBER = {1685585},
      ZBLNUMBER = {0976.53038},
     }
  • [GiTr] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second ed., New York: Springer-Verlag, 1983, vol. 224.
    @book {GiTr, MRKEY = {0737190},
      AUTHOR = {Gilbarg, David and Trudinger, Neil S.},
      TITLE = {Elliptic Partial Differential Equations of Second Order},
      SERIES = {Grundlehren Math. Wissen.},
      VOLUME = {224},
      EDITION = {Second},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {xiii+513},
      ISBN = {3-540-13025-X},
      MRCLASS = {35Jxx (35-01)},
      MRNUMBER = {0737190},
      MRREVIEWER = {O. John},
      ZBLNUMBER = {0562.35001},
      }
  • [G] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Boston, MA: Birkhäuser, 2007.
    @book {G, MRKEY = {2307192},
      AUTHOR = {Gromov, Misha},
      TITLE = {Metric Structures for {R}iemannian and Non-{R}iemannian Spaces},
      SERIES = {Mod. Birkhäuser Class.},
      NOTE = {with appendices by M. Katz, P. Pansu and S. Semmes},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2007},
      PAGES = {xx+585},
      ISBN = {978-0-8176-4582-3; 0-8176-4582-9},
      MRCLASS = {53C23 (53-02)},
      MRNUMBER = {2307192},
      ZBLNUMBER = {1113.53001},
      }
  • [GLP] M. Gromov, J. Lafontaine, and P. Pansu, Structures Metriques pour les Varieties Riemanniennces, Paris: Cedid/Fernand Nathan, 1981.
    @book{GLP,
      author={Gromov, Misha and Lafontaine, J. and Pansu, P.},
      TITLE={Structures Metriques pour les Varieties Riemanniennces},
      PUBLISHER={Cedid/Fernand Nathan},
      ADDRESS={Paris},
      YEAR={1981},
     }
  • [GvPe] Go to document K. Grove and P. Petersen, "Manifolds near the boundary of existence," J. Differential Geom., vol. 33, iss. 2, pp. 379-394, 1991.
    @article {GvPe, MRKEY = {1094462},
      AUTHOR = {Grove, Karsten and Petersen, Peter},
      TITLE = {Manifolds near the boundary of existence},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {33},
      YEAR = {1991},
      NUMBER = {2},
      PAGES = {379--394},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53C23 (53C20)},
      MRNUMBER = {1094462},
      MRREVIEWER = {Gerard Walschap},
      URL = {http://projecteuclid.org/euclid.jdg/1214446323},
      ZBLNUMBER = {0729.53045},
      }
  • [H] Go to document S. Honda, "Bishop-Gromov type inequality on Ricci limit spaces," J. Math. Soc. Japan, vol. 63, iss. 2, pp. 419-442, 2011.
    @article {H, MRKEY = {2793106},
      AUTHOR = {Honda, Shouhei},
      TITLE = {Bishop-{G}romov type inequality on {R}icci limit spaces},
      JOURNAL = {J. Math. Soc. Japan},
      FJOURNAL = {Journal of the Mathematical Society of Japan},
      VOLUME = {63},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {419--442},
      ISSN = {0025-5645},
      CODEN = {NISUBC},
      MRCLASS = {53C23 (49Q15 53C21)},
      MRNUMBER = {2793106},
      MRREVIEWER = {Yu Ding},
      DOI = {10.2969/jmsj/06320419},
      ZBLNUMBER = {05914719},
      }
  • [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},
     }
  • [M1] Go to document X. Menguy, "Noncollapsing examples with positive Ricci curvature and infinite topological type," Geom. Funct. Anal., vol. 10, iss. 3, pp. 600-627, 2000.
    @article {M1, MRKEY = {1779615},
      AUTHOR = {Menguy, X.},
      TITLE = {Noncollapsing examples with positive {R}icci curvature and infinite topological type},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {10},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {600--627},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {53C21 (53C23)},
      MRNUMBER = {1779615},
      MRREVIEWER = {S{é}rgio J. Mendon{ç}a},
      DOI = {10.1007/PL00001632},
      ZBLNUMBER = {0971.53030},
      }
  • [M2] Go to document X. Menguy, "Examples of strictly weakly regular points," Geom. Funct. Anal., vol. 11, iss. 1, pp. 124-131, 2001.
    @article {M2, MRKEY = {1829644},
      AUTHOR = {Menguy, X.},
      TITLE = {Examples of strictly weakly regular points},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {11},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {124--131},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {53C20},
      MRNUMBER = {1829644},
      MRREVIEWER = {S{é}rgio J. Mendon{ç}a},
      DOI = {10.1007/PL00001667},
      ZBLNUMBER = {0990.53025},
      }
  • [M3] Go to document X. C. Menguy, Examples of Manifolds and Spaces with Positive Ricci Curvature, ProQuest LLC, Ann Arbor, MI, 2000.
    @book {M3, MRKEY = {2700637},
      AUTHOR = {Menguy, Xavier C.},
      TITLE = {Examples of Manifolds and Spaces with Positive {R}icci Curvature},
      NOTE = {Ph.D. thesis, Courant Institute, New York University},
      PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
      YEAR = {2000},
      PAGES = {83},
      ISBN = {978-0599-75968-8},
      MRCLASS = {Thesis},
      MRNUMBER = {2700637},
      DOI = {10.1353/ajm.2000.0041},
      }
  • [M4] Go to document X. Menguy, "Examples of nonpolar limit spaces," Amer. J. Math., vol. 122, iss. 5, pp. 927-937, 2000.
    @article {M4, MRKEY = {1781925},
      AUTHOR = {Menguy, X.},
      TITLE = {Examples of nonpolar limit spaces},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {122},
      YEAR = {2000},
      NUMBER = {5},
      PAGES = {927--937},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {53C23},
      MRNUMBER = {1781925},
      MRREVIEWER = {Andrea Sambusetti},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.5menguy.pdf},
      ZBLNUMBER = {0981.53019},
      }
  • [P1] G. Perelman, "Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers," in Comparison Geometry, Cambridge: Cambridge Univ. Press, 1997, vol. 30, pp. 157-163.
    @incollection {P1, MRKEY = {1452872},
      AUTHOR = {Perelman, G.},
      TITLE = {Construction of manifolds of positive {R}icci curvature with big volume and large {B}etti numbers},
      BOOKTITLE = {Comparison Geometry},
      VENUE={{B}erkeley, {CA},
      1993--94},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {30},
      PAGES = {157--163},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      MRCLASS = {53C21 (53C20)},
      MRNUMBER = {1452872},
      MRREVIEWER = {Zhongmin Shen},
      ZBLNUMBER = {0890.53038},
      }
  • [P2] G. Perelman, "A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone," in Comparison Geometry, Cambridge: Cambridge Univ. Press, 1997, vol. 30, pp. 165-166.
    @incollection {P2, MRKEY = {1452873},
      AUTHOR = {Perelman, G.},
      TITLE = {A complete {R}iemannian manifold of positive {R}icci curvature with {E}uclidean volume growth and nonunique asymptotic cone},
      BOOKTITLE = {Comparison Geometry},
      VENUE={{B}erkeley, {CA},
      1993--94},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {30},
      PAGES = {165--166},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      MRCLASS = {53C21 (53C23)},
      MRNUMBER = {1452873},
      MRREVIEWER = {Zhongmin Shen},
      ZBLNUMBER = {0887.53038},
      }
  • [Pn] Go to document A. Petrunin, "Parallel transportation for Alexandrov space with curvature bounded below," Geom. Funct. Anal., vol. 8, iss. 1, pp. 123-148, 1998.
    @article {Pn, MRKEY = {1601854},
      AUTHOR = {Petrunin, A.},
      TITLE = {Parallel transportation for {A}lexandrov space with curvature bounded below},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {8},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {123--148},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {53C23},
      MRNUMBER = {1601854},
      MRREVIEWER = {Takashi Shioya},
      DOI = {10.1007/s000390050050},
      ZBLNUMBER = {0903.53045},
      }
  • [Ti] 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 {Ti, MRKEY = {1055713},
      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},
      CODEN = {INVMBH},
      MRCLASS = {32L07 (32F07 53C25 53C55)},
      MRNUMBER = {1055713},
      MRREVIEWER = {M. Kalka},
      DOI = {10.1007/BF01231499},
      ZBLNUMBER = {0716.32019},
      }
  • [W] G. Wei, "Manifolds with a lower Ricci curvature bound," in Surveys in Differential Geometry. Vol. XI, Somerville, MA: International Press, 2007, pp. 203-227.
    @incollection {W, MRKEY = {2408267},
      AUTHOR = {Wei, Guofang},
      TITLE = {Manifolds with a lower {R}icci curvature bound},
      BOOKTITLE = {Surveys in Differential Geometry. {V}ol. {XI}},
      SERIES = {Surv. Differ. Geom.},
      NUMBER = {11},
      PAGES = {203--227},
      PUBLISHER = {International Press},
      ADDRESS={Somerville, MA},
      YEAR = {2007},
      MRCLASS = {53C21 (53C23)},
      MRNUMBER = {2408267},
      MRREVIEWER = {David J. Wraith},
      ZBLNUMBER = {1151.53036},
      }

Authors

Tobias H. Colding

Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139-4307

Aaron Naber

Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139-4307