Volumes of balls in large Riemannian manifolds

Abstract

We prove two lower bounds for the volumes of balls in a Riemannian manifold. If $(M^n, g)$ is a complete Riemannian manifold with filling radius at least $R$, then it contains a ball of radius $R$ and volume at least $\delta(n) R^n$. If $(M^n, \mathrm{hyp})$ is a closed hyperbolic manifold and if $g$ is another metric on $M$ with volume no greater than $\delta(n) \mathrm{Vol}(M, \mathrm{hyp})$, then the universal cover of $(M,g)$ contains a unit ball with volume greater than the volume of a unit ball in hyperbolic $n$-space.

  • [BCG] Go to document G. Besson, G. Courtois, and S. Gallot, "Entropies et rigidités des espaces localement symétriques de courbure strictement négative," Geom. Funct. Anal., vol. 5, iss. 5, pp. 731-799, 1995.
    @article {BCG, MRKEY = {1354289},
      AUTHOR = {Besson, G. and Courtois, G. and Gallot, S.},
      TITLE = {Entropies et rigidités des espaces localement symétriques de courbure strictement négative},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {5},
      YEAR = {1995},
      NUMBER = {5},
      PAGES = {731--799},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {58F17 (53C21 53C35)},
      MRNUMBER = {1354289},
      MRREVIEWER = {Boris Hasselblatt},
      DOI = {10.1007/BF01897050},
      ZBLNUMBER = {0851.53032},
      }
  • [BS] E. Bombieri and L. Simon, "On the Gehring link problem," in Seminar on Minimal Submanifolds, Princeton, NJ: Princeton Univ. Press, 1983, vol. 103, pp. 271-274.
    @incollection {BS, MRKEY = {795242},
      AUTHOR = {Bombieri, Enrico and Simon, Leon},
      TITLE = {On the {G}ehring link problem},
      BOOKTITLE = {Seminar on {{M}}inimal {{S}}ubmanifolds},
      SERIES = {{\it Ann. of Math. Stud.}},
      VOLUME = {103},
      PAGES = {271--274},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1983},
      MRCLASS = {28A75 (57R99)},
      MRNUMBER = {795242},
      ZBLNUMBER = {0529.53005},
      }
  • [FF] Go to document H. Federer and W. H. Fleming, "Normal and integral currents," Ann. of Math., vol. 72, pp. 458-520, 1960.
    @article {FF, MRKEY = {0123260},
      AUTHOR = {Federer, Herbert and Fleming, Wendell H.},
      TITLE = {Normal and integral currents},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {72},
      YEAR = {1960},
      PAGES = {458--520},
      ISSN = {0003-486X},
      MRCLASS = {28.80 (53.45)},
      MRNUMBER = {23 \#A588},
      MRREVIEWER = {L. C. Young},
      DOI = {10.2307/1970227},
      ZBLNUMBER = {0187.31301},
      }
  • [G1] Go to document M. Gromov, "Large Riemannian manifolds," in Curvature and Topology of Riemannian Manifolds, New York: Springer-Verlag, 1986, vol. 1201, pp. 108-121.
    @incollection {G1, MRKEY = {859578},
      AUTHOR = {Gromov, M.},
      TITLE = {Large {R}iemannian manifolds},
      BOOKTITLE = {Curvature and {{T}}opology of {R}iemannian {{M}}anifolds},
      VENUE={{K}atata, 1985},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1201},
      PAGES = {108--121},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1986},
      MRCLASS = {53C20 (53C21)},
      MRNUMBER = {859578},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      DOI = {10.1007/BFb0075649},
      ZBLNUMBER = {0601.53038},
      }
  • [G2] Go to document M. Gromov, "Filling Riemannian manifolds," J. Differential Geom., vol. 18, iss. 1, pp. 1-147, 1983.
    @article {G2, MRKEY = {697984},
      AUTHOR = {Gromov, M.},
      TITLE = {Filling {R}iemannian manifolds},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {18},
      YEAR = {1983},
      NUMBER = {1},
      PAGES = {1--147},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53C20 (53C21 57R99)},
      MRNUMBER = {697984},
      MRREVIEWER = {Yu. Burago},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214509283},
      ZBLNUMBER = {0515.53037},
      }
  • [G3] M. Gromov, "Systoles and intersystolic inequalities," in Actes de la Table Ronde de Géométrie Différentielle, Paris: Soc. Math. France, 1996, vol. 1, pp. 291-362.
    @incollection {G3, MRKEY = {1427763},
      AUTHOR = {Gromov, M.},
      TITLE = {Systoles and intersystolic inequalities},
      BOOKTITLE = {Actes de la {T}able {R}onde de {G}éométrie {D}ifférentielle},
      VENUE={{L}uminy, 1992},
      SERIES = {Sémin. Congr.},
      VOLUME = {1},
      PAGES = {291--362},
      PUBLISHER = {Soc. Math. France},
      ADDRESS = {Paris},
      YEAR = {1996},
      MRCLASS = {53C22 (53-02 53C20 53C23)},
      MRNUMBER = {99a:53051},
      MRREVIEWER = {Andrea Sambusetti},
      ZBLNUMBER = {0877.53002},
      }
  • [G4] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Boston, MA: Birkhäuser, 1999, vol. 152.
    @book {G4, MRKEY = {1699320},
      AUTHOR = {Gromov, M.},
      TITLE = {Metric {{S}}tructures for {R}iemannian and {{N}}on-{R}iemannian {{S}}paces},
      SERIES = {Progr. Math.},
      VOLUME= {152},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1999},
      PAGES = {xx+585},
      ISBN = {0-8176-3898-9},
      MRCLASS = {53C23 (53-02)},
      MRNUMBER = {1699320},
      MRREVIEWER = {Igor Belegradek},
      ZBLNUMBER = {0953.53002},
      }
  • [G5] Go to document M. Gromov, "Volume and bounded cohomology," Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 5-99 (1983), 1982.
    @article {G5, MRKEY = {686042},
      AUTHOR = {Gromov, M.},
      TITLE = {Volume and bounded cohomology},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {56},
      YEAR = {1982},
      PAGES = {5--99 (1983)},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {53C20 (53C21 57R99 58E99)},
      MRNUMBER = {686042},
      MRREVIEWER = {Karsten Grove},
      URL = {http://www.numdam.org/item?id=PMIHES_1982__56__5_0},
      ZBLNUMBER = {0516.53046},
      }
  • [G6] M. Gromov, "Positive curvature, macroscopic dimension, spectral gaps and higher signatures," in Functional Analysis on the Eve of the 21st Century, Vol. II, Boston, MA: Birkhäuser, 1996, vol. 132, pp. 1-213.
    @incollection {G6, MRKEY = {1389019},
      AUTHOR = {Gromov, M.},
      TITLE = {Positive curvature, macroscopic dimension, spectral gaps and higher signatures},
      BOOKTITLE = {Functional {{A}}nalysis on the {{E}}ve of the 21st {{C}}entury, {V}ol. {{{\rm II}}}},
      VENUE={{N}ew {B}runswick, {NJ},
      1993},
      SERIES = {Progr. Math.},
      VOLUME= {132},
      PAGES = {1--213},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1996},
      MRCLASS = {53C21 (53C20 57R20)},
      MRNUMBER = {1389019},
      MRREVIEWER = {Christopher W. Stark},
      ZBLNUMBER = {0945.53022},
      }
  • [GL] Go to document M. Gromov and B. H. Lawson Jr., "Positive scalar curvature and the Dirac operator on complete Riemannian manifolds," Inst. Hautes Études Sci. Publ. Math., iss. 58, pp. 83-196 (1984), 1983.
    @article {GL, MRKEY = {720933},
      AUTHOR = {Gromov, M. and Lawson, Jr., H. Blaine},
      TITLE = {Positive scalar curvature and the {D}irac operator on complete {R}iemannian manifolds},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {58},
      YEAR = {1983},
      PAGES = {83--196 (1984)},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {58G10 (53C20 57N10 57R99)},
      MRNUMBER = {720933},
      MRREVIEWER = {Howard D. Rees},
      URL = {http://www.numdam.org/item?id=PMIHES_1983__58__83_0},
      ZBLNUMBER = {0538.53047},
      }
  • [K] Go to document M. Katz, "The filling radius of two-point homogeneous spaces," J. Differential Geom., vol. 18, iss. 3, pp. 505-511, 1983.
    @article {K, MRKEY = {723814},
      AUTHOR = {Katz, Mikhail},
      TITLE = {The filling radius of two-point homogeneous spaces},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {18},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {505--511},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53C20 (53C35)},
      MRNUMBER = {85h:53030},
      MRREVIEWER = {Rolf Sulanke},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214437785},
      ZBLNUMBER = {0529.53032},
      }
  • [K2] Go to document M. Katz, "The first diameter of $3$-manifolds of positive scalar curvature," Proc. Amer. Math. Soc., vol. 104, iss. 2, pp. 591-595, 1988.
    @article {K2, MRKEY = {962834},
      AUTHOR = {Katz, Mikhail},
      TITLE = {The first diameter of {$3$}-manifolds of positive scalar curvature},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {104},
      YEAR = {1988},
      NUMBER = {2},
      PAGES = {591--595},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {53C20},
      MRNUMBER = {962834},
      MRREVIEWER = {Viktor Schroeder},
      DOI = {10.2307/2047018},
      ZBLNUMBER = {0693.53013},
      }
  • [LB] C. LeBrun, "Four-manifolds without Einstein metrics," Math. Res. Lett., vol. 3, iss. 2, pp. 133-147, 1996.
    @article {LB, MRKEY = {1386835},
      AUTHOR = {LeBrun, Claude},
      TITLE = {Four-manifolds without {E}instein metrics},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {3},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {133--147},
      ISSN = {1073-2780},
      MRCLASS = {53C25 (14J29 58E11)},
      MRNUMBER = {1386835},
      MRREVIEWER = {Tedi C. Draghici},
      ZBLNUMBER = {0856.53035},
      }
  • [M] A. Malcev, "On isomorphic matrix representations of infinite groups," Rec. Math. [Mat. Sbornik], vol. 8 (50), pp. 405-422, 1940.
    @article {M, MRKEY = {0003420},
      AUTHOR = {Malcev, A.},
      TITLE = {On isomorphic matrix representations of infinite groups},
      JOURNAL = {Rec. Math. [Mat. Sbornik]},
      VOLUME = {8 (50)},
      YEAR = {1940},
      PAGES = {405--422},
      MRCLASS = {20.0X},
      MRNUMBER = {3420},
      MRREVIEWER = {L. Zippin},
      }
  • [MS] Go to document J. H. Michael and L. M. Simon, "Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$," Comm. Pure Appl. Math., vol. 26, pp. 361-379, 1973.
    @article {MS, MRKEY = {0344978},
      AUTHOR = {Michael, J. H. and Simon, L. M.},
      TITLE = {Sobolev and mean-value inequalities on generalized submanifolds of {$R\sp{n}$}},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {26},
      YEAR = {1973},
      PAGES = {361--379},
      ISSN = {0010-3640},
      MRCLASS = {49F10 (46E35)},
      MRNUMBER = {49 \#9717},
      MRREVIEWER = {David Kinderlehrer},
      DOI = {10.1002/cpa.3160260305},
      }
  • [S] Go to document R. M. Schoen, "Variational theory for the total scalar curvature functional for Riemannian metrics and related topics," in Topics in Calculus of Variations, New York: Springer-Verlag, 1989, vol. 1365, pp. 120-154.
    @incollection {S, MRKEY = {994021},
      AUTHOR = {Schoen, Richard M.},
      TITLE = {Variational theory for the total scalar curvature functional for {R}iemannian metrics and related topics},
      BOOKTITLE = {Topics in {{C}}alculus of {{V}}ariations},
      VENUE={{M}ontecatini {T}erme, 1987},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1365},
      PAGES = {120--154},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      MRCLASS = {58E11 (49F99 53C20 58D17 58G30)},
      MRNUMBER = {994021},
      MRREVIEWER = {Hubert Gollek},
      DOI = {10.1007/BFb0089180},
      }

Authors

Larry Guth

Department of Mathematics
University of Toronto
Toronto Ontario
Canada M5S 2E4