July  2012, 6(3): 377-403. doi: 10.3934/jmd.2012.6.377

Compact asymptotically harmonic manifolds

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

Received  May 2012 Published  October 2012

A complete Riemannian manifold without conjugate points is said to be asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank-one locally symmetric spaces of noncompact type. In this paper we show that this list exhausts the compact asymptotically harmonic manifolds under a variety of assumptions including nonpositive curvature or Gromov-hyperbolic fundamental group. We then present a new characterization of symmetric spaces amongst the set of all visibility manifolds.
Citation: Andrew M. Zimmer. Compact asymptotically harmonic manifolds. Journal of Modern Dynamics, 2012, 6 (3) : 377-403. doi: 10.3934/jmd.2012.6.377
References:
[1]

Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2), 121 (1985), 429-461.  Google Scholar

[2]

Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc., 8 (1976), iii+102.  Google Scholar

[3]

Werner Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609.  Google Scholar

[4]

Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2), 122 (1985), 171-203.  Google Scholar

[5]

G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., 5 (1995), 731-799. doi: 10.1007/BF01897050.  Google Scholar

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Yves Benoist, Patrick Foulon and François Labourie, Flots d'Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750.  Google Scholar

[7]

J. Bolton, Conditions under which a geodesic flow is Anosov, Math. Ann., 240 (1979), 103-113. doi: 10.1007/BF01364627.  Google Scholar

[8]

Keith Burns and Ralf Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35-59.  Google Scholar

[9]

Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math., 159 (1993), 241-270.  Google Scholar

[10]

Ewa Damek and Fulvio Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 139-142.  Google Scholar

[11]

Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc., 167 (1972), 151-170. doi: 10.1090/S0002-9947-1972-0295387-4.  Google Scholar

[12]

Patrick Eberlein, When is a geodesic flow of Anosov type? I, J. Differential Geometry, 8 (1973), 437-463.  Google Scholar

[13]

P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.  Google Scholar

[14]

Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Z., 153 (1977), 237-251. doi: 10.1007/BF01214477.  Google Scholar

[15]

J.-H. Eschenburg, A note on symmetric and harmonic spaces, J. London Math. Soc. (2), 21 (1980), 541-543.  Google Scholar

[16]

Patrick Foulon and François Labourie, Sur les variétés compactes asymptotiquement harmoniques, Invent. Math., 109 (1992), 97-111. doi: 10.1007/BF01232020.  Google Scholar

[17]

A. Freire and R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392. doi: 10.1007/BF01389360.  Google Scholar

[18]

Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285-311. doi: 10.1016/0022-1236(83)90015-0.  Google Scholar

[19]

L. W. Green, A theorem of E. Hopf, Michigan Math. J., 5 (1958), 31-34. doi: 10.1307/mmj/1028998009.  Google Scholar

[20]

Alexander Grigor'yan, "Heat Kernel and Analysis on Manifolds," AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.  Google Scholar

[21]

M. Gromov, Hyperbolic groups, in "Essays in Group Theory," Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987), 75-263.  Google Scholar

[22]

J. Heber, On harmonic and asymptotically harmonic homogeneous spaces, Geom. Funct. Anal., 16 (2006), 869-890. doi: 10.1007/s00039-006-0569-4.  Google Scholar

[23]

Jens Heber, Gerhard Knieper and Hemangi M. Shah, Asymptotically harmonic spaces in dimension 3, Proc. Amer. Math. Soc., 135 (2007), 845-849.  Google Scholar

[24]

Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[25]

Gerhard Knieper, New results on noncompact harmonic manifolds, Comment. Math. Helv., 87 (2012), 669-703. doi: 10.4171/CMH/265.  Google Scholar

[26]

A. J. Ledger, Symmetric harmonic spaces, J. London Math. Soc., 32 (1957), 53-56. doi: 10.1112/jlms/s1-32.1.53.  Google Scholar

[27]

F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math., 71 (1990), 275-287. doi: 10.1007/BF02773746.  Google Scholar

[28]

François Ledrappier, Linear drift and entropy for regular covers, Geom. Funct. Anal., 20 (2010), 710-725. doi: 10.1007/s00039-010-0080-9.  Google Scholar

[29]

André Lichnerowicz, Sur les espaces riemanniens complètement harmoniques, Bull. Soc. Math. France, 72 (1944), 146-168.  Google Scholar

[30]

François Ledrappier and Lin Shu, Entropy rigidity of symmetric spaces without focal points, preprint, 2012. Google Scholar

[31]

François Ledrappier and Xiaodong Wang, An integral formula for the volume entropy with applications to rigidity, J. Differential Geom., 85 (2010), 461-477.  Google Scholar

[32]

Anthony Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  Google Scholar

[33]

Akhil Ranjan and Hemangi Shah, Busemann functions in a harmonic manifold, Geom. Dedicata, 101 (2003), 167-183. doi: 10.1023/A:1026369930269.  Google Scholar

[34]

Rafael O. Ruggiero, "Dynamics and Global Geometry of Manifolds Without Conjugate Points," Ensaios Matemáticos [Mathematical Surveys], 12, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007.  Google Scholar

[35]

Hemangi Shah, On 3-dimensional asymptotically harmonic manifolds with minimal horospheres, preprint, 2011. Google Scholar

[36]

Viktor Schroeder and Hemangi Shah, On 3-dimensional asymptotically harmonic manifolds, Arch. Math. (Basel), 90 (2008), 275-278. doi: 10.1007/s00013-008-2611-2.  Google Scholar

[37]

Z. I. Szabó, The Lichnerowicz conjecture on harmonic manifolds, J. Differential Geom., 31 (1990), 1-28.  Google Scholar

[38]

Jordan Watkins, The higher rank rigidity theorem for manifolds with no focal points, preprint, 2011. Google Scholar

show all references

References:
[1]

Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2), 121 (1985), 429-461.  Google Scholar

[2]

Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc., 8 (1976), iii+102.  Google Scholar

[3]

Werner Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609.  Google Scholar

[4]

Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2), 122 (1985), 171-203.  Google Scholar

[5]

G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., 5 (1995), 731-799. doi: 10.1007/BF01897050.  Google Scholar

[6]

Yves Benoist, Patrick Foulon and François Labourie, Flots d'Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750.  Google Scholar

[7]

J. Bolton, Conditions under which a geodesic flow is Anosov, Math. Ann., 240 (1979), 103-113. doi: 10.1007/BF01364627.  Google Scholar

[8]

Keith Burns and Ralf Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35-59.  Google Scholar

[9]

Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math., 159 (1993), 241-270.  Google Scholar

[10]

Ewa Damek and Fulvio Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 139-142.  Google Scholar

[11]

Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc., 167 (1972), 151-170. doi: 10.1090/S0002-9947-1972-0295387-4.  Google Scholar

[12]

Patrick Eberlein, When is a geodesic flow of Anosov type? I, J. Differential Geometry, 8 (1973), 437-463.  Google Scholar

[13]

P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.  Google Scholar

[14]

Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Z., 153 (1977), 237-251. doi: 10.1007/BF01214477.  Google Scholar

[15]

J.-H. Eschenburg, A note on symmetric and harmonic spaces, J. London Math. Soc. (2), 21 (1980), 541-543.  Google Scholar

[16]

Patrick Foulon and François Labourie, Sur les variétés compactes asymptotiquement harmoniques, Invent. Math., 109 (1992), 97-111. doi: 10.1007/BF01232020.  Google Scholar

[17]

A. Freire and R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392. doi: 10.1007/BF01389360.  Google Scholar

[18]

Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285-311. doi: 10.1016/0022-1236(83)90015-0.  Google Scholar

[19]

L. W. Green, A theorem of E. Hopf, Michigan Math. J., 5 (1958), 31-34. doi: 10.1307/mmj/1028998009.  Google Scholar

[20]

Alexander Grigor'yan, "Heat Kernel and Analysis on Manifolds," AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.  Google Scholar

[21]

M. Gromov, Hyperbolic groups, in "Essays in Group Theory," Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987), 75-263.  Google Scholar

[22]

J. Heber, On harmonic and asymptotically harmonic homogeneous spaces, Geom. Funct. Anal., 16 (2006), 869-890. doi: 10.1007/s00039-006-0569-4.  Google Scholar

[23]

Jens Heber, Gerhard Knieper and Hemangi M. Shah, Asymptotically harmonic spaces in dimension 3, Proc. Amer. Math. Soc., 135 (2007), 845-849.  Google Scholar

[24]

Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[25]

Gerhard Knieper, New results on noncompact harmonic manifolds, Comment. Math. Helv., 87 (2012), 669-703. doi: 10.4171/CMH/265.  Google Scholar

[26]

A. J. Ledger, Symmetric harmonic spaces, J. London Math. Soc., 32 (1957), 53-56. doi: 10.1112/jlms/s1-32.1.53.  Google Scholar

[27]

F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math., 71 (1990), 275-287. doi: 10.1007/BF02773746.  Google Scholar

[28]

François Ledrappier, Linear drift and entropy for regular covers, Geom. Funct. Anal., 20 (2010), 710-725. doi: 10.1007/s00039-010-0080-9.  Google Scholar

[29]

André Lichnerowicz, Sur les espaces riemanniens complètement harmoniques, Bull. Soc. Math. France, 72 (1944), 146-168.  Google Scholar

[30]

François Ledrappier and Lin Shu, Entropy rigidity of symmetric spaces without focal points, preprint, 2012. Google Scholar

[31]

François Ledrappier and Xiaodong Wang, An integral formula for the volume entropy with applications to rigidity, J. Differential Geom., 85 (2010), 461-477.  Google Scholar

[32]

Anthony Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  Google Scholar

[33]

Akhil Ranjan and Hemangi Shah, Busemann functions in a harmonic manifold, Geom. Dedicata, 101 (2003), 167-183. doi: 10.1023/A:1026369930269.  Google Scholar

[34]

Rafael O. Ruggiero, "Dynamics and Global Geometry of Manifolds Without Conjugate Points," Ensaios Matemáticos [Mathematical Surveys], 12, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007.  Google Scholar

[35]

Hemangi Shah, On 3-dimensional asymptotically harmonic manifolds with minimal horospheres, preprint, 2011. Google Scholar

[36]

Viktor Schroeder and Hemangi Shah, On 3-dimensional asymptotically harmonic manifolds, Arch. Math. (Basel), 90 (2008), 275-278. doi: 10.1007/s00013-008-2611-2.  Google Scholar

[37]

Z. I. Szabó, The Lichnerowicz conjecture on harmonic manifolds, J. Differential Geom., 31 (1990), 1-28.  Google Scholar

[38]

Jordan Watkins, The higher rank rigidity theorem for manifolds with no focal points, preprint, 2011. Google Scholar

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