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. Google Scholar

[2]

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

[3]

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

[4]

Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I,, Ann. of Math. (2), 122 (1985), 171. 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. 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. doi: 10.2307/2152750. Google Scholar

[7]

J. Bolton, Conditions under which a geodesic flow is Anosov,, Math. Ann., 240 (1979), 103. 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. 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. Google Scholar

[10]

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

[11]

Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points,, Trans. Amer. Math. Soc., 167 (1972), 151. 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. Google Scholar

[13]

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

[14]

Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow,, Math. Z., 153 (1977), 237. 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. Google Scholar

[16]

Patrick Foulon and François Labourie, Sur les variétés compactes asymptotiquement harmoniques,, Invent. Math., 109 (1992), 97. 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. doi: 10.1007/BF01389360. Google Scholar

[18]

Lucy Garnett, Foliations, the ergodic theorem and Brownian motion,, J. Funct. Anal., 51 (1983), 285. 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. doi: 10.1307/mmj/1028998009. Google Scholar

[20]

Alexander Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (2009). Google Scholar

[21]

M. Gromov, Hyperbolic groups,, in, 8 (1987), 75. Google Scholar

[22]

J. Heber, On harmonic and asymptotically harmonic homogeneous spaces,, Geom. Funct. Anal., 16 (2006), 869. 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. 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, 54 (1995). Google Scholar

[25]

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

[26]

A. J. Ledger, Symmetric harmonic spaces,, J. London Math. Soc., 32 (1957), 53. 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. doi: 10.1007/BF02773746. Google Scholar

[28]

François Ledrappier, Linear drift and entropy for regular covers,, Geom. Funct. Anal., 20 (2010), 710. 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. 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. Google Scholar

[32]

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

[33]

Akhil Ranjan and Hemangi Shah, Busemann functions in a harmonic manifold,, Geom. Dedicata, 101 (2003), 167. 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 (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. 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. 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. Google Scholar

[2]

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

[3]

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

[4]

Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I,, Ann. of Math. (2), 122 (1985), 171. 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. 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. doi: 10.2307/2152750. Google Scholar

[7]

J. Bolton, Conditions under which a geodesic flow is Anosov,, Math. Ann., 240 (1979), 103. 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. 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. Google Scholar

[10]

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

[11]

Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points,, Trans. Amer. Math. Soc., 167 (1972), 151. 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. Google Scholar

[13]

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

[14]

Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow,, Math. Z., 153 (1977), 237. 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. Google Scholar

[16]

Patrick Foulon and François Labourie, Sur les variétés compactes asymptotiquement harmoniques,, Invent. Math., 109 (1992), 97. 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. doi: 10.1007/BF01389360. Google Scholar

[18]

Lucy Garnett, Foliations, the ergodic theorem and Brownian motion,, J. Funct. Anal., 51 (1983), 285. 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. doi: 10.1307/mmj/1028998009. Google Scholar

[20]

Alexander Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (2009). Google Scholar

[21]

M. Gromov, Hyperbolic groups,, in, 8 (1987), 75. Google Scholar

[22]

J. Heber, On harmonic and asymptotically harmonic homogeneous spaces,, Geom. Funct. Anal., 16 (2006), 869. 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. 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, 54 (1995). Google Scholar

[25]

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

[26]

A. J. Ledger, Symmetric harmonic spaces,, J. London Math. Soc., 32 (1957), 53. 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. doi: 10.1007/BF02773746. Google Scholar

[28]

François Ledrappier, Linear drift and entropy for regular covers,, Geom. Funct. Anal., 20 (2010), 710. 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. 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. Google Scholar

[32]

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

[33]

Akhil Ranjan and Hemangi Shah, Busemann functions in a harmonic manifold,, Geom. Dedicata, 101 (2003), 167. 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 (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. 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. Google Scholar

[38]

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

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