# American Institute of Mathematical Sciences

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. [2] Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. II,, Mem. Amer. Math. Soc., 8 (1976). [3] Werner Ballmann, Nonpositively curved manifolds of higher rank,, Ann. of Math. (2), 122 (1985), 597. [4] Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I,, Ann. of Math. (2), 122 (1985), 171. [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. [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. [7] J. Bolton, Conditions under which a geodesic flow is Anosov,, Math. Ann., 240 (1979), 103. doi: 10.1007/BF01364627. [8] Keith Burns and Ralf Spatzier, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35. [9] Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov,, Pacific J. Math., 159 (1993), 241. [10] Ewa Damek and Fulvio Ricci, A class of nonsymmetric harmonic Riemannian spaces,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 139. [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. [12] Patrick Eberlein, When is a geodesic flow of Anosov type? I,, J. Differential Geometry, 8 (1973), 437. [13] P. Eberlein and B. O'Neill, Visibility manifolds,, Pacific J. Math., 46 (1973), 45. [14] Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow,, Math. Z., 153 (1977), 237. doi: 10.1007/BF01214477. [15] J.-H. Eschenburg, A note on symmetric and harmonic spaces,, J. London Math. Soc. (2), 21 (1980), 541. [16] Patrick Foulon and François Labourie, Sur les variétés compactes asymptotiquement harmoniques,, Invent. Math., 109 (1992), 97. doi: 10.1007/BF01232020. [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. [18] Lucy Garnett, Foliations, the ergodic theorem and Brownian motion,, J. Funct. Anal., 51 (1983), 285. doi: 10.1016/0022-1236(83)90015-0. [19] L. W. Green, A theorem of E. Hopf,, Michigan Math. J., 5 (1958), 31. doi: 10.1307/mmj/1028998009. [20] Alexander Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (2009). [21] M. Gromov, Hyperbolic groups,, in, 8 (1987), 75. [22] J. Heber, On harmonic and asymptotically harmonic homogeneous spaces,, Geom. Funct. Anal., 16 (2006), 869. doi: 10.1007/s00039-006-0569-4. [23] Jens Heber, Gerhard Knieper and Hemangi M. Shah, Asymptotically harmonic spaces in dimension 3,, Proc. Amer. Math. Soc., 135 (2007), 845. [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). [25] Gerhard Knieper, New results on noncompact harmonic manifolds,, Comment. Math. Helv., 87 (2012), 669. doi: 10.4171/CMH/265. [26] A. J. Ledger, Symmetric harmonic spaces,, J. London Math. Soc., 32 (1957), 53. doi: 10.1112/jlms/s1-32.1.53. [27] F. Ledrappier, Harmonic measures and Bowen-Margulis measures,, Israel J. Math., 71 (1990), 275. doi: 10.1007/BF02773746. [28] François Ledrappier, Linear drift and entropy for regular covers,, Geom. Funct. Anal., 20 (2010), 710. doi: 10.1007/s00039-010-0080-9. [29] André Lichnerowicz, Sur les espaces riemanniens complètement harmoniques,, Bull. Soc. Math. France, 72 (1944), 146. [30] François Ledrappier and Lin Shu, Entropy rigidity of symmetric spaces without focal points,, preprint, (2012). [31] François Ledrappier and Xiaodong Wang, An integral formula for the volume entropy with applications to rigidity,, J. Differential Geom., 85 (2010), 461. [32] Anthony Manning, Topological entropy for geodesic flows,, Ann. of Math. (2), 110 (1979), 567. [33] Akhil Ranjan and Hemangi Shah, Busemann functions in a harmonic manifold,, Geom. Dedicata, 101 (2003), 167. doi: 10.1023/A:1026369930269. [34] Rafael O. Ruggiero, "Dynamics and Global Geometry of Manifolds Without Conjugate Points,", Ensaios Matemáticos [Mathematical Surveys], 12 (2007). [35] Hemangi Shah, On 3-dimensional asymptotically harmonic manifolds with minimal horospheres,, preprint, (2011). [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. [37] Z. I. Szabó, The Lichnerowicz conjecture on harmonic manifolds,, J. Differential Geom., 31 (1990), 1. [38] Jordan Watkins, The higher rank rigidity theorem for manifolds with no focal points,, preprint, (2011).

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. [2] Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. II,, Mem. Amer. Math. Soc., 8 (1976). [3] Werner Ballmann, Nonpositively curved manifolds of higher rank,, Ann. of Math. (2), 122 (1985), 597. [4] Werner Ballmann, Misha Brin and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I,, Ann. of Math. (2), 122 (1985), 171. [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. [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. [7] J. Bolton, Conditions under which a geodesic flow is Anosov,, Math. Ann., 240 (1979), 103. doi: 10.1007/BF01364627. [8] Keith Burns and Ralf Spatzier, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35. [9] Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov,, Pacific J. Math., 159 (1993), 241. [10] Ewa Damek and Fulvio Ricci, A class of nonsymmetric harmonic Riemannian spaces,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 139. [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. [12] Patrick Eberlein, When is a geodesic flow of Anosov type? I,, J. Differential Geometry, 8 (1973), 437. [13] P. Eberlein and B. O'Neill, Visibility manifolds,, Pacific J. Math., 46 (1973), 45. [14] Jost-Hinrich Eschenburg, Horospheres and the stable part of the geodesic flow,, Math. Z., 153 (1977), 237. doi: 10.1007/BF01214477. [15] J.-H. Eschenburg, A note on symmetric and harmonic spaces,, J. London Math. Soc. (2), 21 (1980), 541. [16] Patrick Foulon and François Labourie, Sur les variétés compactes asymptotiquement harmoniques,, Invent. Math., 109 (1992), 97. doi: 10.1007/BF01232020. [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. [18] Lucy Garnett, Foliations, the ergodic theorem and Brownian motion,, J. Funct. Anal., 51 (1983), 285. doi: 10.1016/0022-1236(83)90015-0. [19] L. W. Green, A theorem of E. Hopf,, Michigan Math. J., 5 (1958), 31. doi: 10.1307/mmj/1028998009. [20] Alexander Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (2009). [21] M. Gromov, Hyperbolic groups,, in, 8 (1987), 75. [22] J. Heber, On harmonic and asymptotically harmonic homogeneous spaces,, Geom. Funct. Anal., 16 (2006), 869. doi: 10.1007/s00039-006-0569-4. [23] Jens Heber, Gerhard Knieper and Hemangi M. Shah, Asymptotically harmonic spaces in dimension 3,, Proc. Amer. Math. Soc., 135 (2007), 845. [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). [25] Gerhard Knieper, New results on noncompact harmonic manifolds,, Comment. Math. Helv., 87 (2012), 669. doi: 10.4171/CMH/265. [26] A. J. Ledger, Symmetric harmonic spaces,, J. London Math. Soc., 32 (1957), 53. doi: 10.1112/jlms/s1-32.1.53. [27] F. Ledrappier, Harmonic measures and Bowen-Margulis measures,, Israel J. Math., 71 (1990), 275. doi: 10.1007/BF02773746. [28] François Ledrappier, Linear drift and entropy for regular covers,, Geom. Funct. Anal., 20 (2010), 710. doi: 10.1007/s00039-010-0080-9. [29] André Lichnerowicz, Sur les espaces riemanniens complètement harmoniques,, Bull. Soc. Math. France, 72 (1944), 146. [30] François Ledrappier and Lin Shu, Entropy rigidity of symmetric spaces without focal points,, preprint, (2012). [31] François Ledrappier and Xiaodong Wang, An integral formula for the volume entropy with applications to rigidity,, J. Differential Geom., 85 (2010), 461. [32] Anthony Manning, Topological entropy for geodesic flows,, Ann. of Math. (2), 110 (1979), 567. [33] Akhil Ranjan and Hemangi Shah, Busemann functions in a harmonic manifold,, Geom. Dedicata, 101 (2003), 167. doi: 10.1023/A:1026369930269. [34] Rafael O. Ruggiero, "Dynamics and Global Geometry of Manifolds Without Conjugate Points,", Ensaios Matemáticos [Mathematical Surveys], 12 (2007). [35] Hemangi Shah, On 3-dimensional asymptotically harmonic manifolds with minimal horospheres,, preprint, (2011). [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. [37] Z. I. Szabó, The Lichnerowicz conjecture on harmonic manifolds,, J. Differential Geom., 31 (1990), 1. [38] Jordan Watkins, The higher rank rigidity theorem for manifolds with no focal points,, preprint, (2011).
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