×

Boundaries of non-compact harmonic manifolds. (English) Zbl 1293.53046

The paper under review concerns the problem of classification of harmonic Riemannian manifolds, c.f. [A. Lichnerowicz, Bull. Soc. Math. Fr. 72, 146–168 (1944; Zbl 0060.38506); A. G. Walker, J. Lond. Math. Soc. 24, 21–28 (1949; Zbl 0032.18801); Z. I. Szabó, J. Differ. Geom. 31, No. 1, 1–28 (1990;, Zbl 0686.53042); E. Damek and F. Ricci, Bull. Am. Math. Soc., New Ser. 27, No. 1, 139–142 (1992; Zbl 0755.53032); J. Heber, Geom. Funct. Anal. 16, No. 4, 869–890 (2006; Zbl 1108.53022); G. Knieper, Comment. Math. Helv. 87, No. 3, 669–703 (2012; Zbl 1287.53056)]. It is shown that for a non-flat, non-compact, simply connected harmonic manifold \(X\) the Martin boundary and the Busemann boundary coincide. Moreover, it is proved that if \(M\) is a non-flat finite volume harmonic manifold without conjugate points and \(X\) is the universal Riemannian cover of \(M\) with deck transformations \(\Gamma=\pi_1(M)\subset \mathrm{Isom} (X)\), then up to scaling there exists a unique \(\Gamma\)-Patterson-Sullivan measure on the Buseman boundary of \(X\); besides, the Patterson-Sullivan measures coincides with the harmonic measures under the natural identification of the Busemann boundary and Martin boundary. As consequence, it is stated that if \(M\) is a non-flat finite volume harmonic manifold without conjugate points, then the geodesic flow is topologically transitive on \(SM\).

MSC:

53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
43A85 Harmonic analysis on homogeneous spaces
53C43 Differential geometric aspects of harmonic maps
53D25 Geodesic flows in symplectic geometry and contact geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ancona, A.: Théorie du potentiel sur les graphes et les variétés. In: École d’été de Probabilités de Saint-Flour XVIII-1988, Volume 1427 of Lecture Notes in Math., pp. 1-112. Springer, Berlin (1990) · Zbl 1066.53086
[2] Anderson, M.T., Schoen, R.: Positive harmonic functions on complete manifolds of negative curvature. Ann. Math. (2), 121(3):429-461 (1985) · Zbl 0587.53045
[3] Ballmann, W.: Lectures on Spaces of Nonpositive Curvature, Volume 25 of DMV Seminar. Birkhäuser Verlag, Basel (1995) (With an appendix by Misha Brin) · Zbl 0834.53003
[4] Benoist, Y., Foulon, P., Labourie, F.: Flots d’Anosov à distributions stable et instable différentiables. J. Am. Math. Soc. 5(1), 33-74 (1992) · Zbl 0759.58035
[5] Besse, A.L.: Manifolds all of Whose Geodesics are Closed, Volume 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer, Berlin (1978) (With appendices by Epstein, D.B.A., Bourguignon, J.-P., Bérard-Bergery, L., Berger, M., Kazdan, J.L.) · Zbl 0387.53010
[6] Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5), 731-799 (1995) · Zbl 0851.53032 · doi:10.1007/BF01897050
[7] Besson, G., Courtois, G., Gallot, S.: Minimal entropy and Mostow’s rigidity theorems. Ergodic Theory Dyn. Syst. 16(4), 623-649 (1996) · Zbl 0887.58030 · doi:10.1017/S0143385700009019
[8] Damek, E., Ricci, F.: A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc. (N.S.), 27(1):139-142 (1992) · Zbl 0755.53032
[9] Eberlein, P.: Geodesic flows on negatively curved manifolds. II. Trans. Am. Math. Soc. 178, 57-82 (1973) · Zbl 0264.53027 · doi:10.1090/S0002-9947-1973-0314084-0
[10] Eberlein, P.: When is a geodesic flow of Anosov type? I. J. Differ. Geom. 8, 437-463 (1973) · Zbl 0285.58008
[11] Eschenburg, J.-H.: Horospheres and the stable part of the geodesic flow. Math. Z. 153(3), 237-251 (1977) · Zbl 0332.53028 · doi:10.1007/BF01214477
[12] Eschenburg, J.-H., O’Sullivan, John J.: Growth of Jacobi fields and divergence of geodesics. Math. Z. 150(3), 221-237 (1976) · Zbl 0318.53044 · doi:10.1007/BF01221148
[13] Foulon, P., Labourie, F.: Sur les variétés compactes asymptotiquement harmoniques. Invent. Math. 109(1), 97-111 (1992) · Zbl 0767.53030 · doi:10.1007/BF01232020
[14] Freire, A., Mañé, R.: On the entropy of the geodesic flow in manifolds without conjugate points. Invent. Math. 69(3), 375-392 (1982) · Zbl 0476.58019 · doi:10.1007/BF01389360
[15] Green, L.W.: A theorem of E. Hopf. Michigan Math. J. 5, 31-34 (1958) · Zbl 0134.39601 · doi:10.1307/mmj/1028998009
[16] Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, Volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI (2009) · Zbl 1206.58008
[17] Heber, J.: On harmonic and asymptotically harmonic homogeneous spaces. Geom. Funct. Anal. 16(4), 869-890 (2006) · Zbl 1108.53022 · doi:10.1007/s00039-006-0569-4
[18] Knieper, Gerhard: New results on noncompact harmonic manifolds. Comment. Math. Helv. 87(3), 669-703 (2012) · Zbl 1287.53056 · doi:10.4171/CMH/265
[19] Ledrappier, F.: Linear drift and entropy for regular covers. Geom. Funct. Anal. 20(3), 710-725 (2010) · Zbl 1208.58033 · doi:10.1007/s00039-010-0080-9
[20] Ledrappier, F., Wang, X.: An integral formula for the volume entropy with applications to rigidity. J. Differ. Geom. 85(3), 461-477 (2010) · Zbl 1222.53040
[21] Lichnerowicz, A.: Sur les espaces riemanniens complètement harmoniques. Bull. Soc. Math. France, 72:146-168 (1944) · Zbl 0060.38506
[22] Manning, A.: Topological entropy for geodesic flows. Ann. of Math. (2), 110(3):567-573 (1979) · Zbl 0426.58016
[23] Nikolayevsky, Y.: Two theorems on harmonic manifolds. Comment. Math. Helv. 80(1), 29-50 (2005) · Zbl 1078.53032 · doi:10.4171/CMH/2
[24] Ranjan, A., Shah, H.: Harmonic manifolds with minimal horospheres. J. Geom. Anal. 12(4), 683-694 (2002) · Zbl 1066.53086 · doi:10.1007/BF02930658
[25] Ranjan, A., Shah, H.: Busemann functions in a harmonic manifold. Geom. Dedicata 101, 167-183 (2003) · Zbl 1045.53023 · doi:10.1023/A:1026369930269
[26] Szabó, Z.I.: The Lichnerowicz conjecture on harmonic manifolds. J. Differ. Geom. 31(1), 1-28 (1990) · Zbl 0686.53042
[27] Walker, A.G.: On Lichnerowicz’s conjecture for harmonic 4-spaces. J. Lond. Math. Soc. 24, 21-28 (1949) · Zbl 0032.18801 · doi:10.1112/jlms/s1-24.1.21
[28] Wang, X.: Compactifications of complete riemannian manifolds and their applications. In: Bray, H.L. , Minicozzi II, W.P. (eds.) Surveys in Geometric Analysis and Relativity, pp. 517-529. International Press of Boston (2011) · Zbl 1268.53045
[29] Willmore, T.J.: Mean value theorems in harmonic Riemannian spaces. J. Lond. Math. Soc. 25, 54-57 (1950) · Zbl 0036.23403 · doi:10.1112/jlms/s1-25.1.54
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.