Volume entropy and integral Ricci curvatures over closed geodesics

Paeng, Seong-Hun

doi:10.1090/S0002-9939-07-08869-7

Volume entropy and integral Ricci curvatures over closed geodesics
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Proc. Amer. Math. Soc. 135 (2007), 3677-3684 Request permission

Abstract:

We obtain an upper bound of the volume entropy and the simplicial volume with integrals of Ricci curvature over closed geodesics and apply it to the real Schwarz lemma by Besson, Courtois and Gallot.

References

Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR 1377265, DOI 10.1007/978-3-0348-9240-7
G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), no. 5, 731–799 (French). MR 1354289, DOI 10.1007/BF01897050
Gérard Besson, Gilles Courtois, and Sylvestre Gallot, Minimal entropy and Mostow’s rigidity theorems, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 623–649. MR 1406425, DOI 10.1017/S0143385700009019
G. Besson, G. Courtois, and S. Gallot, A real Schwarz lemma and some applications, Rend. Mat. Appl. (7) 18 (1998), no. 2, 381–410 (English, with English and Italian summaries). MR 1659814
Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
Christopher B. Croke and Nurlan S. Dairbekov, Lengths and volumes in Riemannian manifolds, Duke Math. J. 125 (2004), no. 1, 1–14. MR 2097355, DOI 10.1215/S0012-7094-04-12511-4
Nurlan S. Dairbekov and Vladimir A. Sharafutdinov, Some problems of integral geometry on Anosov manifolds, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 59–74. MR 1971196, DOI 10.1017/S0143385702000822
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320
Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031–1045. MR 1487753, DOI 10.1007/s000390050036
Peter Petersen and Guofang Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds. II, Trans. Amer. Math. Soc. 353 (2001), no. 2, 457–478. MR 1709777, DOI 10.1090/S0002-9947-00-02621-0