Abstract
Let M be a Riemannian manifold with dimension greater than or equal to 3 which admits a complete, finite-volume Riemannian metric \(g_0\) locally isometric to a rank one symmetric space of non-compact type. The volume entropy rigidity theorem (Besson et al. in Geom Funct Anal 5(5), 731–799, 1995, Theorémè principal) asserts that \(g_0\) minimizes a normalized volume growth entropy among all complete, finite-volume, Riemannian metric on M. We will repair a gap in the proof when \(g_0\) is locally isometric to the Cayley hyperbolic space.
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References
Albuquerque, P.: Patterson–Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9(1), 1–28 (1999)
Baez, J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39(2), 145–205 (2002)
Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. (French) [Entropy and rigidity of locally symmetric spaces with strictly negative curvature]. Geom. Funct. Anal. 5(5), 731–799 (1995)
Besson, G., Courtois, G., Gallot, S.: Minimal entropy and Mostow’s rigidity theorems. Ergodic Theory Dyn. Syst. 16(4), 623–649 (1996)
Boland, J., Connell, C., Souto, J.: Volume rigidity for finite volume manifolds. Am. J. Math. 127(3), 535–550 (2005)
Connell, C., Farb, B.: Minimal entropy rigidity for lattices in products of rank one symmetric spaces. Commun. Anal. Geom. 11(5), 1001–1026 (2003)
Connell, C., Farb, B.: The degree theorem in higher rank. J. Differ. Geom. 65(1), 19–59 (2003). (Erratum for The degree theorem in higher rank. J. Differential Geom. 105(1), 21-32 (2017))
Mostow, G.D.: Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies, vol. 78. Princeton University Press, University of Tokyo Press, Princeton (1973)
Parker, J.R.: Hyperbolic Spaces. Jyväskylä Lectures in Mathematics 2 (2008)
Ruan, Y.: Filling volume minimality and boundary rigidity of metrics close to a negatively curved symmetric metric. arXiv:2201.09175 [math.DG]
Springer, T.A., Veldkamp, F.D.: Elliptic and hyperbolic octave planes I, II, III, vol. 66, pp. 413–451. Springer, Berlin (1963)
Springer, T.A., Veldkamp, F.D.: Octonions, Jordan Algebras and Exceptional Groups. Springer Monographs in Mathematics. Springer, Berlin (2000). (ISBN: 3-540-66337-1)
Storm, P.A.: The minimal entropy conjecture for nonuniform rank one lattices. Geom. Funct. Anal. 16(4), 959–980 (2006)
Acknowledgements
I would like to heartily thank my advisor Ralf Spatzier for his support during the entire work. I sincerely thank Gérald Besson, Gilles Courtois and Sylvain Gallot for patiently proofreading this paper and providing valuable suggestions, especially for pointing out a mistake in an earlier version of the manuscript. I am also very grateful to Chris Connell for helpful and thorough discussions on this subject.
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Ruan, Y. The Cayley hyperbolic space and volume entropy rigidity. Math. Z. 306, 4 (2024). https://doi.org/10.1007/s00209-023-03398-0
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DOI: https://doi.org/10.1007/s00209-023-03398-0