Quantitative maximal volume entropy rigidity on Alexandrov spaces

Chen, Lina

doi:10.1090/proc/15904

Quantitative maximal volume entropy rigidity on Alexandrov spaces
HTML articles powered by AMS MathViewer

by Lina Chen PDF

Proc. Amer. Math. Soc. 150 (2022), 3103-3123 Request permission

Abstract:

We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given $N, D$, there exists $\epsilon (N, D)>0$, such that for $\epsilon <\epsilon (N, D)$, if $X$ is an $N$-dimensional Alexandrov space with curvature $\geq -1$, $\operatorname {diam}(X)\leq D, h(X)\geq N-1-\epsilon$, then $X$ is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity provided by Chen, Rong, and Xu [J. Differential Geom. 113 (2019), pp. 227–272] to Alexandrov spaces. And we will also give a quantitative maximal volume entropy rigidity for $\operatorname {RCD}^*$-spaces in the non-collapsing case.

References

Luigi Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 301–340. MR 3966731
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), no. 7, 1405–1490. MR 3205729, DOI 10.1215/00127094-2681605
Luigi Ambrosio and Dario Trevisan, Well-posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDE 7 (2014), no. 5, 1179–1234. MR 3265963, DOI 10.2140/apde.2014.7.1179
G. Besson, G. Courtois, S. Gallot, and A. Sambusetti, Curvature-free Margulis-lemma for Gromov-hyperbolic spaces, arXiv:1712.08386, 2017.
Chris Connell, Xianzhe Dai, Jesús Núñez-Zimbrón, Raquel Perales, Pablo Suárez-Serrato, and Guofang Wei, Maximal volume entropy rigidity for $\mathsf {RCD}^*(-(N -1),N)$ spaces, J. Lond. Math. Soc. (2) 104 (2021), no. 4, 1615–1681. MR 4339946, DOI 10.1112/jlms.12470
J. Cheeger, Degeneration of Riemannian metrics under Ricci curvature bound, Edizioni della Normale, (2001).
Fabio Cavalletti and Emanuel Milman, The globalization theorem for the curvature-dimension condition, Invent. Math. 226 (2021), no. 1, 1–137. MR 4309491, DOI 10.1007/s00222-021-01040-6
Fabio Cavalletti and Andrea Mondino, New formulas for the Laplacian of distance functions and applications, Anal. PDE 13 (2020), no. 7, 2091–2147. MR 4175820, DOI 10.2140/apde.2020.13.2091
Lina Chen, Xiaochun Rong, and Shicheng Xu, Quantitative volume space form rigidity under lower Ricci curvature bound I, J. Differential Geom. 113 (2019), no. 2, 227–272. MR 4023292, DOI 10.4310/jdg/1571882427
Fabio Cavalletti and Karl-Theodor Sturm, Local curvature-dimension condition implies measure-contraction property, J. Funct. Anal. 262 (2012), no. 12, 5110–5127. MR 2916062, DOI 10.1016/j.jfa.2012.02.015
Guido De Philippis and Nicola Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018), 613–650 (English, with English and French summaries). MR 3852263, DOI 10.5802/jep.80
Kenji Fukaya, Theory of convergence for Riemannian orbifolds, Japan. J. Math. (N.S.) 12 (1986), no. 1, 121–160. MR 914311, DOI 10.4099/math1924.12.121
Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253–333. MR 1185120, DOI 10.2307/2946606
Kenji Fukaya and Takao Yamaguchi, Isometry groups of singular spaces, Math. Z. 216 (1994), no. 1, 31–44. MR 1273464, DOI 10.1007/BF02572307
N. Gigli, The splitting theorem in non-smooth context, Mathematics, arXiv:1302.5555, 2013.
Nicola Gigli, An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature, Anal. Geom. Metr. Spaces 2 (2014), no. 1, 169–213. MR 3210895, DOI 10.2478/agms-2014-0006
Nicola Gigli, On the differential structure of metric measure spaces and applications, Mem. Amer. Math. Soc. 236 (2015), no. 1113, vi+91. MR 3381131, DOI 10.1090/memo/1113
Nicola Gigli, Andrea Mondino, and Giuseppe Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. (3) 111 (2015), no. 5, 1071–1129. MR 3477230, DOI 10.1112/plms/pdv047
Luis Guijarro and Jaime Santos-Rodríguez, On the isometry group of $RCD^*(K,N)$-spaces, Manuscripta Math. 158 (2019), no. 3-4, 441–461. MR 3914958, DOI 10.1007/s00229-018-1010-7
Nicola Gigli, Tapio Rajala, and Karl-Theodor Sturm, Optimal maps and exponentiation on finite-dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal. 26 (2016), no. 4, 2914–2929. MR 3544946, DOI 10.1007/s12220-015-9654-y
Yin Jiang, Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry, Math. Z. 291 (2019), no. 1-2, 55–84. MR 3936060, DOI 10.1007/s00209-018-2073-6
John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI 10.4007/annals.2009.169.903
François Ledrappier and Xiaodong Wang, An integral formula for the volume entropy with applications to rigidity, J. Differential Geom. 85 (2010), no. 3, 461–477. MR 2739810
Anthony Manning, Topological entropy for geodesic flows, Ann. of Math. (2) 110 (1979), no. 3, 567–573. MR 554385, DOI 10.2307/1971239
Andrea Mondino and Guofang Wei, On the universal cover and the fundamental group of an $\textrm {RCD}^*(K,N)$-space, J. Reine Angew. Math. 753 (2019), 211–237. MR 3987869, DOI 10.1515/crelle-2016-0068
Andrea Mondino and Aaron Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 6, 1809–1854. MR 3945743, DOI 10.4171/JEMS/874
Shin-ichi Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), no. 4, 805–828. MR 2341840, DOI 10.4171/CMH/110
Anton Petrunin, Alexandrov meets Lott-Villani-Sturm, Münster J. Math. 4 (2011), 53–64. MR 2869253
Guido De Philippis and Nicola Gigli, From volume cone to metric cone in the nonsmooth setting, Geom. Funct. Anal. 26 (2016), no. 6, 1526–1587. MR 3579705, DOI 10.1007/s00039-016-0391-6
J. Pan and G. Wei, Semi-local simple connectedness of non-collapsing Ricci limit spaces, J. Euro. Math. Soc., to appear, DOI 10.411/JEMS/1166, 2021.
Guillemette Reviron, Rigidité topologique sous l’hypothèse “entropie majorée” et applications, Comment. Math. Helv. 83 (2008), no. 4, 815–846 (French, with English and French summaries). MR 2442964, DOI 10.4171/CMH/144
Willi Rinow, Die innere Geometrie der metrischen Räume, Die Grundlehren der mathematischen Wissenschaften, Band 105, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0123969, DOI 10.1007/978-3-662-11499-5
Gerardo Sosa, The isometry group of an $\mathsf {RCD}^*$ space is Lie, Potential Anal. 49 (2018), no. 2, 267–286. MR 3824962, DOI 10.1007/s11118-017-9656-4
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. MR 2237206, DOI 10.1007/s11511-006-0002-8
Karl-Theodor Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. MR 2237207, DOI 10.1007/s11511-006-0003-7
Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
S. Xu and X. Yao, Margulis lemma and Hurwicz fibration theorem on Alexandrov spaces, Commun. Contemp. Math., to appear, DOI 10.1142/S0219199721500486, 2021.
Hui-Chun Zhang and Xi-Ping Zhu, Ricci curvature on Alexandrov spaces and rigidity theorems, Comm. Anal. Geom. 18 (2010), no. 3, 503–553. MR 2747437, DOI 10.4310/CAG.2010.v18.n3.a4