Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 12, 2016

Tangent Lines and Lipschitz Differentiability Spaces

  • Fabio Cavalletti and Tapio Rajala

Abstract

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces.We show that any tangent space of a Lipschitz differentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space.

References

[1] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with ff-finite measure, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4661–4701. Search in Google Scholar

[2] L. Ambrosio, N Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Annals of Probab. 43 (2015), no. 1, 339–404. Search in Google Scholar

[3] L. Ambrosio, N Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405–1490. 10.1215/00127094-2681605Search in Google Scholar

[4] L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527–555. 10.1007/s002080000122Search in Google Scholar

[5] L. Ambrosio, A. Mondino and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, preprint arXiv:1509.07273. Search in Google Scholar

[6] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces. Oxford University press, Oxford Lecture Series in Mathematics and Its Applications, 2004. Search in Google Scholar

[7] D. Bate, Structure of measures in Lipschitz differentiability spaces, Journal Amer. Math. Soc. 28 (2015), 421–482. 10.1090/S0894-0347-2014-00810-9Search in Google Scholar

[8] D. Bate and S. Li, Characterizations of rectifiable metric measure spaces, preprint, arXiv:1409.4242. Search in Google Scholar

[9] M. Bourdon and H. Pajot, Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2315–2324. Search in Google Scholar

[10] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428–517. 10.1007/s000390050094Search in Google Scholar

[11] J. Cheeger and B. Kleiner and A. Schioppa, Infinitesimal structure of differentiability spaces, and metric differentiation, preprint arXiv:1503.07348. Search in Google Scholar

[12] G.C. David, Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces, Geom. Funct. Anal. 25 (2015), no. 2, 553–579. Search in Google Scholar

[13] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61. 10.1007/BF02392747Search in Google Scholar

[14] MErbar, Kuwada and K.T. Sturm, On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner’s Inequality on Metric Measure Space, Invent. Math. 201 (2015), no. 3, 993 – 1071. Search in Google Scholar

[15] N. Gigli, The splitting theorem in non-smooth context, preprint, arXiv:1302.5555. Search in Google Scholar

[16] N. Gigli, A. Mondino and T. Rajala, Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below, J. Reine Angew. Math. 705 (2015), 233–244. Search in Google Scholar

[17] N. Gigli, A. Mondino and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, to appear in Proc. London Math. Soc. doi: 10.1112/plms/pdv047. 10.1112/plms/pdv047Search in Google Scholar

[18] N. Juillet, Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Int.Math. Res. Notices 2009 (2009), 2347–2373. Search in Google Scholar

[19] S. Keith, Measurable Differentiable Structures and the Poincaré Inequality, Indiana Univ. Math. J. 53 (2004), 1127–1150. 10.1512/iumj.2004.53.2417Search in Google Scholar

[20] C. Ketterer and T. Rajala, Failure of topological rigidity results for the measure contraction property, Potential Anal. 42 (2015), no. 3, 645–655. Search in Google Scholar

[21] B. Kirchheim, Rectifiable metric space: local structure and regularity of the Hausdorff measure, Proc. Am. Math. Soc. 121 (1994), 113–123. 10.1090/S0002-9939-1994-1189747-7Search in Google Scholar

[22] R. Korte, Geometric implications of the Poincaré inequality, Result. Math. 50 (2007), 93–107. 10.1007/s00025-006-0237-xSearch in Google Scholar

[23] B. Kleiner and J. Mackay, Differentiable structures on metric measure spaces: A Primer preprint, arXiv:1108.1324. Search in Google Scholar

[24] T. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000), 111–123. 10.1007/s000390050003Search in Google Scholar

[25] E. Le Donne, Metric spaces with unique tangents, Ann. Acad. Sci. Fenn. Math. 36 (2011), 683–694. 10.5186/aasfm.2011.3636Search in Google Scholar

[26] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. ofMath. (2) 169 (2009), 903–991. 10.4007/annals.2009.169.903Search in Google Scholar

[27] A. Mondino and A. Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I, preprint, arXiv:1405.2222. Search in Google Scholar

[28] S.-I. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805–828. 10.4171/CMH/110Search in Google Scholar

[29] S.-I. Ohta, Splitting theorems for Finsler manifolds of nonnegative Ricci curvature, J. Reine Angew. Math. 700 (2015), 155– 174. Search in Google Scholar

[30] D. Preiss, Geometry of measures in Rn: distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537–643. 10.2307/1971410Search in Google Scholar

[31] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations 44 (2012), 477–494. 10.1007/s00526-011-0442-7Search in Google Scholar

[32] T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm, J. Funct. Anal. 263 (2012), 896–924. 10.1016/j.jfa.2012.05.006Search in Google Scholar

[33] W. Rudin, Real and complex analysis, McGraw-Hill Book Company, International Edition 1987. Search in Google Scholar

[34] A. Schioppa Derivations and Alberti representations, preprint, arXiv:1311.2439. Search in Google Scholar

[35] A. Schioppa On the relationship between derivations and measurable differentiable structures, Ann. Acad. Sci. Fenn.Math., 39 (2014), no. 1, 275–304. 10.5186/aasfm.2014.3910Search in Google Scholar

[36] S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. 2 (1996), 155–295. 10.1007/BF01587936Search in Google Scholar

[37] S. Semmes, Some novel types of fractal geometry, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001. Search in Google Scholar

[38] K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65–131. 10.1007/s11511-006-0002-8Search in Google Scholar

[39] K.T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133–177. 10.1007/s11511-006-0003-7Search in Google Scholar

[40] C. Villani, Optimal transport. Old and new, Grundlehren derMathematischenWissenschaften, 338, Springer-Verlag, Berlin, (2009). 10.1007/978-3-540-71050-9Search in Google Scholar

Received: 2015-8-21
Accepted: 2016-4-15
Published Online: 2016-5-12

© 2016 Fabio Cavalletti and Tapio Rajala

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/agms-2016-0004/html
Scroll to top button