Non-collapsed spaces with Ricci curvature bounded from below

De Philippis, Guido; Gigli, Nicola

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 613-650.

Nous proposons une définition d’espace « non-collapsed » avec courbure de Ricci minorée et nous généralisons aux espaces RCD le théorème de convergence du volume de Colding et l’estimation de l’écart de dimension de Cheeger-Colding. En particulier, ceci prouve la stabilité des espaces RCD « non-collapsed » par rapport à la convergence de Gromov-Hausdorff « non-collapsed ».

We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding’s volume convergence theorem and of Cheeger-Colding dimension gap estimate for $\mathsf{RCD}$ spaces. In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence.

Classification : 53C23, 53C21
Keywords: Ricci curvature bounded from below, non-collapsed spaces
Mot clés : Courbure de Ricci minorée, espace “non-collapsed”

[1] Almgren, F. J. Jr. Almgren’s big regularity paper: $Q$-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension $2$, World Scientific Monograph Series in Mathematics, 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000 | Zbl

[2] Ambrosio, L.; De Philippis, G.; Kirchheim, B. Regularity of optimal transport maps and partial differential inclusions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl., Volume 22 (2011) no. 3, pp. 311-336 | DOI | MR | Zbl

[3] Ambrosio, L.; Gigli, N.; Savaré, G. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., Volume 195 (2014) no. 2, pp. 289-391 | MR | Zbl

[4] Ambrosio, L.; Gigli, N.; Savaré, G. Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., Volume 163 (2014) no. 7, pp. 1405-1490 | DOI | MR | Zbl

[5] Ambrosio, L.; Gigli, N.; Savaré, G. Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probability, Volume 43 (2015) no. 1, pp. 339-404 | DOI | Zbl

[6] Ambrosio, L.; Honda, S.; Tewodrose, D. Short-time behavior of the heat kernel and Weyl’s law on ${\mathrm{RCD}}^{*}(K,N)$ spaces (2017) (arXiv:1701.03906)

[7] Ambrosio, L.; Kirchheim, B. Rectifiable sets in metric and Banach spaces, Math. Ann., Volume 318 (2000) no. 3, pp. 527-555 | DOI | MR | Zbl

[8] Ambrosio, L.; Mondino, A.; Savaré, G. On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of ${RCD}^{*}(K,N)$ metric measure spaces, J. Geom. Anal., Volume 26 (2014) no. 1, pp. 1-33

[9] Ambrosio, L.; Mondino, A.; Savaré, G. Nonlinear diffusion equations and curvature conditions in metric measure spaces, Mem. Amer. Math. Soc., American Mathematical Society, Providence, RI, to appear (arXiv:1509.07273)

[10] Ambrosio, L.; Tilli, P. Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, 25, Oxford University Press, Oxford, 2004 | MR | Zbl

[11] Bacher, K.; Sturm, K.-T. Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal., Volume 259 (2010) no. 1, pp. 28-56 | DOI | MR | Zbl

[12] Bruè, E.; Semola, D. Constancy of the dimension for $\mathrm{RCD}(K,N)$ spaces via regularity of Lagrangian flows (2018) (arXiv:1804.07128)

[13] Burago, D.; Burago, Y.; Ivanov, S. A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001 | MR | Zbl

[14] Cavalletti, F.; Milman, E. The globalization theorem for the curvature dimension condition (2016) (arXiv:1612.07623)

[15] Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 428-517 | DOI | MR | Zbl

[16] Cheeger, J.; Colding, T. H. Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), Volume 144 (1996) no. 1, pp. 189-237 | DOI | MR | Zbl

[17] Cheeger, J.; Colding, T. H. On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., Volume 46 (1997) no. 3, pp. 406-480 | DOI | MR | Zbl

[18] Cheeger, J.; Colding, T. H. On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom., Volume 54 (2000) no. 1, pp. 13-35 | DOI | MR | Zbl

[19] Cheeger, J.; Colding, T. H. On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom., Volume 54 (2000) no. 1, pp. 37-74 | DOI | MR | Zbl

[20] Colding, T. H. Large manifolds with positive Ricci curvature, Invent. Math., Volume 124 (1996) no. 1-3, pp. 193-214 | DOI | MR | Zbl

[21] Colding, T. H. Shape of manifolds with positive Ricci curvature, Invent. Math., Volume 124 (1996) no. 1-3, pp. 175-191 | DOI | MR | Zbl

[22] Colding, T. H. Ricci curvature and volume convergence, Ann. of Math. (2), Volume 145 (1997) no. 3, pp. 477-501 | DOI | MR | Zbl

[23] Colding, T. H.; Naber, A. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. (2), Volume 176 (2012) no. 2, pp. 1173-1229 | DOI | Zbl

[24] De Philippis, G.; Gigli, N. From volume cone to metric cone in the nonsmooth setting, Geom. Funct. Anal., Volume 26 (2016) no. 6, pp. 1526-1587 | DOI | MR | Zbl

[25] Erbar, M.; Kuwada, K.; Sturm, K.-T. On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math., Volume 201 (2014) no. 3, pp. 1-79 | MR | Zbl

[26] Federer, H. Geometric measure theory, Grundlehren Math. Wiss., 153, Springer-Verlag New York Inc., New York, 1969 | MR | Zbl

[27] Gigli, N. On the heat flow on metric measure spaces: existence, uniqueness and stability, Calc. Var. Partial Differential Equations, Volume 39 (2010) no. 1-2, pp. 101-120 | DOI | MR | Zbl

[28] Gigli, N. The splitting theorem in non-smooth context (2013) (arXiv:1302.5555)

[29] Gigli, N. Nonsmooth differential geometry - an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc., 251, no. 1196, American Mathematical Society, Providence, RI, 2014

[30] Gigli, N. An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature, Analysis and Geometry in Metric Spaces, Volume 2 (2014), pp. 169-213 | DOI | MR | Zbl

[31] Gigli, N. On the differential structure of metric measure spaces and applications, Mem. Amer. Math. Soc., 236, no. 1113, American Mathematical Society, Providence, RI, 2015 | Zbl

[32] Gigli, N.; Kuwada, K.; Ohta, S.-i. Heat Flow on Alexandrov Spaces, Comm. Pure Appl. Math., Volume 66 (2013) no. 3, pp. 307-331 | DOI | MR | Zbl

[33] Gigli, N.; Mondino, A.; Rajala, T. Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below, J. reine angew. Math., Volume 705 (2015), pp. 233-244 | MR | Zbl

[34] Gigli, N.; Mondino, A.; Savaré, G. Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. London Math. Soc. (3), Volume 111 (2015) no. 5, pp. 1071-1129 | MR | Zbl

[35] Gigli, N.; Mosconi, S. The abstract Lewy-Stampacchia inequality and applications, J. Math. Pures Appl. (9), Volume 104 (2014) no. 2, pp. 258-275 | DOI | MR | Zbl

[36] Gigli, N.; Pasqualetto, E. Behaviour of the reference measure on $\mathrm{RCD}$ spaces under charts, Comm. Anal. Geom. (to appear) (arXiv:1607.05188)

[37] Gigli, N.; Pasqualetto, E. Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces (2016) (arXiv:1611.09645)

[38] Gigli, N.; Rajala, T.; Sturm, K.-T. Optimal maps and exponentiation on finite-dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal., Volume 26 (2016) no. 4, pp. 2914-2929 | DOI | MR | Zbl

[39] Giusti, E. Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984 | MR | Zbl

[40] Gromov, M. Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007

[41] Han, B.-X. Ricci tensor on ${\mathrm{RCD}}^{*}(K,N)$ spaces, J. Geom. Anal., Volume 28 (2018) no. 2, pp. 1295-1314 | DOI | MR

[42] Kell, M.; Mondino, A. On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016) (arXiv:1607.02036) | Zbl

[43] Kitabeppu, Y. A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below, Proc. Amer. Math. Soc., Volume 145 (2017) no. 7, pp. 3137-3151 | DOI | MR | Zbl

[44] Lott, J.; Villani, C. Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), Volume 169 (2009) no. 3, pp. 903-991 | DOI | MR | Zbl

[45] Mondino, A.; Naber, A. Structure theory of metric-measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. (JEMS) (to appear) (arXiv:1405.2222)

[46] Ohta, S.-i. On the measure contraction property of metric measure spaces, Comment. Math. Helv., Volume 82 (2007) no. 4, pp. 805-828 | DOI | MR | Zbl

[47] Petrunin, A. Applications of quasigeodesics and gradient curves, Comparison geometry (Berkeley, CA, 1993–94) (Math. Sci. Res. Inst. Publ.), Volume 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203-219 | MR | Zbl

[48] Petrunin, A. Semiconcave functions in Alexandrov’s geometry (Surv. Differ. Geom.), Volume 11, Int. Press, Somerville, MA, 2007, pp. 137-201 | MR | Zbl

[49] Shanmugalingam, N. Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana, Volume 16 (2000) no. 2, pp. 243-279 | DOI | MR | Zbl

[50] Sturm, K.-T. On the geometry of metric measure spaces. I, Acta Math., Volume 196 (2006) no. 1, pp. 65-131

[51] Sturm, K.-T. On the geometry of metric measure spaces. II, Acta Math., Volume 196 (2006) no. 1, pp. 133-177 | MR

[52] Villani, C. Optimal transport. Old and new, Grundlehren Math. Wiss., 338, Springer-Verlag, Berlin, 2009 | Zbl

[53] Villani, C. Synthetic theory of Ricci curvature bounds, Japan. J. Math., Volume 11 (2016) no. 2, pp. 219-263 | DOI | MR | Zbl

[54] Villani, C. Inégalités isopérimétriques dans les espaces métriques mesurés [d’après F. Cavalletti & A. Mondino] (2017) (Séminaire Bourbaki, available at: http://www.bourbaki.ens.fr/TEXTES/1127.pdf)

[55] White, B. Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. reine angew. Math., Volume 488 (1997), pp. 1-35 | MR | Zbl