Limits of manifolds with a Kato bound on the Ricci curvature

Carron, Gilles¹ ; Mondello, Ilaria² ; Tewodrose, David³

¹ Laboratoire de Mathématiques Jean Leray, UMR 6629, Université de Nantes, Nantes, France
² Laboratoire d’Analyse et Mathématiques Appliquées, UMR CNRS 8050, Université Paris Est Créteil, Créteil, France
³ Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, Nantes, France

Geometry & topology, Tome 28 (2024) no. 6, pp. 2635-2745.

Résumé

We study the structure of Gromov–Hausdorff limits of sequences of Riemannian manifolds ${\{(M_{\alpha }^n,g_{\alpha })\}}_{\alpha \in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy, and show that tangent cones of such limits satisfy the $\mathrm{RCD}(0,n)$ condition. Under a noncollapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff $n$–measure.

Détail

DOI : 10.2140/gt.2024.28.2635

Keywords: Gromov–Hausdorff convergence, Ricci curvature, Kato class, volume convergence

Bibliographie

[1] K Akutagawa, Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl. 4 (1994) 239 | DOI

[2] C L Aldana, G Carron, S Tapie, A∞–weights and compactness of conformal metrics under Ln∕2 curvature bounds, Anal. PDE 14 (2021) 2163 | DOI

[3] L Ambrosio, N Gigli, G Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014) 1405 | DOI

[4] L Ambrosio, N Gigli, G Savaré, Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015) 339 | DOI

[5] P Auscher, T Coulhon, Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 531

[6] K Bacher, K T Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28 | DOI

[7] D Bakry, M Ledoux, A logarithmic Sobolev form of the Li–Yau parabolic inequality, Rev. Mat. Iberoam. 22 (2006) 683 | DOI

[8] R H Bamler, Structure theory of singular spaces, J. Funct. Anal. 272 (2017) 2504 | DOI

[9] R Bamler, Convergence of Ricci flows with bounded scalar curvature, Ann. of Math. 188 (2018) 753 | DOI

[10] R H Bamler, Q S Zhang, Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature, Adv. Math. 319 (2017) 396 | DOI

[11] A Beurling, J Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959) 208 | DOI

[12] P Billingsley, Convergence of probability measures, Wiley (1999) | DOI

[13] E Bruè, E Pasqualetto, D Semola, Rectifiability of RCD(K,N) spaces via δ–splitting maps, Ann. Fenn. Math. 46 (2021) 465 | DOI

[14] E Bruè, D Semola, Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows, Comm. Pure Appl. Math. 73 (2020) 1141 | DOI

[15] D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) | DOI

[16] G Carron, Inégalités isopérimétriques de Faber–Krahn et conséquences, from: "Actes de la Table Ronde de Géométrie Différentielle" (editor A L Besse), Sémin. Congr. 1, Soc. Math. France (1996) 205

[17] G Carron, Geometric inequalities for manifolds with Ricci curvature in the Kato class, Ann. Inst. Fourier (Grenoble) 69 (2019) 3095

[18] G Carron, I Mondello, D Tewodrose, Limits of manifolds with a Kato bound on the Ricci curvature, II, preprint (2022)

[19] G Carron, C Rose, Geometric and spectral estimates based on spectral Ricci curvature assumptions, J. Reine Angew. Math. 772 (2021) 121 | DOI

[20] G Carron, D Tewodrose, A rigidity result for metric measure spaces with Euclidean heat kernel, J. Éc. polytech. Math. 9 (2022) 101 | DOI

[21] J Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428 | DOI

[22] J Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds, Scuola Normale Superiore (2001)

[23] J Cheeger, T H Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189 | DOI

[24] J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406

[25] J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000) 13

[26] J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000) 37

[27] J Cheeger, W Jiang, A Naber, Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below, Ann. of Math. 193 (2021) 407 | DOI

[28] J Cheeger, A Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013) 321 | DOI

[29] J Cheeger, A Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. 182 (2015) 1093 | DOI

[30] L Chen, Segment inequality and almost rigidity structures for integral Ricci curvature, Int. Math. Res. Not. 2022 (2022) 11732 | DOI

[31] T H Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477 | DOI

[32] T H Colding, W P Minicozzi Ii, Liouville theorems for harmonic sections and applications, Comm. Pure Appl. Math. 51 (1998) 113 | DOI

[33] T Coulhon, Itération de Moser et estimation gaussienne du noyau de la chaleur, J. Operator Theory 29 (1993) 157

[34] T Coulhon, A Grigoryan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J. 89 (1997) 133 | DOI

[35] T Coulhon, L Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoam. 9 (1993) 293 | DOI

[36] X Dai, G Wei, Z Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, Adv. Math. 325 (2018) 1 | DOI

[37] G De Philippis, N Gigli, From volume cone to metric cone in the nonsmooth setting, Geom. Funct. Anal. 26 (2016) 1526 | DOI

[38] G De Philippis, N Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018) 613 | DOI

[39] A F M Ter Elst, D W Robinson, A Sikora, Small time asymptotics of diffusion processes, J. Evol. Equ. 7 (2007) 79 | DOI

[40] M Erbar, K Kuwada, K T Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015) 993 | DOI

[41] K Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987) 517 | DOI

[42] M Fukushima, Y Ōshima, M Takeda, Dirichlet forms and symmetric Markov processes, 19, de Gruyter (1994) | DOI

[43] S Gallot, Volumes, courbure de Ricci et convergence des variétés (d’après T H Colding et Cheeger–Colding), from: "Séminaire Bourbaki, 1997/98", Astérisque 252, Soc. Math. France (1998)

[44] M Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, 105, Princeton Univ. Press (1983)

[45] N Gigli, The splitting theorem in non-smooth context, preprint (2013)

[46] N Gigli, An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature, Anal. Geom. Metr. Spaces 2 (2014) 169 | DOI

[47] N Gigli, On the differential structure of metric measure spaces and applications, 1113, Amer. Math. Soc. (2015) | DOI

[48] A A Grigoryan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991) 55

[49] A Grigoryan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoam. 10 (1994) 395 | DOI

[50] A Grigoryan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999) 135 | DOI

[51] A Grigoryan, Heat kernels on metric measure spaces with regular volume growth, from: "Handbook of geometric analysis, II" (editors L Ji, P Li, R Schoen, L Simon), Adv. Lect. Math. 13, International (2010) 1

[52] A Grigoryan, J Hu, K S Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc. 355 (2003) 2065 | DOI

[53] A Grigoryan, A Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab. 40 (2012) 1212 | DOI

[54] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) | DOI

[55] B Güneysu, Covariant Schrödinger semigroups on Riemannian manifolds, 264, Birkhäuser (2017) | DOI

[56] E Hebey, Sobolev spaces on Riemannian manifolds, 1635, Springer (1996) | DOI

[57] W Hebisch, L Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001) 1437 | DOI

[58] J Heinonen, P Koskela, N Shanmugalingam, J T Tyson, Sobolev spaces on metric measure spaces: an approach based on upper gradients, 27, Cambridge Univ. Press (2015) | DOI

[59] H Hess, R Schrader, D A Uhlenbrock, Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds, J. Differential Geom. 15 (1980) 27

[60] G Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285

[61] W Jiang, A Naber, L2 curvature bounds on manifolds with bounded Ricci curvature, Ann. of Math. 193 (2021) 107 | DOI

[62] A Kasue, Convergence of metric measure spaces and energy forms, from: "Selected papers on differential equations and analysis", Amer. Math. Soc. Transl. (2) 215, Amer. Math. Soc. (2005) 79 | DOI

[63] S Keith, X Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. 167 (2008) 575 | DOI

[64] C Ketterer, Stability of metric measure spaces with integral Ricci curvature bounds, J. Funct. Anal. 281 (2021) 109142 | DOI

[65] P Koskela, N Shanmugalingam, Y Zhou, Geometry and analysis of Dirichlet forms, II, J. Funct. Anal. 267 (2014) 2437 | DOI

[66] P Koskela, Y Zhou, Geometry and analysis of Dirichlet forms, Adv. Math. 231 (2012) 2755 | DOI

[67] K Kuwae, T Shioya, Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom. 11 (2003) 599 | DOI

[68] J Lott, C Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009) 903 | DOI

[69] P Mattila, Geometry of sets and measures in Euclidean spaces : fractals and rectifiability, 44, Cambridge Univ. Press (1995) | DOI

[70] A Mondino, A Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019) 1809 | DOI

[71] U Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994) 368 | DOI

[72] P Petersen, G Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997) 1031 | DOI

[73] P Petersen, G Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds, II, Trans. Amer. Math. Soc. 353 (2001) 457 | DOI

[74] J A Ramírez, Short-time asymptotics in Dirichlet spaces, Comm. Pure Appl. Math. 54 (2001) 259 | DOI

[75] C Rose, Li–Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class, Ann. Global Anal. Geom. 55 (2019) 443 | DOI

[76] C Rose, P Stollmann, The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature, Proc. Amer. Math. Soc. 145 (2017) 2199 | DOI

[77] L Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Int. Math. Res. Not. 1992 (1992) 27 | DOI

[78] L Saloff-Coste, Aspects of Sobolev-type inequalities, 289, Cambridge Univ. Press (2002) | DOI

[79] A Sikora, Sharp pointwise estimates on heat kernels, Q. J. Math. 47 (1996) 371 | DOI

[80] B Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982) 447 | DOI

[81] L Simon, Lectures on geometric measure theory, 3, Austral. Nat. Univ. (1983)

[82] M Simon, Extending four dimensional Ricci flows with bounded scalar curvature, Comm. Anal. Geom. 28 (2020) 1683 | DOI

[83] M Simon, Some integral curvature estimates for the Ricci flow in four dimensions, Comm. Anal. Geom. 28 (2020) 707 | DOI

[84] K T Sturm, Analysis on local Dirichlet spaces, I : Recurrence, conservativeness and Lp–Liouville properties, J. Reine Angew. Math. 456 (1994) 173 | DOI

[85] K T Sturm, Analysis on local Dirichlet spaces, III : The parabolic Harnack inequality, J. Math. Pures Appl. 75 (1996) 273

[86] K T Sturm, On the geometry of metric measure spaces, I, Acta Math. 196 (2006) 65 | DOI

[87] K T Sturm, On the geometry of metric measure spaces, II, Acta Math. 196 (2006) 133 | DOI

[88] R Tessera, Volume of spheres in doubling metric measured spaces and in groups of polynomial growth, Bull. Soc. Math. France 135 (2007) 47 | DOI

[89] G Tian, J Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005) 357 | DOI

[90] G Tian, J Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2005) 346 | DOI

[91] G Tian, J Viaclovsky, Volume growth, curvature decay, and critical metrics, Comment. Math. Helv. 83 (2008) 889 | DOI

[92] G Tian, Z Zhang, Regularity of Kähler–Ricci flows on Fano manifolds, Acta Math. 216 (2016) 127 | DOI

[93] N T Varopoulos, Hardy–Littlewood theory for semigroups, J. Funct. Anal. 63 (1985) 240 | DOI

[94] C Villani, Optimal transport: old and new, 338, Springer (2009) | DOI

[95] J Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann. 229 (1977) 163 | DOI

[96] B White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997) 1 | DOI

[97] Q S Zhang, M Zhu, Li–Yau gradient bounds on compact manifolds under nearly optimal curvature conditions, J. Funct. Anal. 275 (2018) 478 | DOI