Download this article
Download this article For screen
For printing
Recent Issues

Volume 30
Issue 3, 835–1201
Issue 2, 389–833
Issue 1, 1–388

Volume 29, 9 issues

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Limits of manifolds with a Kato bound on the Ricci curvature

Gilles Carron, Ilaria Mondello and David Tewodrose
Geometry & Topology 28 (2024) 2635–2745
DOI: 10.2140/gt.2024.28.2635
Bibliography
1 K Akutagawa, Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl. 4 (1994) 239 MR1299397
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 MR4353568
3 L Ambrosio, N Gigli, G Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014) 1405 MR3205729
4 L Ambrosio, N Gigli, G Savaré, Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015) 339 MR3298475
5 P Auscher, T Coulhon, Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 531 MR2185868
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 MR2610378
7 D Bakry, M Ledoux, A logarithmic Sobolev form of the Li–Yau parabolic inequality, Rev. Mat. Iberoam. 22 (2006) 683 MR2294794
8 R H Bamler, Structure theory of singular spaces, J. Funct. Anal. 272 (2017) 2504 MR3603307
9 R Bamler, Convergence of Ricci flows with bounded scalar curvature, Ann. of Math. 188 (2018) 753 MR3866886
10 R H Bamler, Q S Zhang, Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature, Adv. Math. 319 (2017) 396 MR3695879
11 A Beurling, J Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959) 208 MR0106365
12 P Billingsley, Convergence of probability measures, Wiley (1999) MR1700749
13 E Bruè, E Pasqualetto, D Semola, Rectifiability of RCD(K,N) spaces via δ–splitting maps, Ann. Fenn. Math. 46 (2021) 465 MR4277822
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 MR4156601
15 D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
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 MR1427759
17 G Carron, Geometric inequalities for manifolds with Ricci curvature in the Kato class, Ann. Inst. Fourier (Grenoble) 69 (2019) 3095 MR4286831
18 G Carron, I Mondello, D Tewodrose, Limits of manifolds with a Kato bound on the Ricci curvature, II, preprint (2022) arXiv:2205.01956
19 G Carron, C Rose, Geometric and spectral estimates based on spectral Ricci curvature assumptions, J. Reine Angew. Math. 772 (2021) 121 MR4227592
20 G Carron, D Tewodrose, A rigidity result for metric measure spaces with Euclidean heat kernel, J. Éc. polytech. Math. 9 (2022) 101 MR4344324
21 J Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428 MR1708448
22 J Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds, Scuola Normale Superiore (2001) MR2006642
23 J Cheeger, T H Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189 MR1405949
24 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 MR1484888
25 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000) 13 MR1815410
26 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000) 37 MR1815411
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 MR4226910
28 J Cheeger, A Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013) 321 MR3010378
29 J Cheeger, A Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. 182 (2015) 1093 MR3418535
30 L Chen, Segment inequality and almost rigidity structures for integral Ricci curvature, Int. Math. Res. Not. 2022 (2022) 11732 MR4458563
31 T H Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477 MR1454700
32 T H Colding, W P Minicozzi II, Liouville theorems for harmonic sections and applications, Comm. Pure Appl. Math. 51 (1998) 113 MR1488297
33 T Coulhon, Itération de Moser et estimation gaussienne du noyau de la chaleur, J. Operator Theory 29 (1993) 157 MR1277971
34 T Coulhon, A Grigoryan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J. 89 (1997) 133 MR1458975
35 T Coulhon, L Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoam. 9 (1993) 293 MR1232845
36 X Dai, G Wei, Z Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, Adv. Math. 325 (2018) 1 MR3742584
37 G De Philippis, N Gigli, From volume cone to metric cone in the nonsmooth setting, Geom. Funct. Anal. 26 (2016) 1526 MR3579705
38 G De Philippis, N Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018) 613 MR3852263
39 A F M ter Elst, D W Robinson, A Sikora, Small time asymptotics of diffusion processes, J. Evol. Equ. 7 (2007) 79 MR2305727
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 MR3385639
41 K Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987) 517 MR0874035
42 M Fukushima, Y Ōshima, M Takeda, Dirichlet forms and symmetric Markov processes, 19, de Gruyter (1994) MR1303354
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) MR1685585
44 M Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, 105, Princeton Univ. Press (1983) MR0717034
45 N Gigli, The splitting theorem in non-smooth context, preprint (2013) arXiv:1302.5555
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 MR3210895
47 N Gigli, On the differential structure of metric measure spaces and applications, 1113, Amer. Math. Soc. (2015) MR3381131
48 A A Grigoryan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991) 55 MR1098839
49 A Grigoryan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoam. 10 (1994) 395 MR1286481
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 MR1659871
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 MR2743439
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 MR1953538
53 A Grigoryan, A Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab. 40 (2012) 1212 MR2962091
54 M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) MR1699320
55 B Güneysu, Covariant Schrödinger semigroups on Riemannian manifolds, 264, Birkhäuser (2017) MR3751359
56 E Hebey, Sobolev spaces on Riemannian manifolds, 1635, Springer (1996) MR1481970
57 W Hebisch, L Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001) 1437 MR1860672
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) MR3363168
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 MR0602436
60 G Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285 MR1030675
61 W Jiang, A Naber, L2 curvature bounds on manifolds with bounded Ricci curvature, Ann. of Math. 193 (2021) 107 MR4199730
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 MR2186552
63 S Keith, X Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. 167 (2008) 575 MR2415381
64 C Ketterer, Stability of metric measure spaces with integral Ricci curvature bounds, J. Funct. Anal. 281 (2021) 109142 MR4271791
65 P Koskela, N Shanmugalingam, Y Zhou, Geometry and analysis of Dirichlet forms, II, J. Funct. Anal. 267 (2014) 2437 MR3250370
66 P Koskela, Y Zhou, Geometry and analysis of Dirichlet forms, Adv. Math. 231 (2012) 2755 MR2970465
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 MR2015170
68 J Lott, C Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009) 903 MR2480619
69 P Mattila, Geometry of sets and measures in Euclidean spaces : fractals and rectifiability, 44, Cambridge Univ. Press (1995) MR1333890
70 A Mondino, A Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019) 1809 MR3945743
71 U Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994) 368 MR1283033
72 P Petersen, G Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997) 1031 MR1487753
73 P Petersen, G Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds, II, Trans. Amer. Math. Soc. 353 (2001) 457 MR1709777
74 J A Ramírez, Short-time asymptotics in Dirichlet spaces, Comm. Pure Appl. Math. 54 (2001) 259 MR1809739
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 MR3936228
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 MR3611331
77 L Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Int. Math. Res. Not. 1992 (1992) 27 MR1150597
78 L Saloff-Coste, Aspects of Sobolev-type inequalities, 289, Cambridge Univ. Press (2002) MR1872526
79 A Sikora, Sharp pointwise estimates on heat kernels, Q. J. Math. 47 (1996) 371 MR1412562
80 B Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982) 447 MR0670130
81 L Simon, Lectures on geometric measure theory, 3, Austral. Nat. Univ. (1983) MR0756417
82 M Simon, Extending four dimensional Ricci flows with bounded scalar curvature, Comm. Anal. Geom. 28 (2020) 1683 MR4184830
83 M Simon, Some integral curvature estimates for the Ricci flow in four dimensions, Comm. Anal. Geom. 28 (2020) 707 MR4124141
84 K T Sturm, Analysis on local Dirichlet spaces, I : Recurrence, conservativeness and Lp–Liouville properties, J. Reine Angew. Math. 456 (1994) 173 MR1301456
85 K T Sturm, Analysis on local Dirichlet spaces, III : The parabolic Harnack inequality, J. Math. Pures Appl. 75 (1996) 273 MR1387522
86 K T Sturm, On the geometry of metric measure spaces, I, Acta Math. 196 (2006) 65 MR2237206
87 K T Sturm, On the geometry of metric measure spaces, II, Acta Math. 196 (2006) 133 MR2237207
88 R Tessera, Volume of spheres in doubling metric measured spaces and in groups of polynomial growth, Bull. Soc. Math. France 135 (2007) 47 MR2430198
89 G Tian, J Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005) 357 MR2138071
90 G Tian, J Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2005) 346 MR2166311
91 G Tian, J Viaclovsky, Volume growth, curvature decay, and critical metrics, Comment. Math. Helv. 83 (2008) 889 MR2442967
92 G Tian, Z Zhang, Regularity of Kähler–Ricci flows on Fano manifolds, Acta Math. 216 (2016) 127 MR3508220
93 N T Varopoulos, Hardy–Littlewood theory for semigroups, J. Funct. Anal. 63 (1985) 240 MR0803094
94 C Villani, Optimal transport: old and new, 338, Springer (2009) MR2459454
95 J Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann. 229 (1977) 163 MR0445335
96 B White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997) 1 MR1465365
97 Q S Zhang, M Zhu, Li–Yau gradient bounds on compact manifolds under nearly optimal curvature conditions, J. Funct. Anal. 275 (2018) 478 MR3802491