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

Volume 30
Issue 7, 2431–2778
Issue 6, 2043–2429
Issue 5, 1611–2042
Issue 4, 1203–1610
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