#### Volume 24, issue 4 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
New differential operator and noncollapsed $\mathrm{RCD}$ spaces

### Shouhei Honda

Geometry & Topology 24 (2020) 2127–2148
##### Bibliography
 1 L Ambrosio, M Colombo, S Di Marino, Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope, from: "Variational methods for evolving objects" (editors L Ambrosio, Y Giga, P Rybka, Y Tonegawa), Adv. Stud. Pure Math. 67, Math. Soc. Japan (2015) 1 MR3587446 2 L Ambrosio, N Gigli, G Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014) 289 MR3152751 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, S Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, from: "Measure theory in non-smooth spaces" (editor N Gigli), de Gruyter (2017) 1 MR3701735 5 L Ambrosio, S Honda, J W Portegies, D Tewodrose, Embedding of RCD∗(K,N) spaces in L2 via eigenfunctions, preprint (2018) arXiv:1812.03712 6 L Ambrosio, S Honda, D Tewodrose, Short-time behavior of the heat kernel and Weyl’s law on RCD∗(K,N) spaces, Ann. Global Anal. Geom. 53 (2018) 97 MR3746517 7 L Ambrosio, A Mondino, G Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, 1270, Amer. Math. Soc. (2019) MR4044464 8 R H Bamler, B Kleiner, Uniqueness and stability of Ricci flow through singularities, preprint (2017) arXiv:1709.04122 9 P Bérard, G Besson, S Gallot, Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal. 4 (1994) 373 MR1280119 10 E Brué, D Semola, Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows, Comm. Pure Appl. Math. 73 (2020) 1141 MR4156601 11 F Cavalletti, E Milman, The globalization theorem for the curvature dimension condition, preprint (2016) arXiv:1612.07623 12 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 MR1484888 13 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000) 13 MR1815410 14 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000) 37 MR1815411 15 T H Colding, A Naber, Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. 176 (2012) 1173 MR2950772 16 G De Philippis, N Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018) 613 MR3852263 17 G De Philippis, A Marchese, F Rindler, On a conjecture of Cheeger, from: "Measure theory in non-smooth spaces" (editor N Gigli), de Gruyter (2017) 145 MR3701738 18 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 19 N Garofalo, A Mondino, Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces, Nonlinear Anal. 95 (2014) 721 MR3130557 20 N Gigli, The splitting theorem in non-smooth context, preprint (2013) arXiv:1302.5555 21 N Gigli, On the differential structure of metric measure spaces and applications, 1113, Amer. Math. Soc. (2015) MR3381131 22 N Gigli, Nonsmooth differential geometry : an approach tailored for spaces with Ricci curvature bounded from below, 1196, Amer. Math. Soc. (2018) MR3756920 23 N Gigli, C Mantegazza, A flow tangent to the Ricci flow via heat kernels and mass transport, Adv. Math. 250 (2014) 74 MR3122163 24 N Gigli, A Mondino, 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 MR3377394 25 N Gigli, E Pasqualetto, Behaviour of the reference measure on RCD spaces under charts, preprint (2016) arXiv:1607.05188 26 P Hajłasz, P Koskela, Sobolev met Poincaré, 688, Amer. Math. Soc. (2000) MR1683160 27 B X Han, Ricci tensor on RCD∗(K,N) spaces, J. Geom. Anal. 28 (2018) 1295 MR3790501 28 B X Han, Conformal transformation on metric measure spaces, Potential Anal. 51 (2019) 127 MR3981445 29 B X Han, Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds, preprint (2019) arXiv:1902.00942 30 S Honda, Bakry-Émery conditions on almost smooth metric measure spaces, Anal. Geom. Metr. Spaces 6 (2018) 129 MR3877323 31 S Honda, I Mondello, Sphere theorems for RCD and stratified spaces, preprint (2019) arXiv:1907.03482v2 32 R Jiang, The Li–Yau inequality and heat kernels on metric measure spaces, J. Math. Pures Appl. 104 (2015) 29 MR3350719 33 R Jiang, H Li, H Zhang, Heat kernel bounds on metric measure spaces and some applications, Potential Anal. 44 (2016) 601 MR3489857 34 V Kapovitch, C Ketterer, Weakly noncollapsed RCD spaces with upper curvature bounds, Anal. Geom. Metr. Spaces 7 (2019) 197 MR4034631 35 V Kapovitch, A Mondino, On the topology and the boundary of N–dimensional RCD(K,N) spaces, preprint (2019) arXiv:1907.02614v2 36 M Kell, A Mondino, On the volume measure of non-smooth spaces with Ricci curvature bounded below, Ann. Sc. Norm. Super. Pisa Cl. Sci. 18 (2018) 593 MR3801291 37 Y Kitabeppu, A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below, Proc. Amer. Math. Soc. 145 (2017) 3137 MR3637960 38 B Kleiner, J Lott, Singular Ricci flows, I, Acta Math. 219 (2017) 65 MR3765659 39 B Kleiner, J Lott, Singular Ricci flows, II, from: "Geometric analysis" (editors J Chen, P Lu, Z Lu, Z Zhang), Progr. Math. 333, Birkhäuser (2020) 137 40 J Lott, C Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009) 903 MR2480619 41 A Mondino, A Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019) 1809 MR3945743 42 T Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations 44 (2012) 477 MR2915330 43 M K von Renesse, On local Poincaré via transportation, Math. Z. 259 (2008) 21 MR2375612 44 K T Sturm, Analysis on local Dirichlet spaces, II : Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995) 275 MR1355744 45 K T Sturm, Analysis on local Dirichlet spaces, III : The parabolic Harnack inequality, J. Math. Pures Appl. 75 (1996) 273 MR1387522 46 K T Sturm, On the geometry of metric measure spaces, I, Acta Math. 196 (2006) 65 MR2237206 47 K T Sturm, On the geometry of metric measure spaces, II, Acta Math. 196 (2006) 133 MR2237207 48 C Villani, Optimal transport: old and new, 338, Springer (2009) MR2459454 49 N Weaver, Lipschitz algebras and derivations, II : Exterior differentiation, J. Funct. Anal. 178 (2000) 64 MR1800791