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On the topology and the boundary of $N$–dimensional $\mathsf{RCD}(K,N)$ spaces

Vitali Kapovitch and Andrea Mondino

Geometry & Topology 25 (2021) 445–495
Bibliography
1 L Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces, from: "Proceedings of the International Congress of Mathematicians, I" (editors B Sirakov, P N de Souza, M Viana), World Sci. (2018) 301 MR3966731
2 L Ambrosio, N Gigli, A Mondino, T Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ–finite measure, Trans. Amer. Math. Soc. 367 (2015) 4661 MR3335397
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, J W Portegies, D Tewodrose, Embedding of RCD(K,N) spaces in L2 via eigenfunctions, preprint (2018) arXiv:1812.03712
5 L Ambrosio, A Mondino, G Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, 1270, Amer. Math. Soc. (2019) MR4044464
6 G Antonelli, E Brué, D Semola, Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces, Anal. Geom. Metr. Spaces 7 (2019) 158 MR4015195
7 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
8 D Barilari, L Rizzi, Sharp measure contraction property for generalized H–type Carnot groups, Commun. Contemp. Math. 20 (2018) MR3848070
9 J Bertrand, Pincement spectral en courbure de Ricci positive, Comment. Math. Helv. 82 (2007) 323 MR2319931
10 S Bianchini, F Cavalletti, The Monge problem for distance cost in geodesic spaces, Comm. Math. Phys. 318 (2013) 615 MR3027581
11 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
12 F Cavalletti, Monge problem in metric measure spaces with Riemannian curvature-dimension condition, Nonlinear Anal. 99 (2014) 136 MR3160530
13 F Cavalletti, E Milman, The globalization theorem for the curvature dimension condition, preprint (2016) arXiv:1612.07623
14 F Cavalletti, A Mondino, Measure rigidity of Ricci curvature lower bounds, Adv. Math. 286 (2016) 430 MR3415690
15 F Cavalletti, A Mondino, Optimal maps in essentially non-branching spaces, Commun. Contemp. Math. 19 (2017) MR3691502
16 F Cavalletti, A Mondino, New formulas for the Laplacian of distance functions and applications, Anal. PDE 13 (2020) 2091 MR4175820
17 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 MR1484888
18 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000) 13 MR1815410
19 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000) 37 MR1815411
20 J Cheeger, W Jiang, A Naber, Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below, preprint (2018) arXiv:1805.07988
21 J Cheeger, A Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013) 321 MR3010378
22 J Cheeger, A Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. 182 (2015) 1093 MR3418535
23 T H Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996) 193 MR1369415
24 T H Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996) 175 MR1369414
25 T H Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477 MR1454700
26 G De Philippis, N Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. Polytech. Math. 5 (2018) 613 MR3852263
27 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
28 S Donaldson, S Sun, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Differential Geom. 107 (2017) 327 MR3707646
29 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
30 M Erbar, K T Sturm, Rigidity of cones with bounded Ricci curvature, J. Eur. Math. Soc. 23 (2021) 219 MR4186467
31 F Galaz-García, M Kell, A Mondino, G Sosa, On quotients of spaces with Ricci curvature bounded below, J. Funct. Anal. 275 (2018) 1368 MR3820328
32 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
33 N Gigli, On the differential structure of metric measure spaces and applications, 1113, Amer. Math. Soc. (2015) MR3381131
34 N Gigli, A Mondino, G Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. 111 (2015) 1071 MR3477230
35 N Gigli, E Pasqualetto, Behaviour of the reference measure on RCD spaces under charts, preprint (2016) arXiv:1607.05188
36 B X Han, Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds, Adv. Math. 373 (2020) MR4129481
37 S Honda, New differential operator and noncollapsed RCD spaces, Geom. Topol. 24 (2020) 2127 MRMR4173928
38 S Honda, I Mondello, Sphere theorems for RCD and stratified spaces, preprint (2019) arXiv:1907.03482
39 N Juillet, Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Int. Math. Res. Not. 2009 (2009) 2347 MR2520783
40 V Kapovitch, Perelman’s stability theorem, from: "Surveys in differential geometry, XI" (editors J Cheeger, K Grove), International (2007) 103 MR2408265
41 V Kapovitch, C Ketterer, CD meets CAT, J. Reine Angew. Math. 766 (2020) 1 MR4145200
42 V Kapovitch, A Lytchak, A Petrunin, Metric-measure boundary and geodesic flow on Alexandrov spaces, J. Eur. Math. Soc. 23 (2021) 29 MR4186463
43 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
44 C Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. 103 (2015) 1228 MR3333056
45 C Ketterer, Obata’s rigidity theorem for metric measure spaces, Anal. Geom. Metr. Spaces 3 (2015) 278 MR3403434
46 R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, 88, Princeton Univ. Press (1977) MR0645390
47 Y Kitabeppu, A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below, Proc. Amer. Math. Soc. 145 (2017) 3137 MR3637960
48 Y Kitabeppu, A sufficient condition to a regular set being of positive measure on RCD spaces, Potential Anal. 51 (2019) 179 MR3983504
49 Y Kitabeppu, S Lakzian, Characterization of low dimensional RCD(K,N) spaces, Anal. Geom. Metr. Spaces 4 (2016) 187 MR3550295
50 J Lott, C Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009) 903 MR2480619
51 A Lytchak, S Stadler, Ricci curvature in dimension 2, preprint (2018) arXiv:1812.08225
52 A Mondino, A Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019) 1809 MR3945743
53 S i Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007) 805 MR2341840
54 S i Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009) 211 MR2546027
55 G Perelman, Alexandrov’s spaces with curvatures bounded from below, II, preprint (1991)
56 P Petersen, On eigenvalue pinching in positive Ricci curvature, Invent. Math. 138 (1999) 1 MR1714334
57 A Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998) 123 MR1601854
58 A Petrunin, Alexandrov meets Lott–Villani–Sturm, Münster J. Math. 4 (2011) 53 MR2869253
59 F Quinn, Ends of maps, III : Dimensions 4 and 5, J. Differential Geom. 17 (1982) 503 MR679069
60 T Rajala, K T Sturm, Non-branching geodesics and optimal maps in strong CD(K,)–spaces, Calc. Var. Partial Differential Equations 50 (2014) 831 MR3216835
61 L Rifford, Ricci curvatures in Carnot groups, Math. Control Relat. Fields 3 (2013) 467 MR3110060
62 L Rizzi, Measure contraction properties of Carnot groups, Calc. Var. Partial Differential Equations 55 (2016) MR3502622
63 L C Siebenmann, Deformation of homeomorphisms on stratified sets, Comment. Math. Helv. 47 (1972) 123 MR319207
64 M Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below, J. Reine Angew. Math. 662 (2012) 59 MR2876261
65 M Simon, P M Topping, Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, preprint (2017) arXiv:1706.09490
66 K T Sturm, On the geometry of metric measure spaces, I, Acta Math. 196 (2006) 65 MR2237206
67 K T Sturm, On the geometry of metric measure spaces, II, Acta Math. 196 (2006) 133 MR2237207
68 C Villani, Optimal transport: old and new, 338, Springer (2009) MR2459454
69 C Villani, Inégalités isopérimétriques dans les espaces métriques mesurés (d’après F Cavalletti & A Mondino), from: "Séminaire Bourbaki, 2016/2017", Astérisque 407, Soc. Math. France (2019) 213 MR3939278
70 J Wong, An extension procedure for manifolds with boundary, Pacific J. Math. 235 (2008) 173 MR2379775