Curvature tensor of smoothable Alexandrov spaces

Lebedeva, Nina¹ ; Petrunin, Anton²

¹ Mathematics Department, Saint Petersburg State University, Saint Petersburg, Russia, Saint Petersburg Department, V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, Saint Petersburg, Russia
² Department of Mathematics, Pennsylvania State University, University Park, PA, United States

Geometry & topology, Tome 28 (2024) no. 8, pp. 3869-3907.

Résumé

We prove weak convergence of curvature tensors of Riemannian manifolds for converging noncollapsing sequences with a lower bound on sectional curvature.

Détail

DOI : 10.2140/gt.2024.28.3869

Keywords: curvature tensor, Alexandrov space

Bibliographie

[1] A D Aleksandrov, V A Zalgaller, Intrinsic geometry of surfaces, 15, Amer. Math. Soc. (1967) | DOI

[2] S Alexander, V Kapovitch, A Petrunin, An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds, Illinois J. Math. 52 (2008) 1031

[3] S Alexander, V Kapovitch, A Petrunin, Alexandrov geometry: foundations, 236, Amer. Math. Soc. (2024)

[4] E Bruè, A Naber, D Semola, Boundary regularity and stability for spaces with Ricci bounded below, Invent. Math. 228 (2022) 777 | DOI

[5] Y Burago, M Gromov, G Perelman, A D Alexandrov spaces with curvature bounded below, Uspekhi Mat. Nauk 47 (1992) 3

[6] N Gigli, Nonsmooth differential geometry : an approach tailored for spaces with Ricci curvature bounded from below, 1196, Amer. Math. Soc. (2018) | DOI

[7] N Gigli, Riemann curvature tensor on RCD spaces and possible applications, C. R. Math. Acad. Sci. Paris 357 (2019) 613 | DOI

[8] 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 | DOI

[9] B X Han, Ricci tensor on RCD∗(K,N) spaces, J. Geom. Anal. 28 (2018) 1295 | DOI

[10] V Kapovitch, Perelman’s stability theorem, from: "Surveys in differential geometry, XI" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, International (2007) 103 | DOI

[11] H B Lawson Jr., M L Michelsohn, Spin geometry, 38, Princeton Univ. Press (1989)

[12] N Lebedeva, A Nepechiy, Alexandrov regions, in preparation

[13] N Li, A Naber, Quantitative estimates on the singular sets of Alexandrov spaces, Peking Math. J. 3 (2020) 203 | DOI

[14] J Lott, Ricci measure for some singular Riemannian metrics, Math. Ann. 365 (2016) 449 | DOI

[15] A Nepechiy, Toward canonical convex functions in Alexandrov spaces, preprint (2019)

[16] G Perelman, Alexandrov’s spaces with curvatures bounded from below, II, preprint (1991)

[17] G Y Perelman, Elements of Morse theory on Aleksandrov spaces, Algebra i Analiz 5 (1993) 232

[18] G Perelman, DC structure on Alexandrov space, preprint (1995)

[19] G Y Perelman, A M Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993) 242

[20] A Petrunin, Applications of quasigeodesics and gradient curves, from: "Comparison geometry" (editors K Grove, P Petersen), Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press (1997) 203

[21] A Petrunin, Polyhedral approximations of Riemannian manifolds, Turkish J. Math. 27 (2003) 173

[22] A Petrunin, Semiconcave functions in Alexandrov’s geometry, from: "Surveys in differential geometry, XI" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, International (2007) 137 | DOI

[23] A M Petrunin, An upper bound for the curvature integral, Algebra i Analiz 20 (2008) 134

[24] A Petrunin, PIGTIKAL (puzzles in geometry that I know and love), 2, Assoc. Math. Res. (2022) 152

[25] Y G Reshetnyak, Two-dimensional manifolds of bounded curvature, from: "Geometry, IV", Encycl. Math. Sci. 70, Springer (1993) 3 | DOI

[26] L Simon, Lectures on geometric measure theory, 3, Australian National Univ. (1983)

[27] K T Sturm, Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes, J. Funct. Anal. 275 (2018) 793 | DOI