Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 017, 29 pages      arXiv:2007.07491
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

Masayuki Aino
RIKEN, Center for Advanced Intelligence Project AIP, 1-4-1 Nihonbashi, Tokyo 103-0027, Japan

Received July 17, 2020, in final form February 07, 2021; Published online February 24, 2021

We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.

Key words: Gromov-Hausdorff distance; Lichnerowicz-Obata estimate; parallel $p$-form.

pdf (515 kb)   tex (31 kb)  


  1. Aino M., Lichnerowicz-Obata estimate, almost parallel $p$-form and almost product manifolds, arXiv:1904.06533.
  2. Ambrosio L., Gigli N., Savaré G., Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014), 289-391, arXiv:1106.2090.
  3. Ambrosio L., Gigli N., Savaré G., Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405-1490, arXiv:1109.0222.
  4. Ambrosio L., Gigli N., Savaré G., Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015), 339-404, arXiv:1209.5786.
  5. Ambrosio L., Honda S., New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, in Measure Theory in Non-Smooth Spaces, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017, 1-51, arXiv:1605.07908.
  6. Ambrosio L., Trevisan D., Well-posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDE 7 (2014), 1179-1234, arXiv:1402.4788.
  7. Ambrosio L., Trevisan D., Lecture notes on the DiPerna-Lions theory in abstract measure spaces, Ann. Fac. Sci. Toulouse Math. 26 (2017), 729-766, arXiv:1505.05292.
  8. Aubry E., Pincement sur le spectre et le volume en courbure de Ricci positive, Ann. Sci. École Norm. Sup. (4) 38 (2005), 387-405, arXiv:math.DG/0505408.
  9. Burago D., Burago Y., Ivanov S., A course in metric geometry, Graduate Studies in Mathematics, Vol. 33, Amer. Math. Soc., Providence, RI, 2001.
  10. Cavalletti F., Milman E., The globalization theorem for the curvature dimension condition, arXiv:1612.07623.
  11. Cheeger J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517.
  12. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406-480.
  13. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), 37-74.
  14. Erbar M., Kuwada K., Sturm K.-T., On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces, Invent. Math. 201 (2015), 993-1071, arXiv:1303.4382.
  15. Gigli N., Nonsmooth differential geometry - an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 (2018), v+161, arXiv:1407.0809.
  16. Gigli N., Pasqualetto E., Lectures on nonsmooth differential geometry, SISSA Springer Series, Vol. 2, Springer, 2020.
  17. Gigli N., Rigoni C., Recognizing the flat torus among ${\rm RCD}^*(0,N)$ spaces via the study of the first cohomology group, Calc. Var. Partial Differential Equations 57 (2018), 104, 39 pages, arXiv:1705.04466.
  18. Grosjean J.-F., A new Lichnerowicz-Obata estimate in the presence of a parallel $p$-form, Manuscripta Math. 107 (2002), 503-520.
  19. Hajłasz P., Koskela P., Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211-1215.
  20. Hajłasz P., Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), x+101 pages.
  21. Honda S., Ricci curvature and almost spherical multi-suspension, Tohoku Math. J. 61 (2009), 499-522.
  22. Honda S., Ricci curvature and $L^p$-convergence, J. Reine Angew. Math. 705 (2015), 85-154, arXiv:1212.2052.
  23. Jiang R., Li H., Zhang H., Heat kernel bounds on metric measure spaces and some applications, Potential Anal. 44 (2016), 601-627, arXiv:1407.5289.
  24. Ketterer C., Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. 103 (2015), 1228-1275, arXiv:1311.1307.
  25. Ketterer C., Obata's rigidity theorem for metric measure spaces, Anal. Geom. Metr. Spaces 3 (2015), 278-295, arXiv:1410.5210.
  26. Petersen P., On eigenvalue pinching in positive Ricci curvature, Invent. Math. 138 (1999), 1-21.
  27. Petersen P., Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, Vol. 171, Springer, Cham, 2016.
  28. Rajala T., Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations 44 (2012), 477-494, arXiv:1107.4842.
  29. Sturm K.-T., Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275-312.
  30. Sturm K.-T., Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. 75 (1996), 273-297.
  31. Sturm K.-T., On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65-131.
  32. Sturm K.-T., On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133-177.

Previous article  Next article  Contents of Volume 17 (2021)