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

SIGMA 17 (2021), 046, 42 pages      arXiv:2010.08207      https://doi.org/10.3842/SIGMA.2021.046
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

On Scalar and Ricci Curvatures

Gerard Besson and Sylvestre Gallot
CNRS-Université Grenoble Alpes, Institut Fourier, CS 40700, 38058 Grenoble cedex 09, France

Received October 19, 2020, in final form April 05, 2021; Published online May 01, 2021

Abstract
The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of ''lower bounds of the Ricci curvature'' on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's inequality.

Key words: scalar curvature; Ricci curvature; Whitehead 3-manifolds; infinite connected sums; Ricci flow; synthetic Ricci curvature; metric spaces; Bishop-Gromov inequality; Gromov-hyperbolic spaces; hyperbolic groups; Busemann spaces; CAT(0)-spaces.

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References

  1. Anderson M.T., Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990), 265-275.
  2. Bessières L., Besson G., Maillot S., Ricci flow on open 3-manifolds and positive scalar curvature, Geom. Topol. 15 (2011), 927-975, arXiv:1001.1458.
  3. Besson G., Courtois G., Gallot S., Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), 731-799.
  4. Besson G., Courtois G., Gallot S., Sambusetti A., Curvature-free Margulis lemma for Gromov-hyperbolic spaces, arXiv:1712.08386.
  5. Besson G., Courtois G., Gallot S., Sambusetti A., Compactness and finiteness theorems for Gromov-hyperbolic spaces, Preprint, 2020.
  6. Bieberbach L., Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann. 70 (1911), 297-336.
  7. Breuillard E., Green B., Tao T., The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115-221, arXiv:1110.5008.
  8. Bridson M.R., Haefliger A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer-Verlag, Berlin, 1999.
  9. Burago Y.D., Zalgaller V.A., Geometric inequalities, Grundlehren der Mathematischen Wissenschaften, Vol. 285, Springer-Verlag, Berlin, 1988.
  10. Buser P., Karcher H., The Bieberbach case in Gromov's almost flat manifold theorem, in Global Differential Geometry and Global Analysis (Berlin, 1979), Lecture Notes in Math., Vol. 838, Springer, Berlin - New York, 1981, 82-93.
  11. Cavalucci N., Sambusetti A., Packing and doubling in metric spaces with curvature bounded above, Preprint, 2020.
  12. Chang S., Weinberger S., Yu G., Taming 3-manifolds using scalar curvature, Geom. Dedicata 148 (2010), 3-14.
  13. Cheeger J., Colding T.H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996), 189-237.
  14. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406-480.
  15. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom. 54 (2000), 13-35.
  16. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), 37-74.
  17. Colding T.H., Minicozzi II W.P., A course in minimal surfaces, Graduate Studies in Mathematics, Vol. 121, Amer. Math. Soc., Providence, RI, 2011.
  18. Coornaert M., Delzant T., Papadopoulos A., Géométrie et théorie des groupes: les groupes hyperboliques de Gromov, Lecture Notes in Math., Vol. 1441, Springer-Verlag, Berlin, 1990.
  19. Courtois G., Lemme de Margulis à courbure de Ricci minorée (d'après Vitali Kapovitch et Burkhard Wilking), Astérisque 367-368 (2015), Exp. No. 1075, 25-56.
  20. Davis M.W., Januszkiewicz T., Hyperbolization of polyhedra, J. Differential Geom. 34 (1991), 347-388.
  21. Edwards Jr. C.H., Open $3$-manifolds which are simply connected at infinity, Proc. Amer. Math. Soc. 14 (1963), 391-395.
  22. Farrell F.T., Jones L.E., Topological rigidity for compact non-positively curved manifolds, in Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Vol. 54, Amer. Math. Soc., Providence, RI, 1993, 229-274.
  23. Ferry S.C., Topological finiteness theorems for manifolds in Gromov-Hausdorff space, Duke Math. J. 74 (1994), 95-106.
  24. Ghys E., de la Harpe P. (Editors), Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988), Progr. Math., Vol. 83, Springer, New York, MA, 1990.
  25. Greenberg M.J., Harper J.R., Algebraic topology: a first course, Mathematics Lecture Note Series, Vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981.
  26. Greene R.E., Petersen V P., Little topology, big volume, Duke Math. J. 67 (1992), 273-290.
  27. Gromov M., Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits), Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-78.
  28. Gromov M., Large Riemannian manifolds, in Curvature and Topology of Riemannian Manifolds (Katata, 1985), Lecture Notes in Math., Vol. 1201, Springer, Berlin, 1986, 108-121.
  29. Gromov M., Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., Vol. 8, Springer, New York, 1987, 75-263.
  30. Gromov M., Sign and geometric meaning of curvature, Rend. Sem. Mat. Fis. Milano 61 (1991), 9-123.
  31. Gromov M., Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007.
  32. Gromov M., A dozen problems, questions and conjectures about positive scalar curvature, in Foundations of Mathematics and Physics one Century after Hilbert, Springer, Cham, 2018, 135-158, arXiv:1710.05946.
  33. Gromov M., Metric inequalities with scalar curvature, Geom. Funct. Anal. 28 (2018), 645-726, arXiv:1710.04655.
  34. Gromov M., Lawson Jr. H.B., Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. 111 (1980), 209-230.
  35. Gromov M., Lawson Jr. H.B., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83-196.
  36. Grove K., Petersen V P., Wu J.Y., Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), 205-213.
  37. Grove K., Petersen V P., Wu J.Y., Erratum to Geometric finiteness theorems via controlled topology, Invent. Math. 104 (1991), 221-222.
  38. Guth L., Volumes of balls in large Riemannian manifolds, Ann. of Math. 173 (2011), 51-76, arXiv:math.DG/0610212.
  39. Hajłasz P., Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), 1-101.
  40. Hamilton R.S., Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), 255-306.
  41. Heintze E., Mannigfaltigkeiten negativer Krümmung, Bonner Mathematische Schriften, Vol. 350, Universität Bonn, Mathematisches Institut, Bonn, 2002, Nachdruck der habilitationsschrift von 1976 edition.
  42. Kapovitch V., Wilking B., Structure of fundamental groups of manifolds with Ricci curvature bounded below, arXiv:1105.5955.
  43. Kazhdan D.A., Margulis G.A., A proof of Selberg's conjecture, Math. USSR Sb. 4 (1968), 147-152.
  44. Kister J.M., McMillan Jr. D.R., Locally euclidean factors of $E^{4}$ which cannot be imbedded in $E^{3}$, Ann. of Math. 76 (1962), 541-546.
  45. Kreck M., Lück W., Topological rigidity for non-aspherical manifolds, Pure Appl. Math. Q. 5 (2009), 873-914, arXiv:math.GT/0509238.
  46. Li C., The dihedral rigidity conjecture for $n$-prisms, arXiv:1907.03855.
  47. Li C., A polyhedron comparison theorem for 3-manifolds with positive scalar curvature, Invent. Math. 219 (2020), 1-37, arXiv:1710.08067.
  48. Liu G., 3-manifolds with nonnegative Ricci curvature, Invent. Math. 193 (2013), 367-375, arXiv:1108.1888.
  49. Lott J., Villani C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903-991, arXiv:math.DG/0412127.
  50. Maillot S., Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions, Algebr. Geom. Topol. 8 (2008), 1795-1810, arXiv:0802.1438.
  51. Margulis G.A., Discrete groups of motions of manifolds of nonpositive curvature, in Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 2, 1975, 21-34.
  52. McMillan Jr. D.R., Some contractible open $3$-manifolds, Trans. Amer. Math. Soc. 102 (1962), 373-382.
  53. Meeks III W.H., Yau S.T., Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. 112 (1980), 441-484.
  54. Perelman G., The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
  55. Perelman G., Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109.
  56. Perelman G., Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245.
  57. Rolfsen D., Characterizing the 3-cell by its metric, Fund. Math. 68 (1970), 215-223.
  58. Rosenberg J., Manifolds of positive scalar curvature: a progress report, in Surveys in Differential Geometry, Vol. XI, Surv. Differ. Geom., Vol. 11, Int. Press, Somerville, MA, 2007, 259-294.
  59. Schoen R., Yau S.T., Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979), 127-142.
  60. Schoen R., Yau S.T., Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, in Seminar on Differential Geometry, Ann. of Math. Stud., Vol. 102, Princeton University Press, Princeton, N.J., 1982, 209-228.
  61. Scott P., Tucker T., Some examples of exotic noncompact $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 40 (1989), 481-499.
  62. Serre J.-P., Trees, Springer-Verlag, Berlin - New York, 1980.
  63. Siebenmann L.C., Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-294.
  64. Souto J., Stover M., A Cantor set with hyperbolic complement, Conform. Geom. Dyn. 17 (2013), 58-67, arXiv:1205.4668.
  65. Sturm K.-T., On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133-177.
  66. Thurston W.P., Three-dimensional geometry and topology, Vol. 1, Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1997.
  67. Veronelli G., Scalar curvature via local extent, Anal. Geom. Metr. Spaces 6 (2018), 146-164, arXiv:1710.07178.
  68. Wang J., Contractible $3$-manifold and positive scalar curvature (I), arXiv:1901.04605.
  69. Wang J., Contractible $3$-manifold and positive scalar curvature (II), arXiv:1906.04128.
  70. Wang J., Geometry of $3$-manifolds with uniformly positive scalar curvature, Preprint University of Augsburg, 2020.
  71. Whitehead J.H.C., Certain theorems about three-dimensional manifolds. I, Quart. J. Math. Oxford 5 (1934), 308-320.
  72. Whitehead J.H.C., Tree-dimensional manifolds (corrigendum), Quart. J. Math. Oxford 6 (1935), 80.
  73. Whitehead J.H.C., A certain open manifold whose group is unity, Quart. J. Math. Oxford 6 (1935), 268-279.
  74. Zassenhaus H., Beweis eines satzes über diskrete gruppen, Abh. Math. Sem. Univ. Hamburg 12 (1937), 289-312.

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