Abstract
We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below. In so far as geometry is concerned these inequalities appear as generalisations of the classical bounds on the distances between conjugates points in surfaces with positive sectional curvatures. The techniques of our proofs is based on the Schoen–Yau descent method via minimal hypersurfaces, while the overall logic of our arguments is inspired by and closely related to the torus splitting argument in Novikov’s proof of the topological invariance of the rational Pontryagin classes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Almeida, Sebastiao: Minimal Hypersurfaces of a Positive Scalar Curvature. Math. Z. 190, 73–82 (1985)
Almgren Jr., F.J.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35, 451–547 (1986)
Lars Andersson, Mingliang Cai and Gregory J. Galloway. Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann. Henri Poincaré (1)9 (2008), 1–33
Atiyah, M.F.: Elliptic operators, discrete groups and von Neumann algebras. Astérisque 32–3, 43–72 (1976)
R. Bamler. A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature. Mathematical Research Letters Volume 23 (2016). Number 2, Pages 325 - 337
Boris Botvinnik, Johannes Ebert, Oscar Randal-Williams. Infinite loop spaces and positive scalar curvature, Inventiones mathematicae (3)209 (2017), 749–835
Brunnbauer, M., Hanke, B.: Large and small group homology. J. Topology 3, 463–486 (2010)
Bourguignon, Jean-Pierre, Hijazi, Oussama, Milhorat, Jean-Louis: Andrei Moroianu and Sergiu Moroianu. A Spinorial Approach to Riemannian and Conformal Geometry, EMS Monographs in Mathematics (2015)
S Brendle, F.C. Marques, A. Neves. Deformations of the hemisphere that increase scalar curvature, arXiv:1004.3088[math.DG]
Dranishnikov, A.N.: Steven C. Ferry and Shmuel Weinberger. Large Riemannian manifolds which are flexible. Annals of Mathematics 157, 919–938 (2003)
A. Dranishnikov. Asymptotic topology. Russian Math. Surveys (6)55 (2000), 71–116
Dranishnikov, A.N.: On hypereuclidean manifolds. Geom. Dedicata 117, 215–231 (2006)
Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33, 199–211 (1980)
Gromov, M., Lawson, B.: Spin and Scalar Curvature in the Presence of a Fundamental Group I. Annals of Mathematics 111, 209–230 (1980)
Gromov, M., Lawson, H.B.: The classification. of simply connected. manifolds of positive scalar curvature. Annals of Mathematics 11, 423–434 (1980)
Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Etudes Sci. Publ. Math. 58, 83–196 (1983)
Gromov, M.: Filling Riemannian manifolds. J. Differential Geom. 18(1), 1–147 (1983)
M. Gromov. Partial differential relations. Springer (1986)
M. Gromov. Positive curvature, macroscopic dimension, spectral gaps and higher signatures. In: Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993) ,volume 132 of Progr. Math., Birkhäuser, (1996), pp. 1–213
M. Gromov. Hilbert volume in metric spaces. Part 1. Cent. Eur. J. Math. (2)10(2012), 371–400
M. Gromov, Dirac and Plateau billiards in domains with corners. Central European Journal of Mathematics (8)12 (2014), 1109–1156
M. Gromov. M. Plateau-Stein manifolds, Cent. Eur. J. Math., 12(7), 923–951 (2014)
M. Gromov. 101 Questions, Problems and Conjectures around Scalar Curvature. http://www.ihes.fr/~gromov/PDF/101-problemsOct1-2017.pdf (2017)
S. Goette, U. Semmelmann. Spin\(^c\) Structures and Scalar Curvature Estimates. Annals of Global Analysis and Geometry. (4)20 (2001) pp 301–324
S. Goette and U. Semmelmann. Scalar curvature estimates for compact symmetric spaces. Differential Geom. Appl. (1)16 (2002), 65–78
B. Hanke. Positive scalar curvature, K-area and essentialness, Global Differential Geometry pp 275–302, (2011)
B. Hanke, T. Schick. Enlargeability and index theory. J. Differential Geom. (2)74 (2006), 293–320
Bernhard Hanke, Daniel Pape and Thomas Schick. Codimension two index obstructions to positive scalar curvature. Annales de l'institut Fourier (6)65 (2015), 2681–2710
Hitchin, N.: Harmonic spinors. Advances in Math. 14, 1–55 (1974)
M. Llarull. Sharp estimates and the Dirac operator. Mathematische AnnalenJanuary (1)310 (1998), 55–71
H.B., Jr Lawson and M.-L. Michelsohn. Approximation by positive mean curvature immersions: frizzing. Inventiones mathematicae 77 (1984), 421–426
Lohkamp, J.: Scalar curvature and hammocks. Math. Ann. 313, 385–407 (1999)
J. Lohkamp. The Higher Dimensional Positive Mass Theorem II, arXiv:1612.07505 (2016)
Donovan McFeron, Gábor Székelyhidi. On the positive mass theorem for manifolds with corners, Communications in Mathematical Physics (2)313 (2012)
P. Miao. Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys., (6)6 (2002), 1163–1182
M. Min-Oo. Scalar curvature rigidity of certain symmetric spaces. In Geometry, topology, and dynamics (Montreal, PQ, 1995), volume 15 of CRM Proc. Lecture Notes , pages 127–136
Min-Oo, M.: Scalar curvature rigidity of asymptotically hyperbolic spin manifolds. Math. Ann. 285, 527–539 (1989)
M. Min-Oo. K-Area, mass and asymptotic geometry, http://ms.mcmaster.ca/minoo/mypapers/crm_es.pdf (2002)
N. Smale. Generic regularity of homologically area minimizing hyper surfaces in eight-dimensional mani- folds. Comm. Anal. Geom. (2)1 (1993), 217–228
J. Rosenberg. Manifolds of positive scalar curvature: a progress report. In: Surveys on Differential Geometry, vol. XI: Metric and Comparison Geometry, International Press (2007)
Thomas Schick. A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture. Topology (6)37 (1998)
Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds of non-negative scalar curvature. Ann. of Math. 110, 127–142 (1979)
Schoen, R., Yau, S.T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28, 159–183 (1979)
Richard Schoen, Shing-Tung Yau. Positive Scalar Curvature and Minimal Hypersurface Singularities, arXiv:1704.05490 (2017)
S. Stolz. Manifolds of a positive scalar curvature, in: T. Farrell etal. (eds.), Topology of high dimensional manifolds, ICTP Lect. Notes, vol. 9, 665–706. 1.1, Trieste (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gromov, M. Metric Inequalities with Scalar Curvature. Geom. Funct. Anal. 28, 645–726 (2018). https://doi.org/10.1007/s00039-018-0453-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-018-0453-z