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

SIGMA 20 (2024), 038, 25 pages      arXiv:2306.02932      https://doi.org/10.3842/SIGMA.2024.038
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Product Inequalities for $\mathbb T^\rtimes$-Stabilized Scalar Curvature

Misha Gromov ab
a) Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185, USA
b) Institut des Hautes Études Scientifiques, 91893 Bures-sur-Yvette, France

Received June 26, 2023, in final form April 26, 2024; Published online May 08, 2024

Abstract
We study metric invariants of Riemannian manifolds $X$ defined via the $\mathbb T^\rtimes$-stabilized scalar curvatures of manifolds $Y$ mapped to $X$ and prove in some cases additivity of these invariants under Riemannian products $X_1\times X_2$.

Key words: scalar curvature; Riemannian manifold.

pdf (662 kb)   tex (36 kb)  

References

  1. Borisov D., Freitas P., The spectrum of geodesic balls on spherically symmetric manifolds, Comm. Anal. Geom. 25 (2017), 507-544, arXiv:1603.02399.
  2. Botvinnik B., Rosenberg J., Positive scalar curvature on manifolds with fibered singularities, J. Reine Angew. Math. 803 (2023), 103-136, arXiv:1808.06007.
  3. Brendle S., Hirsch S., Johne F., A generalization of Geroch's conjecture, Comm. Pure Appl. Math. 77 (2024), 441-456, arXiv:2207.08617.
  4. Brunnbauer M., Hanke B., Large and small group homology, J. Topol. 3 (2010), 463-486, arXiv:0902.0869.
  5. Cecchini S., Hanke B., Schick T., Lipschitz rigidity for scalar curvature, arXiv:2206.11796.
  6. Cecchini S., Zeidler R., Scalar and mean curvature comparison via the Dirac operator, Geom. Topol., to appear, arXiv:2103.06833.
  7. Chavel I., Feldman E.A., The first eigenvalue of the Laplacian on manifolds of non-negative curvature, Compositio Math. 29 (1974), 43-53.
  8. Cheeger J., On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. USA 76 (1979), 2103-2106.
  9. Chodosh O., Mantoulidis C., Schulze F., Generic regularity for minimizing hypersurfaces in dimensions 9 and 10, arXiv:2302.02253.
  10. Fischer-Colbrie D., Schoen R., The structure of complete stable minimal surfaces in $3$-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211.
  11. Goette S., Semmelmann U., ${\rm Spin}^c$ structures and scalar curvature estimates, Ann. Global Anal. Geom. 20 (2001), 301-324, arXiv:math.DG/9905089.
  12. Goette S., Semmelmann U., Scalar curvature estimates for compact symmetric spaces, Differential Geom. Appl. 16 (2002), 65-78, arXiv:math.DG/0010199.
  13. Gromov M., 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), Progr. Math., Vol. 132, Birkhäuser, Boston, MA, 1996, 1-213.
  14. Gromov M., Manifolds: Where do we come from? What are we? Where are we going?, in The Poincaré Conjecture, Clay Math. Proc., Vol. 19, American Mathematical Society, Providence, RI, 2014, 81-144.
  15. Gromov M., Hilbert volume in metric spaces, arXiv:1811.04332.
  16. Gromov M., Mean curvature in the light of scalar curvature, Ann. Inst. Fourier 69 (2019), 3169-3194, arXiv:1812.09731.
  17. Gromov M., Four lectures on scalar curvature, in Perspectives in Scalar Curvature. Vol. 1, World Scientific Publishing, Hackensack, NJ, 2023, 1-514, arXiv:1902.10612.
  18. Gromov M., Scalar curvature, injectivity radius and immersions with small second fundamental, arXiv:2203.14013.
  19. Gromov M., Hanke B., Torsion obstructions to positive scalar curvature, arXiv:2112.04825.
  20. 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.
  21. 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.
  22. Gromov M., Zhu J., Area and Gauss-Bonnet inequalities with scalar curvature, Comment. Math. Helv. 99 (2024), 355-395, arXiv:2112.07245.
  23. Hanke B., Kotschick D., Roe J., Schick T., Coarse topology, enlargeability, and essentialness, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 471-493, arXiv:0707.1999.
  24. Herzlich M., Refined Kato inequalities in Riemannian geometry, in Journées ''Équations aux Dérivées Partielles'' (La Chapelle sur Erdre, 2000), Université de Nantes, Nantes, 2000, 1-11.
  25. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
  26. Kazdan J.L., Warner F.W., Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113-134.
  27. Lawson Jr. H.B., Michelsohn M.-L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  28. Liokumovich Y., Maximo D., Waist inequality for 3-manifolds with positive scalar curvature, in Perspectives in Scalar Curvature. Vol. 2, World Scientific Publishing, Hackensack, NJ, 2023, 799-831, arXiv:2012.12478.
  29. Listing M., Scalar curvature on compact symmetric spaces, arXiv:1007.1832.
  30. Llarull M., Sharp estimates and the Dirac operator, Math. Ann. 310 (1998), 55-71.
  31. Lohkamp J., Minimal smoothings of area minimizing cones, arXiv:1810.03157.
  32. Lohkamp J., Contracting maps and scalar curvature, arXiv:1812.11839.
  33. Marques F.C., Neves A., Rigidity of min-max minimal spheres in three-manifolds, Duke Math. J. 161 (2012), 2725-2752, arXiv:1105.4632.
  34. Min-Oo M., Scalar curvature rigidity of certain symmetric spaces, in Geometry, Topology, and Dynamics (Montreal, PQ, 1995), CRM Proc. Lecture Notes, Vol. 15, American Mathematical Society, Providence, RI, 1998, 127-136.
  35. O'Neill B., The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469.
  36. Qu C.K., Wong R., ''Best possible'' upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc. 351 (1999), 2833-2859.
  37. Richard T., On the 2-systole of stretched enough positive scalar curvature metrics on $\mathbb S^2\times\mathbb S^2$, SIGMA 16 (2020), 136, 7 pages, arXiv:2007.02705.
  38. Schoen R., Yau S.-T., Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with non-negative scalar curvature, Ann. of Math. 110 (1979), 127-142.
  39. Schoen R., Yau S.-T., Positive scalar curvature and minimal hypersurface singularities, in Differential Geometry, Calabi-Yau Theory, and General Relativity. Part 2, Surv. Differ. Geom., Vol. 24, International Press, Boston, MA, 2022, 441-480, arXiv:1704.05490.
  40. Smale N., Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds, Comm. Anal. Geom. 1 (1993), 217-228.
  41. Stolz S., Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), 511-540.
  42. Su G., Wang X., Zhang W., Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds, J. Reine Angew. Math. 790 (2022), 85-113, arXiv:2104.03472.
  43. Wang J., Xie Z., Yu G., A proof of Gromov's cube inequality on scalar curvature, arXiv:2105.12054.
  44. Zeidler R., Width, largeness and index theory, SIGMA 16 (2020), 127, 15 pages, arXiv:2002.13754.
  45. Zeidler R., Band width estimates via the Dirac operator, J. Differential Geom. 122 (2022), 155-183, arXiv:1905.08520.
  46. Zhu J., Rigidity of area-minimizing $2$-spheres in $n$-manifolds with positive scalar curvature, Proc. Amer. Math. Soc. 148 (2020), 3479-3489, arXiv:1903.05785.
  47. Zhu J., Rigidity results for complete manifolds with nonnegative scalar curvature, J. Differential Geom. 125 (2023), 623-644, arXiv:2008.07028.

Previous article  Next article  Contents of Volume 20 (2024)