Extremality and rigidity for scalar curvature in dimension four

• Renato G. Bettiol ^[2] ; McFeely Jackson Goodman ^[1]
  1. [1] Colby College

     Colby College

     City of Waterville, Estados Unidos

     
  2. [2] Department of Mathematics, CUNY Lehman College, USA Department of Mathematics, CUNY Graduate Center, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 1, 2024, 29 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-023-00892-5
• Enlaces
  • Texto completo
• Resumen

  • Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4.

• Referencias bibliográficas
  • Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)....
  • Bär, C., Brendle, S., Hanke, B., Wang, Y.: Scalar curvature rigidity of warped product metrics. arXiv:2306.04015
  • Bettiol, R.G., Krishnan, A.M.: Ricci flow does not preserve positive sectional curvature in dimension four. Calc. Var. Part. Differ. Equ....
  • Bettiol, R.G., Kummer, M., Mendes, R.A.E.: Convex algebraic geometry of curvature operators. SIAM J. Appl. Algebra Geom. 5(2), 200–228 (2021)....
  • Bettiol, R.G., Kummer, M., Mendes, R.A.E.: Geography of pinched four-manifolds. Commun. Anal. Geom. (to appear). arXiv:2106.02138
  • Bettiol, R.G., Mendes, R.A.E.: Sectional curvature and Weitzenböck formulae. Indiana Univer. Math. J. 71(3), 1209–1242 (2022)
  • Brendle, S., Marques, F.C., Neves, A.: Deformations of the hemisphere that increase scalar curvature. Invent. Math. 185(1), 175–197 (2011)....
  • Cecchini, S., Zeidler, R.: Scalar and mean curvature comparison via the Dirac operator. Geom. Topol. (to appear). arXiv:2103.06833
  • Cheeger, J.: Some examples of manifolds of nonnegative curvature. J. Differ. Geom. 8, 623–628 (1973). (MR 341334)
  • Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189...
  • Eschenburg, J.-H.: Comparison Theorems in Riemannian Geometry. Lecture Notes (1994)
  • Finsler, P.: Über das vorkommen definiter und semidefiniter formen in scharen quadratischer formen. Comment. Math. Helv. 9(1), 188–192 (1936)....
  • Goette, S., Semmelmann, U.: Spinc structures and scalar curvature estimates. Ann. Glob. Anal. Geom. 20(4), 301–324 (2001). (MR 1876863)
  • Goette, S., Semmelmann, U.: Scalar curvature estimates for compact symmetric spaces. Differ. Geom. Appl. 16(1), 65–78 (2002). (MR 1877585)
  • Gromov, M.: A dozen problems, questions and conjectures about positive scalar curvature. Foundations of Mathematics and Physics One Century...
  • Gromov, M.: Positive curvature, macroscopic dimension, spectral gaps and higher signatures. Functional analysis on the eve of the 21st century,...
  • Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. (2) 152(1), 331–367 (2000). (MR 1792298 (2001i:53047))
  • Grubb, G.: Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems. Commun. Part. Differ. Equ. 17(11–12),...
  • Lawson, H.B., Jr., Michelsohn, M.-L.: Spin Geometry, Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)....
  • Llarull, M.: Sharp estimates and the Dirac operator. Math. Ann. 310(1), 55–71 (1998). (MR 1600027)
  • Lott, J.: Index theory for scalar curvature on manifolds with boundary. Proc. Am. Math. Soc. 149(10), 4451–4459 (2021). (MR 4305995)
  • Miao, P., Tam, L.-F.: Scalar curvature rigidity with a volume constraint. Commun. Anal. Geom. 20(1), 1–30 (2012). (MR 2903099)
  • Min-Oo, M.: Scalar curvature rigidity of certain symmetric spaces. Geometry, topology, and dynamics (Montreal, PQ, 1995), CRM Proc. Lecture...
  • Noronha, M.H.: A splitting theorem for complete manifolds with nonnegative curvature operator. Proc. Am. Math. Soc. 105(4), 979–985 (1989)....
  • Pólik, I., Terlaky, T.: A survey of the S-lemma. SIAM Rev. 49(3), 371–418 (2007). (MR 2353804)
  • Shanahan, P.: The Atiyah-Singer index theorem. An introduction. Lecture Notes in Mathematics, vol. 638. Springer, Berlin (1978). (MR 487910)
  • Thorpe, J.A.: On the curvature tensor of a positively curved 4-manifold. Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical...
  • Wilking, B.: Nonnegatively and positively curved manifolds. Surveys in Differential Geometry, Vol. XI, Surv. Differ. Geom., vol. 11, pp. 25–62....
  • Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411...