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, págs. 1-29
• Idioma: inglés
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• 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.