Extremality and rigidity for scalar curvature in dimension four
arXiv:2205.00543 · doi:10.1007/s00029-023-00892-5
Abstract
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.
LaTeX2e, 24 pages, final (revised) version. To appear in Selecta Math