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arXiv · 2205.00543

Extremality and rigidity for scalar curvature in dimension four

Renato G. Bettiol·McFeely Jackson Goodman
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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.

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BibTeXRIS

Renato G. Bettiol, McFeely Jackson Goodman. 2022-05-01. Extremality and rigidity for scalar curvature in dimension four. https://doi.org/10.1007/s00029-023-00892-5

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