Extremality and rigidity for scalar curvature in dimension four

TL;DR

This paper proves that certain 4-dimensional manifolds with nonnegative or positive sectional curvature are area-extremal, meaning their scalar curvature cannot be increased without decreasing the area of some tangent 2-plane, using advanced geometric analysis techniques.

Contribution

It establishes the area-extremality of broad classes of 4-manifolds with nonnegative or positive sectional curvature, extending Gromov's concept through novel analytical methods.

Findings

Large classes of 4-manifolds with nonnegative sectional curvature are area-extremal.

Regions of positive sectional curvature in 4-manifolds are locally area-extremal.

Analysis of twisted Dirac operators and Finsler--Thorpe trick are key tools.

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.