Area extremality and Nonnegative Curvature in Dimension 4
Abstract: Following Gromov, a Riemannian manifold is called "area extremal" if any modification of which increases scalar curvature must decrease the area of a 2-plane. Previous work of Llarull and Goette-Semmelmann has established area extremality for certain metrics with nonnegative curvature operator, and Kahler metrics with positive Ricci curvature. We show that in dimension 4 a larger class of nonnegatively curved metrics are area extremal, including on manifolds which do not admit metrics with nonnegative curvature operator or Kahler metrics. Following Lott, we examine area extremality on 4-manifolds with boundary, proving that all positively curved metrics are "locally" area extremal. This is joint work with Renato Bettiol.