Title: 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.