Curvature Functionals, Optimal Metrics, and the Differential Topology of 4-Manifolds

This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called optimal; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor methods, optimal metrics of the second type are constructed on the connected sums kCP2 for k > 5. However, related constructions also show that large classes of simply connected 4-manifolds do not admit any optimal metrics at all. Interestingly, the difference between existence and non-existence turns out to delicately depend on one’s choice of smooth structure; there are smooth 4-manifolds which carry optimal metrics, but which are homeomorphic to infinitely many distinct smooth 4-manifolds on which no optimal metric exists.