- 6635 South Hall
Abstract: The L2 norm of the Riemannian curvature tensor is a natural energy to associate to a Riemannian manifold, especially in dimension 4. A natural path for understanding the structure of this functional and its minimizers is via its gradient flow, the "L2 flow." This is a quasi-linear fourth order parabolic equation for a Riemannian metric, which one might hope shares behavior in common with the Yang-Mills flow. We verify this idea by exhibiting structural results for finite time singularities of this flow resembling results on Yang-Mills flow. We also exhibit a new short-time existence statement for the flow exhibiting a lower bound for the existence time purely in terms of a measure of the volume growth of the initial data. As corollaries we establish new compactness and diffeomorphism finiteness theorems for four-manifolds generalizing known results to ones with have effectively minimal hypotheses/dependencies. These results all rely on a new technique for controlling the growth of distances along a geometric flow, which is especially well-suited to the L2 flow.