A 4-manifold is said to have positive biorthogonal curvature if the average of sectional curvatures of any pair of orthogonal planes is positive. In this talk, I will describe a construction of metrics with positive biorthogonal curvature on the product of spheres, and then combine it with recent surgery stability results of Hoelzel to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a metric with positive biorthogonal curvature.

We consider the Dirichlet problem for curve shortening flow on surfaces of constant curvature and show long-time existence of the flow when the initial curve is embedded in a convex region. Furthermore, the limit curve of the flow is a geodesic. The proof relies on an adaptation of Huisken's distance comparison estimate for planar curves, a maximum principle of Angenent, and a blow-up analysis of singularities.