Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(πLk) in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds for families of relative dimension one. As a corollary, we show that when X is smooth, the line bundle η associated to X → B, which was introduced in Phong-Sturm [12], coincides with the CM bundle defined by Paul-Tian [10, 11]. In a second and third corollaries, we establish asymptotics for the K-energy along Bergman rays generalizing the formulas obtained in [11].

In a previous paper we have constructed an invariant of fourdimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four- manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a discrete Yamabe flow with surgery.

Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the C1 topology.

In this paper we construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of S2 × S3. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A K¨ahler–Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi–Boyer–Galicki method.