We investigate the rigidity of hyperbolic cone metrics on 3manifolds which are isometric gluing of ideal and hyperideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyperideal hyperbolic polyhedral metrics. It is shown that a hyperideal hyperbolic polyhedral metric is determined up to isometry by its curvature and a decorated ideal hyperbolic polyhedral metric is determined up to isometry and change of decorations by its curvature. The main tool used in the proof is the Fenchel dual of the volume function.

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

We study the Seiberg-Witten invariant λSW(X) of smooth spin 4-manifolds X with integral homology of S1×S3 defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Frøyshov invariant h(X) and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology 3-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.

We define a new combinatorial class of triangulations of closed 3manifolds, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.

It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the conclusion still holds if some of the tetrahedra in the complete metric are flat.