We verify a conjecture of Lutwak, Yang, and Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

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We consider 3-dimensional hyperbolic cone-manifolds which are “convex co-compact” in a natural sense, with cone singularities along infinite lines. Such singularities are sometimes used by physicists as models for massive spinless point particles. We prove an infinitesimal rigidity statement when the angles around the singular lines are less than π: any infinitesimal deformation changes either these angles, or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure. These results hold also when the singularities are along a graph, i.e., for “interacting particles”.

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