Thurston's sphere packings on 3-dimensional manifolds, I

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity that Thurston's Euclidean sphere packing can not be deformed (except by scaling) while keeping the combinatorial Ricci curvature fixed.Comment: Arguments are simplife

Calderon-Type Uniqueness Theorem for Stochastic Partial Differential Equations

In this Note, we present a Calder\'on-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.Comment: