A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

In this note a proof is given for global existence and uniqueness of minimal Lorentzian surface maps from a cylinder into globally hyperbolic Lorentzian manifolds for given initial values up to the first derivatives.

In this paper we introduce the notion of a “stack” of geodesic spaces. Loosely speaking, this consists of a geodesic space decomposed into a sequence of “sheets” indexed by a set of consecutive integers. A stack is said to be “hyperbolic” if it is Gromov hyperbolic and its sheets are uniformly Gromov hyperbolic. We define a Cannon-Thurston map for such a stack, and show that the boundary of a o...

Time discounting is the phenomenon that a desired result in the future is perceived as less valuable than the same result now. Economic theories can take this psychological fact into account in several ways. In the economic literature the most widely used type of additive time discounting is exponential discounting. In exponential discounting, the fall of valuation depends by a constant factor ...

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

We show that there is a universal constant, k, such that the curve graph associated to any compact orientable surface is k-hyperbolic. An independent proof of this has been found by Clay, Rafi and Schleimer.

The paper continues an earlier work of the author. An extension theorem is proved for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Carathéodory hyperbolic Riemann surface of finite type.

We use an elliptic differential equation of Ţiţeica (or Toda) type to construct a minimal Lagrangian surface in CH from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero sect...

In this paper, we find lower bounds for the volumes of certain hyperbolic Haken 3manifolds. The theory of Jorgensen and Thurston shows that the volumes of hyperbolic 3-manifolds are well-ordered, but no one has been able to find the smallest one. The best known result for closed manifolds is that the smallest closed hyperbolic 3-manifold has volume > 0.16668, proven by Gabai, Meyerhoff, and Thu...