Initiated by the work of Uhlenbeck in late 1970s, we study questions about existence, multiplicity and asymptotic behavior for minimal immersions closed surface some hyperbolic three-manifold, with prescribed conformal structure on second fundamental form immersion. We prove several results these directions. In particular, determine when exactly solution is unique multiple solutions appear. Mor...

We consider surface homeomorphisms f : M ! M which leave invariant a set M and satisfy conditions which generalise the situation where is a hyperbolic invariant set or where f is Anosov or pseudo-Anosov. We deene a C r structure on which gives a notion of smoothness for pseudo-Anosov diieomorphisms with singularities. We prove that the holonomy maps are C 1+ for such systems. We show that there...

in this paper, we prove that every metric line in the poincare ball model of hyperbolic geometry is exactly a classical line of itself. we also proved nonexistence of periodic lines in the poincare ball model of hyperbolic geometry.

We generalize McShane’s identity for the length series of simple closed geodesics on a cusped hyperbolic surface [17] to hyperbolic cone-surfaces (with all cone angles ≤ π), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotient of the one-holed torus by its elliptic involution, and the closed genus tw...