No abstract uploaded!

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

We nd a one-parameter family of coordinates f hgh2R which is a deformation of Penner's simplicial coordinate of the decorated Teichmuller space of an ideally triangulated punctured surface (S; T) of negative Euler characteristic. If h > 0, the decorated Teichmuller space in the h coordinate becomes an explicit convex polytope P(T) independent of h, and if h < 0, the decorated Teichmuller space becomes an explicit bounded convex polytope Ph(T) so that Ph(T)  Ph0 (T) if h < h0. As a consequence, Bowditch-Epstein and Penner's cell decomposition of the decorated Teichmuller space is reproduced.

We consider inverse curvature ows in Hn+1 with star-shaped initial hypersurfaces and prove that the ows exist for all time, and that the leaves converge to innity, become strongly convex exponentially fast and also more and more totally umbilic. Afteran appropriate rescaling the leaves converge in C1 to a sphere.

Contrast to the hyperbolic mean curvature flows studied in [HKL], [LS], and [KW], a new hyperbolic curvature flow is proposed for convex hypersurfaces. This flow is most suited when the Gauss curvature is involved. The equation satisfied by the graph of the hypersurface under this flow gives rise to a new class of fully nonlinear Euclidean invariant hyperbolic equations.