Local and Global Parametrizations of Hyperbolic Space and Hyperbolic Manifolds via Heat Kernel

This thesis proposes a simple construction of a system of coordinates in the hyperbolic space of any dimension and on hyperbolic manifolds via heat kernels. It is a global system of coordinates in hyperbolic space. On manifolds the only restriction we need to make is that the range of this system of coordinates is up to some positive distance away of the boundary of the Dirichlet domain of one of the basepoints. Each coordinate function in our construction is obtained by taking the logarithm of a ratio of two heat kernels at a small value of time parameter t.;We also provide estimates on the hyperbolic heat kernel in any dimension. In particular we prove that the heat kernel is a log-concave function of the hyperbolic distance for small values of time parameter t.