Role of fundamental solutions for optimal Lipschitz extensions on hyperbolic space

In this paper we consider the problem of finding the relation between absolutely minimizing Lipschitz extension of a given function defined over a subset of the hyperbolic space and the viscosity solution of the PDE that appears from the associated variational problem. Here we have shown that the absolute minimizers can be fully characterized by a comparison principle (comparison with cones) with the fundamental solutions of the associated PDE. We have finally proved that the three properties, (i) comparison with cones, (ii) absolutely minimizing Lipschitz extension and (iii) viscosity solution of associated PDE, are equivalent.