| This thesis examines infinite harmonic functions in the viscosity sense on the Heisenberg group. As an example, the homogeneous norm induced by the infinite harmonic gauge in groups of Heisenberg-type is shown to be infinite harmonic itself. Existence of infinite harmonic functions in the viscosity sense is proved following the scheme of Jensen's proof with variations added by Juutinen, Lindqvist, and Manfredi. Uniqueness of infinite harmonic functions is conjectured with a partial proof as motivation for the conjecture. In the case of smooth functions, uniqueness is shown. The uniqueness conjecture and proof in the smooth case utilize the concept of subelliptic jets. By establishing a natural relationship between Euclidean and subelliptic jets, the technology of viscosity solutions can be used. |
