Ancient Caloric Functions on Pseudohermitian Manifolds

For any Riemannian manifold with polynomial volume growth, Colding and Minicozzi obtained a sharp bound on the dimension of the space of ancient caloric functions with polynomial growth. For any pseudohermitian manifold satisfies doubling volume property and parabolic mean value property, we obtain in this paper a sharp bound on the dimension of the space of ancient pseudohermitian caloric functions with polynomial growth.