On the number of omitted values by a meromorphic function of finite energy and heat diffusions

We give a bound of the number of omitted values by a meromorphic function of finite energy on parabolic manifolds in terms of Ricci curvature and the energy of the functions. An analogy of Nevanlinna's theorems based on heat diffusions is used. We also show that meromorphic functions whose energy satisfies some growth condition on algebraic varieties can omit at most two points as a corollary to our main theorems.