On Green functions of second-order elliptic operators on Riemannian manifolds: The critical case

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation $Pu=0$ in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive (super)solution, then such a solution is called a ground state of P in M, and P is said to be critical in M.We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and $P^{?}$. Under some further assumptions, we describe the behavior at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace–Beltrami operator on a smooth and complete Riemannian manifold M.