Sharp bounds for the Green’s function and the heat kernel

The main purpose of this note is two-fold. The first is to give simple proofs to some recent theorems of Colding-Minicozzi in C-M on the asymptotic behavior of the Green’s function on manifolds with non-negative Ricci curvature and maximal volume growth. The second is to give sharp upper and lower estimates for the heat kernel on this class of manifolds. In fact, after integrating these estimates with respect to the time variable, they yield sharp pointwise bounds for the Green’s function. The interested reader should refer to C-M for a detailed history of this subject. Throughout this paper, we assume that M is an n-dimensional complete noncompact manifold with non-negative Ricci curvature. With the exception of Theorem 2.1 and Corollary 2.3, we will assume that n ≥ 3. Let us denote the distance between x, y ∈M by d(x, y). If Bx(r) denotes the geodesic ball of radius r centered at x, then we denote Vx(r) and Ax(r) to be the volume and the area of Bx(r) and ∂Bx(r), respectively. In all the theorems, with the exception of Theorem 1.2, we will assume that M has maximal volume growth. This means that for some fixed point p ∈M ,