PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 96, Number 2. February 1986 ON THE EXISTENCE OF GREEN'S FUNCTION IN RIEMANNIAN MANIFOLDS JOSÉ L. FERNÁNDEZ Abstract. This note provides a sufficient condition of geometric character for the existence of Green's function in an arbitrary complete Riemannian manifold. 1. The purpose of this note is to present a sufficient condition for an arbitrary complete C00 Riemannian manifold M to possess Green's function. The condition is given in terms of some isoperimetric inequalities which we now describe. Define cpM(t) by (t\ — ■ f/^(9^): ^> smooth relatively compact domain) I in M with volume > t, I where A denotes the ((dimM) — l)-dimensional measure induced by the metric of M. In particular <pM(V(Q)) < A(dQ) for each such Í2. In §2 we will prove: Theorem. /// V(M)dt/<pM(t)2 < oo, then M has a Green's function. Recently, Dodziuk has proved in [D] that if <pM(t) > ct, i.e. if the isoperimetric inequality holds in M, then M has a Green's function (see also [T, p. 438]). In [V2], Varopoulos has shown by extending a classical result of Ahlfors [A], that if we let L(t) = A(dB(x0, t)) for a point x0 g M, where B(x0, t) denotes the ball around x0 of radius t, then fxdt/L(t) < oo if M has a Green's function. To see the relation with the theorem above define cpw by taking the infimum only on balls around x0, i.e. yM(V(B(x0,r)))=A(dB(x0,r)). But then, since (d/dr)V(B(x0, r)) = A(dB(x0, ry), we have that rV(M) dt r°° dV(B(x0,t)) /•*> dt J MO'' <pM(V(B(x0,t)))2~J Lit)' Thus our sufficient condition is close to the Ahlfors-Varopoulos necessary condition. We should remark that, as shown in [VI], the latter is also sufficient if the Ricci curvature is (semi-)positive definite but not in general; and that Milnor [M] (see also [GW]) showed that it is necessary and sufficient for 2-dimensional models. Received by the editors November 19, 1984 and, in revised form, February 26, 1985. 1980 Mathematics Subject Classification. Primary 31C12; Secondary 30F20. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 284 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use