One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the usual Riemannian structure).
At the very beginning of "Green's functions, harmonic functions, and volume comparison" by P. Li and L.-F. Tam (J. Differential Geom. 41(2): 277-318 (1995)) one can read the following:
"A sharp necessary condition was later proved by Varopoulos in [4], which states that if a complete manifold $M$ has a positive Green's function, then $$\int _1 ^\infty \frac t {V_p (t)} \ \mathrm d t < \infty \ ,$$ where $V_p (t)$ is the volume of the geodesic ball of radius $t$ with center at $p$."
The reference [4], which I do not have access to, is: N. Varopoulos, "Potential theory and diffusion on Riemannian manifolds", Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Vols. I, II, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, 821-837.
And this is my first problem, because in $\mathbb R$ it is obvious that $V_p (t) = 2t$ and the above integral is $\infty$:
were there any constraints on $\dim M$ in Varopoulos' article, that Li and Tam have forgotten to mention?
- Furthermore, it is a deep geometric fact that, on many Riemannian manifolds, the (minimal, positive) Green function of the Laplace-Beltrami operator is the integral in time of the heat kernel, i.e.
$$G(x,y) = \int _0 ^\infty h(t,x,y) \ \mathrm d t \ ,$$
where $h$ is the heat kernel on $M$.
This, though, does not happen on $\mathbb R$, because $$\int _0 ^\infty \frac 1 {\sqrt{4 \pi t}} \mathrm e ^{- \frac {|x-y|^2} {4t}} \ \mathrm d t = \infty \ .$$
Can one recognize those Riemannian manifolds that admit a positive Green function which is not equal to the integral in time of the heat kernel?
