Let $n$ - dimension $\geq 3$.


Consider a compact manifold (M,g). Let $\epsilon\_0$ denote the injectivity radius of $(M,g)$. Let $B\_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \epsilon\_0$.


Consider the Green's function on $B\_\epsilon(0)$ ( i.g. verifies that $\Delta G = \delta\_y$ and $G=0$ on the boundary. G is also positive, smooth and well defined of the diagonal).


Is it possible to get the following upper bound 


$$ G(x,y) \leq C(n) \rho(x,y)^{2-n}. $$


It is known that this estimate holds near the singularity (even for a general compact subdomain of $(M,g)$); see Schoen and Yau for instance.


Is it true for all $x\neq y$, not just near the singularity? 


N.B. In $\mathbb{R}^n$, we know that $G(x,y) \leq \mbox{ Fundamental solution } \leq C r^{2-n}$; it is a consequence of the max principle. In short, I am trying to get such estimates for a geodesic ball on manifolds.


Do we have an explicit formula for the Green's function of a geodesic ball ? Can we derive such bound from it?



Estimates for Green's function
I suggest you look at:
<http://www.jstor.org/stable/2374588>


