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?