  We show existence, uniqueness and positivity for the Green's function of the
operator $(\Delta_g + \alpha)^k$ in a closed Riemannian manifold $(M,g)$, of
dimension $n>2k$, $k\in \mathbb{N}$, $k\geq 1$, with Laplace-Beltrami operator
$\Delta_g = -\operatorname{div}_g(\nabla \cdot)$, and where $\alpha >0$. We are
interested in the case where $\alpha$ is large : We prove pointwise estimates
with explicit dependence on $\alpha$ for the Green's function and its
derivatives. We highlight a region of exponential decay for the Green's
function away from the diagonal, for large $\alpha$.
