In the following, let $M$ be a compact Riemannian manifold and $\Delta = \nabla \cdot\nabla $ a Laplace-Beltrami operator on $M$.
Let $G_y(x)$ be Green's function for $\Delta$ (a.k.a fundamental solution for Laplace equation) defined by the condition $$ \Delta G_y(x) = \delta_y(x) - V^{-1},$$ where $V$ is the Riemannian volume of $M$.
It is known that singularity of $G$ is of the same form as in $\mathbb{R}^n$, that is (up to constants) $$ G_y(x) = u(x, y) + v(x, y),$$ where $u(x, y)= \begin{cases} {d(x,y)}^{2-n} \text{ for } n > 2\\ \log(d(x,y)) \text{ for } n = 2 \end{cases}$ and $v(x, y)$ is continuous/smooth.
The question is: is anything known about the asymtotics of $v(x, y)$ in the limit $d(x, y) \rightarrow 0 $?
I am aware of the expansion of $G_y$ in terms of eigendecomposition of $\Delta$, as well as number of results on bounds for eigenvalues (Cheeger-Yau, Lichnerowicz, ...). However, none of them seem to provide anything about the eigenfunctions, which is what really needed to answer the question.