The Green's function of polyharmonic operators with diverging coefficients: Construction and sharp asymptotics

We show existence, uniqueness and positivity for the Green's function of the operator (g +) ᵏ in a closed Riemannian manifold (M, g), of dimension n>2k, k N, k 1, with Laplace-Beltrami operator g = -divg (), and where >0. We are interested in the case where is large: We prove pointwise estimates with explicit dependence on 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.