Title: $H$-compactness of elliptic operators on weighted Riemannian Manifolds,
Abstract: In this paper we study the asymptotic behavior of second-order uniformly
elliptic operators on weighted Riemannian manifolds. We appeal to the notion of
\mbox{$H$-convergence} introduced by Murat and Tartar. In our main result we
establish an \mbox{$H$-compactness} result that applies to elliptic operators
with measurable, uniformly elliptic coefficients on weighted Riemannian
manifolds. We further discuss the special case of "locally periodic"
coefficients and study the asymptotic behavior of the Laplace-Beltrami operator
on families of weighted manifolds obtained from a reference manifold by a
conformal (rapidly oscillating) change of metrics.