  In this paper we study the asymptotic behavior of second-order uniformly
elliptic operators on weighted Riemannian manifolds. They naturally emerge when
studying spectral properties of the Laplace-Beltrami operator on families of
manifolds with rapidly oscillating metrics. We appeal to the notion of
H-convergence introduced by Murat and Tartar. In our main result we establish
an 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 spectral behavior of compact submanifolds of $\mathbb R^n$ with
rapidly oscillating geometry.
