The behavior of the spectral gap under growing drift

Abstract

We analyze the behavior of the spectral gap of the Laplace- Beltrami operator on a compact Riemannian manifold when a divergence-free drift vector field is added. We increase the drift by multiplication with a large constant c c and ask the question how the spectral gap behaves as c c goes to infinity. It turns out that the spectral gap stays bounded if and only if the drift-vector field has eigenfunctions in H 1 H^1 . In that case the spectral gaps converge and we determine the limit.