The behavior of the spectral gap under growing drift

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 and ask the question how the spectral gap behaves as 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 . In that case the spectral gaps converge and we determine the limit.

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