Under positive Ricci curvature on the total space and Einstein fibers with constant less than the total Ricci bound, the first eigenvalue of the canonical variation satisfies an explicit lower bound of the form constant plus t^{-2}, so the scale-invariant product tends to infinity.
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Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers
Under positive Ricci curvature on the total space and Einstein fibers with constant less than the total Ricci bound, the first eigenvalue of the canonical variation satisfies an explicit lower bound of the form constant plus t^{-2}, so the scale-invariant product tends to infinity.