Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

Abstract

Given a Riemannian submersion $(M,g) \to (B,j)$ each of whose fibers is connected and totally geodesic, we consider a certain 1-parameter family of Riemannian metrics $(g_{t})_{t > 0}$ on $M$, which is called the canonical variation. Let $\lambda_{1}(g_{t})$ be the first positive eigenvalue of the Laplace--Beltrami operator $\Delta^{M}_{g_{t}}$ and $\mbox{Vol}(M,g_{t})$ the volume of $(M, g_{t})$. In 1982, B\'{e}rard-Bergery and Bourguignon showed that the scale-invariant quantity $\lambda_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $0$ with $t$. In this paper, we show that if each fiber is Einstein and $(M,g)$ satisfies a certain condition about its Ricci curvature, then bounds for $\lambda_{1}(g_{t})$ can be obtained. In particular this implies $\lambda_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $\infty$ with $t$. Moreover, using the bounds, we consider stability of critical points of the Yamabe functional. We will see that our results can be applied to many examples. In particular, we consider the twistor fibration of a quaternionic K\"{a}hler manifold of positive scalar curvature.