Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

TL;DR

This paper investigates the behavior of the first eigenvalue of the Laplacian under a canonical variation of metrics on Riemannian submersions with totally geodesic fibers, especially when fibers are Einstein, revealing conditions under which certain scale-invariant quantities diverge or converge.

Contribution

It provides bounds for the first eigenvalue in specific geometric settings and analyzes the stability of Yamabe functional critical points under these conditions.

Findings

The scale-invariant eigenvalue quantity diverges to infinity under certain conditions.

Bounds for the first eigenvalue are established when fibers are Einstein and Ricci curvature conditions are met.

Applications include analysis of the twistor fibration of quaternionic Kähler manifolds.

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.