Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers
Kazumasa Narita
TL;DR
The paper studies how the first Laplacian eigenvalue behaves under the canonical variation of a Riemannian submersion with totally geodesic fibers, deriving explicit two-sided bounds for $\lambda_{1}(g_t)$ under Ricci curvature conditions and showing that the scale-invariant quantity $\Lambda_{1}(M,t)$ can grow like $O(t^{2(n-p)/n})$ as $t\to\infty$. The core analysis combines a key lemma based on Bochner and Hessian arguments with a decomposition of the horizontal/vertical Laplacians, yielding both a sharp left bound and a right bound via pullbacks from the base. These eigenvalue estimates feed into a quantitative stability criterion for the Yamabe problem under canonical variation: for $\Gamma = \frac{n^{2}+1}{n+1}(\widetilde{c}-c) + pc$, stability holds for all $t\ge \max\{1,\sqrt{\Gamma}/|A|\}$. The framework is illustrated through multiple geometric examples, notably the twistor fibration of quaternionic Kähler manifolds with positive scalar curvature, tying the results to prior work on Sasaki-Einstein and related fibrations and providing new stable instances. Overall, the work links Ricci-geometry of the total space and the A-tensor to spectral and Yamabe properties under canonical variations, broadening the scope of stability phenomena in geometric analysis.
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 $λ_{1}(g_{t})$ be the first positive eigenvalue of the Laplace--Beltrami operator $Δ^{M}_{g_{t}}$ and $\mbox{Vol}(M,g_{t})$ the volume of $(M, g_{t})$. In 1982, Bérard-Bergery and Bourguignon showed that the scale-invariant quantity $λ_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ 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 $λ_{1}(g_{t})$ can be obtained. In particular this implies $λ_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ 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 Kähler manifold of positive scalar curvature.