Volume spectrum of fiber bundles and the widths of Berger spheres

TL;DR

This paper establishes bounds on the volume spectrum of fiber bundles, computes low widths of Berger spheres, and explores the relationship between minimal surfaces, isoperimetric profiles, and spectral invariants.

Contribution

It provides new upper bounds for volume spectra of Riemannian fiber bundles and computes specific widths of Berger spheres, linking spectral geometry with minimal surface theory.

Findings

Volume spectrum of fiber bundles is bounded by base spectrum and fiber volume.

Low widths of Berger spheres are explicitly computed for small parameters.

The equatorial sphere in Berger spheres attains certain widths but not lower ones.

Abstract

We establish that for a fiber bundle $\pi: E \to B$, which is a Riemannian submersion, the volume spectrum of $E$ is bounded above by the product of the volume spectrum of $B$ and the volume of the largest fiber. Specifically, we prove the following inequality: $$\omega_p(E,g_E) \leq \left( \sup_{b \in B} \operatorname{vol}_{g_E}(\pi^{-1}(b)) \right) \omega_p(B,g_B). $$ Furthermore, we extend this result to the phase transition spectrum. In addition, we also obtain lower bounds for the isoperimetric profile of Riemannian fibrations with totally geodesic, spherical fibers in terms of the isoperimetric profile of the product of the base and a sphere. By exploiting connections between volume spectrum, least area minimal surfaces, and the isoperimetric profile, we employ these bounds to compute the low widths of Berger spheres and product of spheres. Notably, our analysis reveals that for sufficiently small $\tau$, the equatorial sphere $S^2$ in the Berger sphere $S^3_{\tau}$ (a $S^1-$bundle over $S^2(\frac{1}{2})$ with fiber length $2\pi \tau$) attains the Simon-Smith $1,2,3,4$ widths but fails to attain any lower widths, in both the Almgren-Pitts setting and the Allen-Cahn setting.