Volume spectrum of fiber bundles and the widths of Berger spheres
Jingwen Chen, Pedro Gaspar
TL;DR
The work establishes a fundamental link between the volume spectrum of a Riemannian fiber bundle and the spectra of its base and fibers, proving an explicit upper bound $ω_p(E,g_E) ≤ (\sup_{b} vol_{g_E}(π^{-1}(b)))\, ω_p(B,g_B)$ and extending this to the phase transition (Allen–Cahn) spectrum. It leverages the fiber-integration map to transfer sweepouts from base to total space and develops a fiberwise isoperimetric comparison to obtain lower bounds, enabling computations of low widths in concrete geometries. The authors apply these tools to Berger spheres and products of spheres, showing that in certain regimes the low widths are realized by Clifford tori, while equatorial spheres fail to realize the smallest widths when the fibers are sufficiently small. Collectively, the results provide new, robust techniques for bounding and computing min-max widths in fibered geometries, with broad implications for the study of minimal hypersurfaces and phase-transition methods in geometric analysis.
Abstract
We establish that for a fiber bundle $π: E \to B$, 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: $$ω_p(E,g_E) \leq \left( \sup_{b \in B} \operatorname{vol}_{g_E}(π^{-1}(b)) \right) ω_p(B,g_B). $$ 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 $τ$, the equatorial sphere $S^2$ in the Berger sphere $S^3_τ$ (a $S^1-$bundle over $S^2(\frac{1}{2})$ with fiber length $2πτ$) 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.