Volume spectrum of fiber bundles and the widths of Berger spheres

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: $${\omega }_p(E,g_E)\le \left(\underset{b\in B}{\sup }{\mathrm{vol}}_{g_E}({\pi }^{-1}(b))\right){\omega }_p(B,g_B).$$ 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 $S_{\tau }^3$ (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.

27 pages. Comments are welcome. The range in Corollary D has been corrected; minor revisions made