The p-widths of a Riemannian manifold are a sequence of geometric invariants defined by min-max methods. Each ω_p captures the “width” of the manifold at a specific level of topological complexity — how much area is needed to sweep through the space using hypersurfaces with increasingly complex topology. The sequence is non-decreasing and encodes deep information about the manifold's geometry, analogous to how eigenvalues of the Laplacian encode spectral information.
Marx-Kuo (arXiv:2603.17734) computes all p-widths of the hemisphere with the round metric, for every p. This is the first example of any manifold with boundary for which the complete sequence of p-widths is known.
Computing even a single p-width is generally hard. The definition involves an optimization over all families of hypersurfaces with sufficient topological complexity, and the critical points of this optimization are minimal (or stationary) hypersurfaces. For the hemisphere, the symmetry of the round metric constrains these critical objects enough to determine the entire sequence.
The result provides a benchmark — a complete computation against which conjectures about p-width asymptotics, growth rates, and relationships to other spectral quantities can be tested. The Weyl law for p-widths, which relates their asymptotic growth to volume, can now be verified exactly in this case rather than estimated.
The hemisphere is one of the simplest non-trivial manifolds with boundary. That its complete set of variational widths can be determined exactly, when most manifolds resist even partial computation, is a measure of how much geometric structure the round metric provides — and of how much remains unknown for less symmetric spaces.