Honda–Mondino spectral gap conjecture for the first Poincaré constant

Let $M^n$ be a closed Riemannian manifold with

$$\mathrm{Ric}\ge -(n-1),\quad \mathrm{diam}\le d,\quad \mathrm{vol}\ge v.$$

Here $C_{P,1}$ denotes the Poincaré constant for $1$-forms. Honda–Mondino's spectral gap conjecture. Under these assumptions,

$$C_{P,1}\le C(n,d,v).$$

This conjecture concerns uniform spectral gaps under lower Ricci curvature, diameter, and volume bounds; the supplied text recalls it as an open conjecture and proves related four-dimensional results.