One-Dimensional Projective Structures, Convex Curves and the Ovals of Benguria and Loss

Abstract

Benguria and Loss have conjectured that, amongst all smooth closed curves in $${\mathbb{R}^2}$$ of length 2π, the lowest possible eigenvalue of the operator $${L=-\Delta+\kappa^2}$$ is 1. They observed that this value was achieved on a two-parameter family, $${\mathcal{O}}$$ , of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in $${\mathcal{O}}$$ as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all three problems.