The Ovals Conjecture by Benguria and Loss

Abstract

Reformulation of the problem: For a smooth closed curve $\Gamma \subset \mathbb{R}^2$, Benguria and Loss conjectured that the lowest possible eigenvalue of the operator $\mathcal{H}_\Gamma = -\Delta_\Gamma + \kappa_\Gamma^2$ is $\lambda = 1$. The problem of finding this eigenvalue can be transformed into the problem of finding the infimum of two geometric functionals. Three improving bounds for $\lambda$ are given, up to $\lambda \ge 0.6085$. Also, a proof of the existence of a minimizing $\Gamma$ is provided, showing that $\lambda = 1$ for the round circle and its degeneration to a two-line segment. It is stated that such $\Gamma$ is a planar, convex, analytic curve with strictly positive curvature. A particular example is given at the end, starting from its curvature.