1 Introduction

In [1], Benguria and Loss conjectured that for any, \(\sigma\), a smooth closed curve in \(\mathbb R^2\) of length \(2\pi\), the lowest eigenvalue, \(\lambda_\sigma\), of the operator \(L_\sigma=-\Delta_\sigma+\kappa^2_\sigma\) satisfied \(\lambda_\sigma\geq 1\). That is, they conjectured that for all such \(\sigma\) and all functions \(f\in H^1(\sigma)\), \[\tag{1.1} \int_{\sigma} |\nabla_\sigma f|^2 +\kappa^2_\sigma f^2 \; \mathrm{d}s \geq \int_{\sigma} f^2 \; \mathrm{d}s,\] where \(\nabla_\sigma f\) is the intrinsic gradient of \(f\), \(\kappa_\sigma\) is the geodesic curvature and \(\mathrm{d}s\) is the length element. This conjecture was motivated by their observation that it was equivalent to a certain one-dimensional Lieb-Thirring inequality with conjectured sharp constant. They further observed that the above inequality is saturated on a two-parameter family of strictly convex curves which contains the round circle and degenerates into a multiplicity-two line segment. The curves in this family look like ovals and so we call them the ovals of Benguria and Loss and denote the family by \(\mathcal{O}\). Finally, they showed that for closed curves \(\lambda_\sigma\geq \frac{1}{2}\).

Further work on the conjecture was carried out by Burchard and Thomas in [3]. They showed that \(\lambda_\sigma\) is strictly minimized in a certain neighborhood of \(\mathcal{O}\) in the space of closed curves – verifying the conjecture in this neighborhood. More globally, Linde [5] improved the lower bound to \(\lambda_\sigma\geq 0.608\) when \(\sigma\) is a planar convex curves. In addition, he showed that \(\lambda_\sigma\geq 1\) when \(\sigma\) satisfied a certain symmetry condition. Recently, Denzler [4] has shown that if the conjecture is false, then the infimum of \(\lambda_\sigma\) over the space of closed curves is achieved by a closed strictly convex planar curve. Coupled with Linde’s work, this implies that for any closed curve \(\lambda_\sigma\geq 0.608\). In a different direction, the first author and Breiner in [2] connected the conjecture to a certain convexity property for the length of curves in a minimal annulus.

In the present article, we consider the family \(\mathcal{O}\) and observe that the curves in this class are the unique minimizers of two natural geometric functionals. To motivate these functionals, we first introduce an energy functional modeled on (1.1). Specifically, for a smooth curve, \(\sigma\), of length \(L(\sigma)\) and function, \(f\in C^\infty(\sigma)\), set \[\tag{1.2} \mathcal{E}_S[\sigma,f]= \int_{\sigma} |\nabla_\sigma f|^2 +\kappa^2_\sigma f^2 -\frac{(2\pi)^2}{L(\sigma)^2} f^2\; \mathrm{d}s.\] Clearly, the conjecture of Benguria and Loss is equivalent to the non-negativity of this functional. For any strictly convex smooth curve, \(\sigma\), set \[\tag{1.3} \mathcal{E}_G[\sigma]=\int_{\sigma} \frac{|\nabla_\sigma \kappa_\sigma|^2}{4 \kappa^3_\sigma} - \frac{(2\pi)^2}{L(\sigma)^2} \frac{1}{ \kappa_\sigma} \; \mathrm{d}s + {2\pi},\] and \[\tag{1.4} \mathcal{E}_G^*[\sigma]=\int_{\sigma} \frac{|\nabla_\sigma \kappa_\sigma|^2}{4 \kappa^2_\sigma}-\kappa^2_\sigma \; \mathrm{d}s + \frac{(2\pi)^2}{L(\sigma)}.\] Notice \(\mathcal{E}_G\) is scale invariant, while \(\mathcal{E}_G^*\) scales inversely with length. We will show that \(\mathcal{E}_G\) and \(\mathcal{E}_G^*\) are dual to each other in a certain sense – justifying the notation.

Our main result is that the functionals (1.3) and (1.4) are always non-negative and are zero only for ovals.

To the best of our knowledge both inequalities are new. Clearly, \[\mathcal{E}_G[\sigma]=\mathcal{E}_S[\sigma, \kappa^{-1/2}_\sigma],\] and so the non-negativity of (1.3) would follow from the non-negativity of (1.2). Hence, Theorem 1.1 provides evidence for the conjecture of Benguria and Loss.

We also discuss the natural symmetry of these functionals. To do so we need appropriate domains for the functionals. To that end, we say a (possibly open) smooth planar curve is degree-one if its unit tangent map is a degree one map from \(\mathbb{S}^1\) to \(\mathbb{S}^1\) – for instance, any closed convex curve. A degree-one curve is strictly convex if the unit tangent map is a diffeomorphism. We show (see §3.3) that there are natural (left and right) actions of \(\mathrm{SL}(2,\mathbb R)\) on \(\mathcal{D}^\infty\), the space of smooth, degree-one curves and on \(\mathcal{D}_+^\infty\), the space of smooth strictly convex degree-one curves, which preserve the functionals.

We observe that \(\mathcal{O}\) is precisely the orbit of the round circle under these actions. Generically, the action does not preserve the condition of being a closed curve. Indeed, the image of the set of closed curves under this action is an open set in the space of curves and so is not well suited for the direct method in the calculus of variations. Arguably, this is the source of the difficulty in answering Benguria and Loss’s conjecture. Indeed, we prove Theorem 1.1 in part by overcoming it.

Acknowledgments

The authors express gratitude to the anonymous referees for their careful reading and many useful suggestions. The first author was partially supported by the NSF Grant DMS-1307953. The second author was supported by Schweizerischer Nationalfonds SNF via the postdoctoral fellowship PA00P2_142053.