Communications in Mathematical Physics

, Volume 336, Issue 2, pp 933–952 | Cite as

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

  • Jacob Bernstein
  • Thomas Mettler


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.


Closed Curf Geodesic Curvature Fractional Linear Transformation Balance Point Schwarzian Derivative 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA
  2. 2.Department of MathematicsZürichSwitzerland

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