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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
Article

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.

Keywords

Closed Curf Geodesic Curvature Fractional Linear Transformation Balance Point Schwarzian Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

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