A variational characterization of the catenoid

Abstract

In this note, we use a result of Osserman and Schiffer [13] to give a variational characterization of the catenoid. Namely, we show that subsets of the catenoid minimize area within a geometrically natural class of minimal annuli. To the best of our knowledge, this fact has gone unremarked upon in the literature. As an application of the techniques, we give a sharp condition on the lengths of a pair of connected, simple closed curves σ 1 and σ 2 lying in parallel planes that precludes the existence of a connected minimal surface Σ with ∂Σ = σ 1 ∪ σ 2 .

On the Area of Minimal Surfaces in a Slab

“…Now we need to find the catenoidal waist in H a −a with smallest area. This was already proved by Bernstein-Breiner [1], but is also proved here for completeness. This is a straightforward computation.…”

“…Pyo showed that the catenoid in a slab is the only minimal annulus meeting the boundary of the slab in a constant angle [10]. Also it was proved by Bernstein and Breiner that all embedded minimal annuli in a slab have area bigger than or equal to the minimum area of the catenoids in the same slab [1]. That minimum is attained by the catenoidal waist along the boundary of which the rays from the center of the slab are tangent to the waist.…”

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

“…Coupled with Linde's work, this implies that for any closed curve λ σ ≥ 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.…”

Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator

“…Remark Very recently, Bernstein and Breiner [31] have introduced a new twist on this problem (i.e., on the conjectured isoperimetric inequality for ovals on the plane). Bernstein and Breiner have considered the Catenoid…”