Closed 1/2-Elasticae in the 2-Sphere

Abstract

We study critical trajectories in the hyperbolic plane for the 1/2-Bernoulli's bending energy with length constraint. Critical trajectories with periodic curvature are classified into three different types according to the causal character of their momentum. We prove that closed trajectories arise only when the momentum is a time-like vector. Indeed, for suitable values of the Lagrange multiplier encoding the conservation of the length during the variation, we show the existence of countably many closed trajectories with time-like momentum, which depend on a pair of relatively prime natural numbers.

Generalized Elastic Translating Solitons

“…In general, the critical curves with nonconstant curvature for Θ p =1,λ , were geometrically described in [31] and their general shape depicted in several figures (the case p = 1/2 and λ ∈ R has also been recently considered in [32]). Observe that in [31] these curves are analyzed according to the value of n = 1/(p − 1).…”