Delaunay surfaces in S3(ρ)

Abstract

Recently, invariant constant mean curvature (CMC) surfaces in real space forms have been characterized locally by using extremal curves of a Blaschke type energy functional [5]. Here, we use this characterization to offer a new approach to some global results for CMC rotational surfaces in the 3-sphere.

Closed Biconservative Hypersurfaces in Spheres

“…Above result for the case n = 3 was shown in [22]. Moreover, for n = 5, i.e., p = 1/2, it was obtained in [4] rewriting I(d) in terms of elliptic integrals (this method will also work for n = 3 since in this case Q(u) is a polynomial of degree four). For n = 5, the result was also shown in [1,27] using the same approach from Complex Analysis.…”

“…In particular, when p = 1/2 the energy Θ 1/2 for curves in R 2 was studied by W. Blaschke, in [7], obtaining that critical curves are catenaries. Some results about 1/2-elastic curves in S 2 (ρ) can be found in [3,4] and about p-elastic curves in H 2 (ρ) in [2].…”

“…For n = 3, these two curves were shown in [22]. Moreover, in the case n = 5, these curves are the profile curves of the simplest minimal rotationally symmetric tori in S 3 (ρ) (other than the Clifford torus), which are well-known ( [1,4,26,27]). In Figures 1 and 2, using the parameterization (13), we show these two curves together with the sphere S 2 (ρ) for different values of n ≥ 4 to see how they deform as n increases.…”

“…For some particular choices of p, it may be possible to obtain explicitly the curvatures κ d (s) solutions of ( 9). One such a case corresponds to p = 1/2 (see [4], where the curvatures were given in terms of trigonometric functions).…”

Critical tori for mean curvature energies in Killing submersions

“…In the case of S 2 (ρ), existence of closed critical curves for any value of λ was proved in [6]. In fact, those are the only closed curves for d > 0.…”