Some examples of critical points for the total mean curvature functional

Abstract

We study the following problem: existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in CP 2 (4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional.

Holography and total charge

“…Moreover, F g is constant on each regular homotopy class of curves in Λ. In studying the variational problem associated to F g , in higher dimensional backgrounds [1], we found that the same happens in any surface with zero Gaussian curvature but, in contrast with this fact, there are no critical points when considering non-zero constant Gaussian curvature surfaces. In fact, we have the following result.…”

“…In order to do that, we first notice that the Gauss-Bonnet formula gives a natural relationship between the total charge functional of a domain, φ(Ω), and the one which measures the total curvature of its boundary, φ(∂Ω), in (M, g). Therefore, we define Λ to be the space of closed curves in M (one can consider immersed curves in a more general context, as we did in [1]) and define…”

Epicycloids generating Hamiltonian minimal surfaces in the complex quadric

“…This variational problem was considered in [11], where it was shown that critical points are just plane curves. However, C only acts on curves in the two sphere (compare with [12]).…”

Letter: Models of Relativistic Particle with Curvature and Torsion Revisited

“…That is, this associated with the action measuring the total curvature of trajectories which does not provide any consistent dynamics, (see [2] for more details). The former one corresponds again with the action giving the total curvature.…”