Oscar J. Garay

Letter: Models of Relativistic Particle with Curvature and Torsion Revisited

Models, describing relativistic particles, where Lagrangian densities depend linearly on both the curvature and the torsion of the trajectories, are revisited in D=3 space forms. The moduli spaces of trajectories are completely and explicitly determined using the Lancret program. The moduli subspaces of closed solitons in the three sphere are also determined.

Closed generalized elastic curves in S2(1)

We study the existence and stability of (closed) curves in S 2 (1) which are critical points of generic curvature energy functionals. Firstly, we compute the first and second variation formulas and obtain first integrals of the Euler–Lagrange equations, then we establish conditions under which critical points close up. We apply the results to analyze two concrete situations: a natural generalization of the classical Euler–Bernoulli elastic functional and a constrained version of the total curvature functional.

A characterisation of Helices and Cornu spirals in real space forms

(A > 0). One sees easily that condition Ax = Ax implies AH = XH,where H is considered as an m-valued function on M. Unlike condition Ax = Ax whichmakes no sense for a submanifold of any Riemannian manifold, condition AH = XHcan be considered in such a more general context as we see in Section 2. Thus, itmakes sense to study the condition AH = XH

Holography and total charge

We solve the variational problem associated with the total charge action on bounded domains in D = 2 background gravitational fields. The solutions of the field equations, stability and solitons are obtained holographically in terms of the massless spinning particles that evolve generating worldlines which play the role of boundaries. Moreover, we construct different background gravitational fields to apply the above mentioned program, thus we describe, in particular, solutions, stable (and unstable) solutions and soliton solutions.

Elastic circles in 2-spheres

By solving the Euler–Lagrange equations, we determine the elastic curves in the two-dimensional sphere which are circular at rest. We characterize the family of closed critical curves by a rational condition and the existence of closed elastic curves for the different possible values of their curvature at rest is proved. In the final part of the paper, we utilize a numerical approach to get a better understanding of the space of closed critical curves. In this manner we analyse their shape, uniqueness and minimizing properties.

CONSTANT MEAN CURVATURE HYPERSURFACES WITH CONSTANT δ-INVARIANT

We completely classify constant mean curvature hypersurfaces (CMC) with constant δ -invariant in the unit 4-sphere S 4 and in the Euclidean 4-space 𝔼 4 .

Closed free hyperelastic curves in the hyperbolic plane and Chen-Willmore rotational hypersurfaces

We show that closed Chen-Willmore rotational hypersurfaces of nonnegative curved real space forms are shaped on closed hyperelastic curves of the hyperbolic plane. Then, we study the variational problem associated to this class of curves, proving that there exist a rationally dependent family of closed solutions. They give rise to the first non-trivial examples of Chen-Willmore hypersurfaces in real space forms.

Relativistic particles with rigidity generating non-standard examples of Willmore–Chen hypersurfaces

We study a natural extension to higher dimensions of the Nambu–Goto– Polyakov action. In particular, those dynamical objects evolving with SO(3) symmetry in four dimensions. We show that this problem is strongly related to that of relativistic particles with rigidity of order three in a hyperbolic plane. The moduli space of solitonic solutions is completely determined in terms of the so-called rotation number. A quantization principle for closed solutions is also obtained and this gives a rational one-parameter family of Willmore–Chen hypersurfaces in the standard conformal structure of dimension four. Moreover, these are the first non-standard examples of this kind of hypersurfaces.

Spectral decomposition of spherical immersions with respect to the Jacobi operator

by J. ARROYO, M. BARROS and O. J. GARA + Y(Received 1 October, 1996; revised 8 January, 1997)Abstract. We study the spectral decomposition with respect to the Jacobi operator, /,of spherical immersions and characterize those with a simple decomposition in terms of theFinite Chen-type submanifolds. As a consequence, we give an application to the inverseproblem for J.1. Introduction. The Jacobi operator J (or the second variation operator) was intro-duced by Simons in [11]. It appears in the study of the second variation formula of the areafunction for a compact minimal submanifold M of a Riemannian manifold M and it is anelliptic operator acting on the normal bundle of M, N{M). A cross-section V of N(M) is aJacobi field if JV = 0, [11]. This definition is a generalization of the Jacobi fields over geo-desies. Spectral properties of this operator have been studied in [7], [9].For another classical elliptic operator, the Laplacian A, the spectral behaviour of anisometric immersion of a Riemannian manifold M in an Euclidean space has been widelystudied: finite type submanifolds (a generalization of minimal submanifolds) were introducedby B.-Y. Chen in the late seventies and can be characterized by a variational minimal prin-ciple. For a recent survey on this subject, see [6].We have found in [2] a relation between both elliptic operators (one can see also [10]).This suggests studying the spectral behaviour with respect to J of the position vector ofspherical submanifolds and relate it to the Chen-type of the submanifold.In this paper, we first define the notion offinite J-type for spherical submanifolds. Then,we analyse those spherical submanifolds with the easiest spectral decomposition with respectto J. Thus, we find that ./-type 1 characterizes minimality in the sphere and therefore isequivalent to the Chen-type 1. The following step is to study /-type 2 spherical immersions.We find that, analogously to the previous case, this family coincides with that of Chen-type 2for hypersurfaces. Finally, we use this to see that there exists a spectral condition for aspherical hypersurface with constant mean and scalar curvatures to be totally geodesic.2. Preliminaries

Euclidean submanifolds with Jacobi mean curvature vector field

We study submanifolds in the Euclidean space whose mean curvature vector field is a Jacobi field. First, we characterize them and produce non-trivial (non-minimal) examples and then, we look for additional conditions which imply minimality.