Willmore tori in non-standard 3-spheres

Barros, Manuel

doi:10.1017/s0305004196001466

Cited by 23 publications

(40 citation statements)

References 5 publications

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“…Assume that γ is an extremum of F λ . Then if W is a vector field along γ, that is, an infinitesimal variation of the curve, we have From this it follows that Λ(s) = − 3 2 k 2 + λ 2 for some constant λ . The vector field J(s) = k 2 −λ 2 T + k s N + kτB is constant along the curve.…”

Section: Classical Bernoulli-euler Elasticamentioning

confidence: 99%

“…What follows is a brief survey of the ideas in geometric variational problems, using the machinery of Optimal Control Theory. 3 A different approach to this type of variational problem, using the theory of exterior differential systems, is developed in [5].…”

Section: Hamiltonian Theorymentioning

confidence: 99%

“…If we apply Theorem 4, this gives an infinite family of embedded Willmore surfaces in R 3 . Variations on this theme can be found in, e.g., [2] and [3].…”

Section: The Hyperbolic Planementioning

confidence: 99%

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Lectures on Elastic Curves and Rods

Singer¹

2008

AIP Conference Proceedings

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Abstract. These five lectures constitute a tutorial on the Euler elastica and the Kirchhoff elastic rod. We consider the classical variational problem in Euclidean space and its generalization to Riemannian manifolds. We describe both the Lagrangian and the Hamiltonian formulation of the rod, with the goal of examining the (Liouville-Arnol'd) integrability. We are particularly interested in determining closed (i.e., periodic) solutions.Keywords: elastica, Kirchhoff rod PACS: 02.30.Xx,02.30.Yy CLASSICAL BERNOULLI-EULER ELASTICAThe classical curve known as the elastica is the solution to a variational problem proposed by Daniel Bernoulli to Leonhard Euler in 1744, that of minimizing the bending energy of a thin inextensible wire (See, e.g., [27], §263). The mathematical idealization of this problem is that of minimizing the integral of the squared curvature for curves of a fixed length satisfying given first order boundary data. In this lecture, we will use the classical techniques of the calculus of variations to derive the equations of the elastica.Consider regular curves (curves with nonvanishing velocity vector) in Euclidean space defined on a fixed interval [a 1 , a 2 ]:We will assume (for technical reasons) that the (geodesic) curvature k of γ is nonvanishing. This will allow us to define the Frenet Frame along the curve.The Frenet frame {T, N, B} is orthonormal and satisfies γ = vT dT ds = kN dN ds = −kT + τB dB ds = −τN The elastica minimizes the bending energywith fixed length and boundary conditions. Accordingly, let α 1 and α 2 be points in R 3 and α 1 and α 2 nonzero vectors. We will consider the space of smooth curves

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Section: Classical Bernoulli-euler Elasticamentioning

confidence: 99%

Section: Hamiltonian Theorymentioning

confidence: 99%

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Lectures on Elastic Curves and Rods

Singer¹

2008

AIP Conference Proceedings

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“…Then the following relationship between the mean curvature function α of N γ in (P,h) and the curvature function κ of γ in (M, u −2 h) holds (see [1])…”

Section: The Main Theoremmentioning

confidence: 99%

“…Examples of nontrivial Willmore surfaces in standard spheres are given in [3], [7], [10], [14] and [17]. Examples of Willmore surfaces in nonstandard spheres can be found in [1] and [8], as well as in spaces with a pseudo-Riemannian global warped product structure in [2] and [4]. The first nontrivial examples of Willmore-Chen submanifolds in standard spheres were given in [6], and later in [4] for conformal structures associated with warped product metrics, and consequently on reducible spaces.…”

Section: Introductionmentioning

confidence: 99%

A criterion for reduction of variables in the Willmore-Chen variational problem and its applications

Barros

Ferrández

Lucas

et al. 2000

Trans. Amer. Math. Soc.

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Abstract. We exhibit a criterion for a reduction of variables for WillmoreChen submanifolds in conformal classes associated with generalized KaluzaKlein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.

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