### Citations

3829 |
eds.): Handbook of mathematical functions
- Abramowitz, Stegun
- 1965
(Show Context)
Citation Context ...tion v0 with initial condition v(0) = 2b/H. The solution with v(0) = 2|b/H| is given by v(y) = ±2|a/H|sn(2i|b|y + F(arcsin(κ −1/2 ) |κ) ∣ ∣ κ). By the complex arguments formula for sn, see 16.21.1 in =-=[1]-=-, this simplifies to v(y) = ±2|b/H|dn(2|b|y |κ ′ ). Choosing the sign in (4.10) so that ±|b/H| = b/H, and using the fact that dn is an even function 16.8.3 [1], proves (4.5) and concludes the proof of... |

232 |
Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints
- Pohlmeyer
- 1976
(Show Context)
Citation Context ... the primitivity condition is vacuous. We combine the cases of interest to us by saying that ϕ is primitive harmonic if k = 2 and ϕ is harmonic or k > 2 and ϕ is primitive. 1.3. The basic observation =-=[14, 23, 24]-=- is that if F frames a primitive harmonic map then defining (1.2) αλ = αk + λ −1 α ′ m + λα ′′ m yields a solution of the Maurer–Cartan equations for each non–zero complex number λ ∈ C ×. Thus, we can... |

220 |
A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Func
- Zakharov, Shabat
- 1974
(Show Context)
Citation Context ... the primitivity condition is vacuous. We combine the cases of interest to us by saying that ϕ is primitive harmonic if k = 2 and ϕ is harmonic or k > 2 and ϕ is primitive. 1.3. The basic observation =-=[14, 23, 24]-=- is that if F frames a primitive harmonic map then defining (1.2) αλ = αk + λ −1 α ′ m + λα ′′ m yields a solution of the Maurer–Cartan equations for each non–zero complex number λ ∈ C ×. Thus, we can... |

142 |
Harmonic maps into Lie groups: classical solutions of the chiral model
- Uhlenbeck
- 1989
(Show Context)
Citation Context ...heory of harmonic maps, especially those from a Riemann surface to a Riemannian symmetric space, has been greatly enriched in recent years by the realisation that they constitute an integrable system =-=[4, 9, 10, 13, 22, 23]-=-. Thus these maps admit a spectral deformation (the associated family); algebro-geometric (finite type) solutions and an interpretation in terms of certain holomorphic maps to a loop group. This last ... |

127 |
A duality theorem for Willmore surfaces
- Bryant
- 1984
(Show Context)
Citation Context ...deformation. A pleasant application of the foregoing theory lies in the fact that several types of surface of classical geometric interest are characterised by harmonicity of an appropriate Gauss map =-=[3, 7, 17]-=-. In particular, the classical theory of constant mean curvature surfaces in R 3 amounts to the study of harmonic maps to a 2-sphere with the link between the loop group approach and the classical sur... |

116 | H.Wu, Weierstrass type representations of harmonic maps into symmetric spaces
- Dorfmeister
(Show Context)
Citation Context ...heory of harmonic maps, especially those from a Riemann surface to a Riemannian symmetric space, has been greatly enriched in recent years by the realisation that they constitute an integrable system =-=[4, 9, 10, 13, 22, 23]-=-. Thus these maps admit a spectral deformation (the associated family); algebro-geometric (finite type) solutions and an interpretation in terms of certain holomorphic maps to a loop group. This last ... |

95 |
Harmonic maps from a 2-torus to the 3-sphere
- Hitchin
- 1990
(Show Context)
Citation Context ...heory of harmonic maps, especially those from a Riemann surface to a Riemannian symmetric space, has been greatly enriched in recent years by the realisation that they constitute an integrable system =-=[4, 9, 10, 13, 22, 23]-=-. Thus these maps admit a spectral deformation (the associated family); algebro-geometric (finite type) solutions and an interpretation in terms of certain holomorphic maps to a loop group. This last ... |

73 | Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras
- Burstall, Ferus, et al.
- 1993
(Show Context)
Citation Context |

68 | Bäcklund transformations and loop group actions
- Terng, Uhlenbeck
(Show Context)
Citation Context |

53 | Harmonic maps via Adler–Kostant–Symes theory, Harmonic maps and integrable systems
- Burstall, Pedit
- 1994
(Show Context)
Citation Context ... arise this way. 1. Primitive harmonic maps and loop groups 1.1. We study primitive harmonic maps of a Riemann surface into a k-symmetric space and so begin by recalling the ingredients of that story =-=[5, 6]-=-. Let G be a compact semisimple Lie group. A (regular) k-symmetric G-space [12] is a coset space N = G/K where (G τ )0 ⊂ K ⊂ G τ for some automorphism τ : G → G of finite order k ≥ 2. In particular, a... |

50 |
The tension field of the Gauss map
- Ruh, Vilms
- 1970
(Show Context)
Citation Context ...deformation. A pleasant application of the foregoing theory lies in the fact that several types of surface of classical geometric interest are characterised by harmonicity of an appropriate Gauss map =-=[3, 7, 17]-=-. In particular, the classical theory of constant mean curvature surfaces in R 3 amounts to the study of harmonic maps to a 2-sphere with the link between the loop group approach and the classical sur... |

47 |
Constant mean curvature surfaces and integrable equations
- Bobenko
- 1991
(Show Context)
Citation Context ...su(2) → Euc(3) of the Euclidean group. An extended frame F(z, λ) gives rise to an associated family of parallel pairs f ± λ of conformal constant mean curvature immersions via the Sym–Bobenko formula =-=[2, 20]-=- which, in our formalism, reads (3.3) f ± λ = − 1 2H Φλ(F) · (∓e1). Here e1 ∈ su(2) is the normal to f = f + 1 at z = 0, and H ∈ R∗ is the mean curvature. Since the normal to the surface f − in (3.3) ... |

45 |
Systems of Toda type, inverse spectral problems and representation
- Symes
- 1980
(Show Context)
Citation Context ...very g ∈ ΛG C τ can be uniquely factored g = F b with F ∈ ΛGτ and b ∈ Λ+Gτ. 1.5. Theorem 1.1 underlies a construction of extended frames which is an infinite-dimensional version of a formula of Symes =-=[21]-=- from integrable systems theory. For this, let Λgτ,Λg C τ be the Lie algebras of ΛGτ,ΛG C τ respectively. Thus Λg C τ = Λg C τ = { ξ : S 1 → g C } smooth : ξ(ωλ) = τξ(λ) ; { ξ ∈ Λg C τ : ξ : S1 } → g ... |

32 | Dressing orbits of harmonic maps
- Burstall, Pedit
(Show Context)
Citation Context ... arise this way. 1. Primitive harmonic maps and loop groups 1.1. We study primitive harmonic maps of a Riemann surface into a k-symmetric space and so begin by recalling the ingredients of that story =-=[5, 6]-=-. Let G be a compact semisimple Lie group. A (regular) k-symmetric G-space [12] is a coset space N = G/K where (G τ )0 ⊂ K ⊂ G τ for some automorphism τ : G → G of finite order k ≥ 2. In particular, a... |

28 |
On the classification of constant mean curvature
- Pinkall, Sterling
- 1989
(Show Context)
Citation Context |

16 |
Helicoidal surfaces with constant mean curvature, Tohoku
- Dajczer, Carmo, et al.
(Show Context)
Citation Context ...ko formula. We apply our general theory to this case which means that we study constant mean curvature surfaces with screw motion symmetry. We find a very simple proof of a result of Do Carmo–Dajczer =-=[8]-=- which asserts that these surfaces are precisely those in the associated family of a Delaunay surface (that is, a constant mean curvature surface of revolution). For this, we provide an interpretation... |

8 |
Generalized symmetric spaces, Lect
- Kowalski
- 1980
(Show Context)
Citation Context ...ve harmonic maps of a Riemann surface into a k-symmetric space and so begin by recalling the ingredients of that story [5, 6]. Let G be a compact semisimple Lie group. A (regular) k-symmetric G-space =-=[12]-=- is a coset space N = G/K where (G τ )0 ⊂ K ⊂ G τ for some automorphism τ : G → G of finite order k ≥ 2. In particular, a Riemannian symmetric space of compact type is the same as a 2-symmetric space.... |

6 |
Soliton surfaces and their appplication (soliton geometry from spectral problems
- Sym
- 1985
(Show Context)
Citation Context ...su(2) → Euc(3) of the Euclidean group. An extended frame F(z, λ) gives rise to an associated family of parallel pairs f ± λ of conformal constant mean curvature immersions via the Sym–Bobenko formula =-=[2, 20]-=- which, in our formalism, reads (3.3) f ± λ = − 1 2H Φλ(F) · (∓e1). Here e1 ∈ su(2) is the normal to f = f + 1 at z = 0, and H ∈ R∗ is the mean curvature. Since the normal to the surface f − in (3.3) ... |

4 |
Computer Graphics of Helicoidal Surfaces with Constant Mean Curvature
- Hitt, Roussos
- 1991
(Show Context)
Citation Context ...we will discuss the closing conditions for helicoidal constant mean curvature cylinders. Other methods have been used to study helicoidal cmc surfaces, as in the investigations of Roussos et al., see =-=[11]-=- and [16] and the references therein. Consider translations δ : z ↦→ z + p + iq with p, q ∈ R . If an extended framing F(z, λ) with F(0, λ) = (6.1) M(δ, λ) = δ ∗ F F −1 = F(δ(0), λ) 0 has a well defin... |

2 |
Harmonic maps in unfashionable geometries Manuscripta Math
- Burstall, Hertrich-Jeromin
- 2002
(Show Context)
Citation Context ...deformation. A pleasant application of the foregoing theory lies in the fact that several types of surface of classical geometric interest are characterised by harmonicity of an appropriate Gauss map =-=[3, 7, 17]-=-. In particular, the classical theory of constant mean curvature surfaces in R 3 amounts to the study of harmonic maps to a 2-sphere with the link between the loop group approach and the classical sur... |

2 |
A geometric characterization of helicoidal surfaces of constant mean curvature
- Roussos
- 1988
(Show Context)
Citation Context ...iscuss the closing conditions for helicoidal constant mean curvature cylinders. Other methods have been used to study helicoidal cmc surfaces, as in the investigations of Roussos et al., see [11] and =-=[16]-=- and the references therein. Consider translations δ : z ↦→ z + p + iq with p, q ∈ R . If an extended framing F(z, λ) with F(0, λ) = (6.1) M(δ, λ) = δ ∗ F F −1 = F(δ(0), λ) 0 has a well defined monodr... |

1 |
Constant mean curvature trinoids in three dimensional space forms
- Schmitt, Kilian, et al.
- 2002
(Show Context)
Citation Context ... □ 5. The extended Delaunay frame To solve the period problem for a helicoidal cmc surface, we need to compute the extended framing of an associated family of a Delaunay surface. This was obtained in =-=[19]-=- in an untwisted setting, but since we need most of the ingredients, we provide the reader with all the details. Let ξdz be a potential in normal form with a, b ∈ R ∗ and a ̸= ±b, and Q and v0 as in T... |