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## Integrable Systems, Harmonic Maps and the Classical Theory of Surfaces (1994)

Venue: | Aspects Math |

Citations: | 12 - 0 self |

### Citations

234 |
Integrable Hamiltonian systems and interactions through quadratic constraints,
- Pohlmeyer
- 1976
(Show Context)
Citation Context ... ; (A ) 00 := A 00 0 +s\Gamma1 A 00 1 ;s2 Cs: (1.5) For reasons to be explained later,swill be referred to as the spectral parameter . The following fundamental observation was first due to Pohlmeyer =-=[27]-=-: Proposition 1.4 A is an element of the space of admissible connections C if and only if its associated loop A satisfies dA + 1 2 [ A ; A ] = 0 ; 8s2 Cs: (1.6) In order to recover the the geometry de... |

137 |
A soliton on a vortex filament,
- Hasimoto
- 1972
(Show Context)
Citation Context ...f a class of surfaces arising from a soliton equation. In this case, the compatability condition is the nonlinear Schrodinger equation, and the corresponding surface is called a Hasimoto surface (see =-=[17]-=- and also [24]). Bianchi also studied classes of surfaces whose compatability equations are of a more general character than the Weingarten surfaces discussed above. These results appear in the origin... |

126 |
A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co.,
- Eisenhart
- 1909
(Show Context)
Citation Context ...e) that is complete and compact necessarily a round sphere? Wente proved that there exist soap bubbles (Wente tori) with the same topological type as a torus. He had learned from Eisenhart's textbook =-=[12]-=- the relationship between soap bubbles and the sinh-Gordon equation discussed in Section 1. Had he read a few more pages in Eisenhart he would have seen that the classical geometers had already found ... |

113 |
Lezioni di Geometria Differenziale, 3-a ed., Zanichelli,
- Bianchi
- 1924
(Show Context)
Citation Context ...eal with the work of Enneper and Dobriner. These include Darboux and Eisenhart. In 1893 Adam (probably Darboux 's student) also studied Enneper surfaces [3]. The Italian edition of Bianchi's textbook =-=[4]-=- is more inclusive than the German translation. For example, Bianchi's multisoliton like surfaces can be found there (Figure 11 and also [30], [31]). A classification of constant mean curvature tori u... |

100 |
On the classification of constant mean curvature tori
- Pinkall, Sterling
- 1989
(Show Context)
Citation Context ...or example, Bianchi's multisoliton like surfaces can be found there (Figure 11 and also [30], [31]). A classification of constant mean curvature tori using soliton theory is given in Pinkall-Sterling =-=[23]-=-. These surfaces can be generated by theta functions using Bobenko's representation [6]. 140 M. Melko and I. Sterling Integrable systems, harmonic maps and the classical theory of surfaces 141 142 M. ... |

75 |
Counterexample to a conjecture of H
- Wente
- 1986
(Show Context)
Citation Context ...of ad G ( =1 ) determines the direction of the axis for the corresponding surface ' =1 . This axis is apparent in the series of pictures at the end of this article. 3 History Wente, in his 1984 paper =-=[37]-=- (Figure 4), solved the long standing problem of Hopf: Is a constant mean curvature surface in R 3 (soap bubble) that is complete and compact necessarily a round sphere? Wente proved that there exist ... |

74 |
All constant mean curvature tori
- Bobenko
- 1991
(Show Context)
Citation Context ...[30], [31]). A classification of constant mean curvature tori using soliton theory is given in Pinkall-Sterling [23]. These surfaces can be generated by theta functions using Bobenko's representation =-=[6]-=-. 140 M. Melko and I. Sterling Integrable systems, harmonic maps and the classical theory of surfaces 141 142 M. Melko and I. Sterling Integrable systems, harmonic maps and the classical theory of sur... |

67 | Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, Harmonic maps and integrable systems 83–127, Aspects Math. E23
- Bobenko
- 1994
(Show Context)
Citation Context ...lts appear in the original version of his book, but not in the German translation. These "Bianchi surfaces" have also been studied by Antoni Sym et. al. These surfaces are discussed in more =-=detail in [7]. Integrab-=-le systems, harmonic maps and the classical theory of surfaces 135 Examples of such "soliton geometries" also arise in affine surface theory [34]. Geometry Soliton P.D.E. Willmore surfaces !... |

56 |
Sur une nouvelle classe de surfaces,
- Tzitzeica
- 1910
(Show Context)
Citation Context ...hese surfaces are discussed in more detail in [7]. Integrable systems, harmonic maps and the classical theory of surfaces 135 Examples of such "soliton geometries" also arise in affine surfa=-=ce theory [34]-=-. Geometry Soliton P.D.E. Willmore surfaces ! xx + ! yy = 2 e \Gamma2! cos j \Gamma 2 e ! j xx + j yy = 2 e \Gamma2! sin j Hasimoto surfaces ! xx = \Gammai! t + !j!j 2 Bianchi surfaces see Bobenko's c... |

54 | Harmonic maps via Adler-Kostant-Symes theory, Harmonic Maps and Integrable Systems
- Burstall, Pedit
- 1994
(Show Context)
Citation Context ... 2 R This equation is a sufficient condition for insuring that [ ; ] R satisfies the Jacobi identity. A linear endomorphism R that satisfies the Yang-Baxter equation is called an "r-matrix" =-=(see also [39, 9]-=-). Integrable systems, harmonic maps and the classical theory of surfaces 137 We denote the canonical Poisson structure associated to ( ~ G ; [ ; ] R ) by f ; gR . This new bracket has the property th... |

47 |
Constant mean curvature surfaces and integrable equations
- Bobenko
- 1991
(Show Context)
Citation Context ...a1 d\Psi is in normal form, then Ad \Psisis a canonical frame for each surface ' . The formulas for the parametrized surface ' given above are originally due to Sym [33] (for K = \Gamma1) and Bobenko =-=[5] (for-=- K = +1 or H = 1=2). (Sym has derived similar expressions for a variety of other examples.) We remind the reader that proofs for the case that " is the Minkowski metric can be found in [21]. As n... |

36 |
Constant mean curvature tori in terms of elliptic functions
- Abresch
- 1987
(Show Context)
Citation Context ...already found the solutions he would spend years reconstructing via analytical methods [38]. These solutions play a central role in Wente's work. In fact, in 1985, after Wente's construction, Abresch =-=[1]-=- rediscovered the methods of Enneper (and his school) [13] and [14], used over a century earlier. Abresch gave explicit formulas, in terms of elliptic integrals, of Wente tori. The simplifying ansatz ... |

30 |
Soliton surfaces and their applications, in Soliton geometry from spectral problems
- Sym
- 1985
(Show Context)
Citation Context ...e, in both cases, if A = \Psi \Gamma1 d\Psi is in normal form, then Ad \Psisis a canonical frame for each surface ' . The formulas for the parametrized surface ' given above are originally due to Sym =-=[33] (for-=- K = \Gamma1) and Bobenko [5] (for K = +1 or H = 1=2). (Sym has derived similar expressions for a variety of other examples.) We remind the reader that proofs for the case that " is the Minkowski... |

19 |
Existence and classification of constant mean curvature multibubbletons of finite and infinite type
- Sterling, Wente
- 1993
(Show Context)
Citation Context ...per surfaces [3]. The Italian edition of Bianchi's textbook [4] is more inclusive than the German translation. For example, Bianchi's multisoliton like surfaces can be found there (Figure 11 and also =-=[30]-=-, [31]). A classification of constant mean curvature tori using soliton theory is given in Pinkall-Sterling [23]. These surfaces can be generated by theta functions using Bobenko's representation [6].... |

18 | Hamonic maps into symmetric spaces and integrable systems
- Wood
- 1993
(Show Context)
Citation Context ...ly if \Delta/ = ae / ; ae : R 2 ! R : (1.1) The idea of the proof is simply to consider / as a map into R 3 , and then to compute the Euler-Lagrange equation with the constraint k/k ffi = 1, see also =-=[39]-=-. It is natural to try to make use of the symmetric space structure of S 2 in order to put the harmonic map equation (1.1) in some tractable form. For that reason we introduce the following notation c... |

15 |
Application of soliton theory to the construction of pseudospherical surfaces in R
- Melko, Sterling
- 1993
(Show Context)
Citation Context ... constructing the preliminary data that is needed for the development in Section 2. This is done via harmonic maps. A more detailed account for the case that " is the Minkowski metric may be foun=-=d in [21]. In what follo-=-ws, we assume that R 2 has a metric, which we denote by ", and we denote the corresponding Laplacian on (R 2 ; ") by \Delta. We are mainly interested in the case that " is either the st... |

11 |
S1-equivariant minimal tori
- Ferus, Pedit
- 1990
(Show Context)
Citation Context ...faces, see [26].) The corresponding compatabilty conditions for the surface can be expressed in terms of the Toda system given in Table 2. The Willmore torus (Figure 9) constructed by Ferus and Pedit =-=[16]-=- is one of the simplest nontrivial examples constructed via this method. If one idealizes a smoke ring as a curve in R 3 , then the surface swept out by its evolution over time (Figure 10) is another ... |

7 |
Über die Zentralflächen der Enneperschen Flächen konstanten Krümmungsmasses
- Sievert
(Show Context)
Citation Context ...ical survey of surfaces of constant curvature is Reckziegel [28]. The earlier works of Bonnet, Dini and Joachimsthal are discussed. Also discussed are the examples of Kuen [18] (Figure 5) and Sievert =-=[29]-=- (Figure 6) which correspond to simple solitons and are given explicitly via trigonometric functions. One of Kuen's beautiful surfaces graces the cover of Gerd Fischer's book (see [28]). The most deta... |

5 |
Analytisch-geometrische Untersuchungen. III, Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen
- Enneper
(Show Context)
Citation Context ...ucting via analytical methods [38]. These solutions play a central role in Wente's work. In fact, in 1985, after Wente's construction, Abresch [1] rediscovered the methods of Enneper (and his school) =-=[13]-=- and [14], used over a century earlier. Abresch gave explicit formulas, in terms of elliptic integrals, of Wente tori. The simplifying ansatz of Enneper was the geometric condition that one family of ... |

4 |
Zur theorie der flächen konstanter Kruümmung
- LIE
(Show Context)
Citation Context ...mation and it's relationship to Enneper surfaces. Recall that the sine-Gordon equation, in characteristic coordinates, is given by @u@ v ! = sin ! ; where ! is a real-valued function on the plane. In =-=[20]-=-, Lie observed that if ! is a solution to this equation, then ~ ! (u; v) := ! (u;s\Gamma1 v) is also a solution for all nonzero real numbers . By virtue of the fundamental theory of surfaces, there is... |

4 | Computational aspects of soap bubble deformations - Pinkall, Sterling - 1993 |

4 |
Über Enneper’sche Flächen und Bäcklund’sche Transformation. (On Enneper surfaces and Bäcklund transformation
- Steuerwald
- 1936
(Show Context)
Citation Context ...onometric functions. One of Kuen's beautiful surfaces graces the cover of Gerd Fischer's book (see [28]). The most detailed and comprehensive work from the classical perspective is that of Steuerwald =-=[32]-=-. This paper also gives a complete discussion of Backlund's transformation and it's relationship to Enneper surfaces. Recall that the sine-Gordon equation, in characteristic coordinates, is given by @... |

3 |
Old and new periodic solutions of the sinh-Gordon equation, Seminar on new results in non-linear partial differential equations
- Abresch
- 1987
(Show Context)
Citation Context ...s paper in 1984 that some of these examples in fact close up to form compact soap bubbles. As early as 1982 Walter [36] attempted to construct compact examples using Enneper's method. In 1986 Abresch =-=[2]-=- extended Enneper's method. This second paper of Abresch played a pivotal role in the modern development of the theory. Enneper's solutions, in the case K = \Gamma1, correspond to the solutions genera... |

3 |
Über die Flächen mit einem System sphärischer Krümmungslinien, J. reine angew
- Dobriner
(Show Context)
Citation Context ... The analogous constant mean curvature surfaces of revolution were found (in 1841) by Delaunay (Figure 2). Integrable systems, harmonic maps and the classical theory of surfaces 139 In 1883, Dobriner =-=[10]-=-, wrote a long paper essentially repeating Enneper's 1868 work. The editor allowed (in the same issue) a scathing rebuttal by Enneper [15], criticizing Dobriner for being 15 years behind the times! Ne... |

3 |
1886 Die Flachen Constanter Krummung mit einem System Spharischer Krummungslinien dargestellt mit Hilfe von Theta Functionen Zweier Variabeln Acta
- Dobriner
(Show Context)
Citation Context ...ally repeating Enneper's 1868 work. The editor allowed (in the same issue) a scathing rebuttal by Enneper [15], criticizing Dobriner for being 15 years behind the times! Nevertheless in 1886 Dobriner =-=[11]-=-, solved a much harder problem posed by Enneper: Characterize those constant Gauss curvature K = \Gamma1 surfaces with one family of spherical lines of curvature. In this elegant paper, Dobriner class... |

2 |
Untersuchungen uber die Flachen mit planen und spharischen
- Enneper
(Show Context)
Citation Context ...a analytical methods [38]. These solutions play a central role in Wente's work. In fact, in 1985, after Wente's construction, Abresch [1] rediscovered the methods of Enneper (and his school) [13] and =-=[14]-=-, used over a century earlier. Abresch gave explicit formulas, in terms of elliptic integrals, of Wente tori. The simplifying ansatz of Enneper was the geometric condition that one family of curvature... |

2 | Applications of Lie Groups to Differental Equations - Olver |

2 |
Renormalized Energy for Multisolitons and the LangerPerline Invariant
- Sterling, Wente
- 1993
(Show Context)
Citation Context ...rfaces [3]. The Italian edition of Bianchi's textbook [4] is more inclusive than the German translation. For example, Bianchi's multisoliton like surfaces can be found there (Figure 11 and also [30], =-=[31]-=-). A classification of constant mean curvature tori using soliton theory is given in Pinkall-Sterling [23]. These surfaces can be generated by theta functions using Bobenko's representation [6]. 140 M... |

1 |
les surfaces isothermique `a lignes de courbure planes dans un syst`eme ou dans les deux syst`emes
- Adam, Sur
(Show Context)
Citation Context ...of the standard references for surface theory deal with the work of Enneper and Dobriner. These include Darboux and Eisenhart. In 1893 Adam (probably Darboux 's student) also studied Enneper surfaces =-=[3]-=-. The Italian edition of Bianchi's textbook [4] is more inclusive than the German translation. For example, Bianchi's multisoliton like surfaces can be found there (Figure 11 and also [30], [31]). A c... |

1 |
Uber die Enneperschen Flachen mit konstantem positivem Krummungsmass, Dissertation, Universitat Gottingen
- Bockwoldt
- 1878
(Show Context)
Citation Context ... H = 1/2. So there are in some sense three cases: K = \Gamma1, K = +1, and H = 1=2. And Enneper had three students: Lenz, Bockwoldt and Voretzsch (resp!). And each wrote a doctoral dissertation [19], =-=[8]-=- and [35]. The dissertation of Voretzsch is the most impressive. The last six pages are tables of numbers giving the x, y, and z coordinates of a constant mean curvature surface. It is important to no... |

1 |
Uber die Flachen mit einem System spharischer Krummungslinien
- Enneper
(Show Context)
Citation Context ...ps and the classical theory of surfaces 139 In 1883, Dobriner [10], wrote a long paper essentially repeating Enneper's 1868 work. The editor allowed (in the same issue) a scathing rebuttal by Enneper =-=[15]-=-, criticizing Dobriner for being 15 years behind the times! Nevertheless in 1886 Dobriner [11], solved a much harder problem posed by Enneper: Characterize those constant Gauss curvature K = \Gamma1 s... |

1 |
Flachen von constantem negativen Krummungsmass nach
- Kuen
- 1884
(Show Context)
Citation Context ... and 8). An excellent historical survey of surfaces of constant curvature is Reckziegel [28]. The earlier works of Bonnet, Dini and Joachimsthal are discussed. Also discussed are the examples of Kuen =-=[18]-=- (Figure 5) and Sievert [29] (Figure 6) which correspond to simple solitons and are given explicitly via trigonometric functions. One of Kuen's beautiful surfaces graces the cover of Gerd Fischer's bo... |

1 |
Ennepersche Flachen konstanten negativen Krummungsmasses, Dissertation, Universitat Gottingen
- Lenz, Uber
- 1879
(Show Context)
Citation Context ...vature H = 1/2. So there are in some sense three cases: K = \Gamma1, K = +1, and H = 1=2. And Enneper had three students: Lenz, Bockwoldt and Voretzsch (resp!). And each wrote a doctoral dissertation =-=[19]-=-, [8] and [35]. The dissertation of Voretzsch is the most impressive. The last six pages are tables of numbers giving the x, y, and z coordinates of a constant mean curvature surface. It is important ... |

1 |
Computational aspects of smoke ring deformations, in preparation
- Pinkall, Sterling
(Show Context)
Citation Context ...urfaces arising from a soliton equation. In this case, the compatability condition is the nonlinear Schrodinger equation, and the corresponding surface is called a Hasimoto surface (see [17] and also =-=[24]-=-). Bianchi also studied classes of surfaces whose compatability equations are of a more general character than the Weingarten surfaces discussed above. These results appear in the original version of ... |

1 |
Surfaces of Constant Curvature, Mathematical Models (edited by Gerd Fischer
- Reckziegel
- 1986
(Show Context)
Citation Context ...rrespond to the solutions generated by (symmetic) genus three hyperelliptic spectral curves [21],[24] (Figures 7 and 8). An excellent historical survey of surfaces of constant curvature is Reckziegel =-=[28]-=-. The earlier works of Bonnet, Dini and Joachimsthal are discussed. Also discussed are the examples of Kuen [18] (Figure 5) and Sievert [29] (Figure 6) which correspond to simple solitons and are give... |

1 |
Untersuchung einer speziellen Flache constanter mittlerer Krummung bei welcher die eine der beiden Schaaren der Krummungslinien von ebenen Curven gebildet wird, Dissertation, Universitat Gottingen
- Voretzsch
- 1883
(Show Context)
Citation Context .... So there are in some sense three cases: K = \Gamma1, K = +1, and H = 1=2. And Enneper had three students: Lenz, Bockwoldt and Voretzsch (resp!). And each wrote a doctoral dissertation [19], [8] and =-=[35]-=-. The dissertation of Voretzsch is the most impressive. The last six pages are tables of numbers giving the x, y, and z coordinates of a constant mean curvature surface. It is important to note that i... |

1 |
Zum H-Satz von H
- Walter
(Show Context)
Citation Context ...nt mean curvature surface. It is important to note that it was not known until Wente's paper in 1984 that some of these examples in fact close up to form compact soap bubbles. As early as 1982 Walter =-=[36]-=- attempted to construct compact examples using Enneper's method. In 1986 Abresch [2] extended Enneper's method. This second paper of Abresch played a pivotal role in the modern development of the theo... |

1 |
personal conversation
- Wente
(Show Context)
Citation Context ...ssed in Section 1. Had he read a few more pages in Eisenhart he would have seen that the classical geometers had already found the solutions he would spend years reconstructing via analytical methods =-=[38]-=-. These solutions play a central role in Wente's work. In fact, in 1985, after Wente's construction, Abresch [1] rediscovered the methods of Enneper (and his school) [13] and [14], used over a century... |