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The affine Toda equations and minimal surfaces
 In Harmonic Maps and Integrable Systems
, 1993
"... this article we consider geometrical interpretations of the twodimensional affine Toda equations for a compact simple Lie group G. These equations originated from the work of Toda [33],[34] over 25 years ago on vibrations of lattices, and they have received considerable attention from both pure and ..."
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this article we consider geometrical interpretations of the twodimensional affine Toda equations for a compact simple Lie group G. These equations originated from the work of Toda [33],[34] over 25 years ago on vibrations of lattices, and they have received considerable attention from both pure and applied mathematicians particularly over the last 15 years. (For the original context of the ideas the reader is referred to [35], [27] and [1] and for a survey of recent work to [29] and [21]).
Discretizing constant curvature surfaces via loop group factorization: the discrete sine and sinhGordon equations
 J. Geom. Phys
, 1995
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Coverings and Integrability of the GaussMainardiCodazzi Equations
"... . Using covering theory approach (zerocurvature representations with the gauge group SL 2 ), we insert the spectral parameter into the Gauss MainardiCodazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained. Introduction When immersed in t ..."
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. Using covering theory approach (zerocurvature representations with the gauge group SL 2 ), we insert the spectral parameter into the Gauss MainardiCodazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained. Introduction When immersed in the Euclidean space E 3 , surfaces with metric g = g 11 dx 2 + 2g 12 dx dy + g 22 dy 2 and the second fundamental form b = b 11 dx 2 + 2b 12 dx dy + b 22 dy 2 satisfy the GaussMainardiCodazzi equations (GMC) R k ijl = b ij b k l \Gamma b il b k j and b ij;k = b ik;j , where R k ijl are components of the curvature tensor. By imposing an algebraic constraint of the form L(g ij ; b ij ; x; y) = 0 we obtain a special instance of what is called reduced GMC equations. Many reduced GMC systems have been found integrable in the sense of soliton theory, e.g., in works [1, 2, 3, 4, 5, 10, 13], thus leading to integrable classes of surfaces. An example is provided by the socalled li...
The Toda Equations and the Geometry of Surfaces
"... The Toda equations are a particularly interesting example of a completely integrable Hamiltonian system and were initially studied in a dynamical context. Two particular forms of Toda equations, the "open" and "affine" equations, can be formulated for any simple Lie algebra and s ..."
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The Toda equations are a particularly interesting example of a completely integrable Hamiltonian system and were initially studied in a dynamical context. Two particular forms of Toda equations, the "open" and "affine" equations, can be formulated for any simple Lie algebra and solutions to both forms arise naturally in a geometrical context from special types of harmonic maps into certain homogeneous spaces associated to the Lie algebra. Generally speaking, the open case is rather easier to deal with since the solutions correspond to holomorphic curves, but the affine case is in some ways more interesting since it has certain special features not enjoyed by the open case. All these aspects are discussed briefly below. The Toda equations are a remarkable phenomenon with many interesting facets. Since their discovery in the 1960s they have engendered an extensive and varied literature, much of which involves a detailed understanding of a number of different areas of mathematics. As a fi...
HARMONIC TORI AND THEIR SPECTRAL DATA.
, 2004
"... One of the earliest applications of modern integrable systems theory (or “soliton theory”) to differential geometry was the solution of the problem of finding all constant mean curvature (CMC) tori in R 3 (and therefore, by taking the Gauss map, finding all nonconformal harmonic maps from a torus t ..."
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One of the earliest applications of modern integrable systems theory (or “soliton theory”) to differential geometry was the solution of the problem of finding all constant mean curvature (CMC) tori in R 3 (and therefore, by taking the Gauss map, finding all nonconformal harmonic maps from a torus to S 2). At its simplest level this proceeds from the recognition that the GaussCodazzi equations of a CMC torus are the elliptic sinhGordon equations (1) uz¯z + sinh(4u) = 0, z = x + iy. It was shown in the late 1980’s ([24, 1]) that each doubly periodic solution of this equation can be written down in terms of the Riemann θfunction for a compact Riemann surface X, called the spectral curve (this also follows from Hitchin’s work [10] on harmonic tori in S3, which used a distinctly different approach). That this is true relies on two observations. First, (1) has a zerocurvature (or Lax pair) representation: it is the condition that − Uζ,
Conserved quantities in the theory of discrete surfaces 1 Background
, 2007
"... Suppose you are given a simple first order smooth ordinary differential equation with a given initial condition. If you cannot write down its solution explicitly, you might find a discrete approximate solution by using the Euler or RungaKutta algorithm, just to have some initial idea how the smooth ..."
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Suppose you are given a simple first order smooth ordinary differential equation with a given initial condition. If you cannot write down its solution explicitly, you might find a discrete approximate solution by using the Euler or RungaKutta algorithm, just to have some initial idea how the smooth solution behaves. In this case, your interest in the approximate solution is only as a stepping stone for understanding the smooth true solution. We can think of the equation (i.e. the algorithm) for the discrete approximate solution as a finite dimensional problem because the full space of objects (a vector space of discrete functions) that can be inserted to test for validity in the equation is finite dimensional. Likewise, we can call the smooth differential equation an infinite dimensional problem (this might be somewhat unconventional), because the objects insertable into the equation form an infinite dimensional vector space. Or you might instead look at a related ordinary difference equation, with little concern that the resulting discrete solution approximates the smooth solution, and rather be more concerned that the difference equation maintains some property found in the smooth differential equation that you deem important. In this case, as your primary interest is the ”finite dimensional” difference equation situation itself, you might discard the smooth equation altogether, or you
From CMC surfaces to Hamiltonian stationary Lagrangian surfaces
"... Minimal surfaces and surfaces with constant mean curvature (CMC) have fascinated differential geometers for over two centuries. Indeed these surfaces are solutions to variational problems whose formulation is elegant, modelling physical situations involving soap films and bubbles; ..."
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Minimal surfaces and surfaces with constant mean curvature (CMC) have fascinated differential geometers for over two centuries. Indeed these surfaces are solutions to variational problems whose formulation is elegant, modelling physical situations involving soap films and bubbles;
Immersions of Arbitrary Constant Mean Curvature in Hyperbolic Space H 3 (−1)
, 2003
"... Our work studies immersions of arbitrary constant mean curvature in H 3 (−1), as integrable surfaces (i.e., associate families of surfaces). We first present new results related to isometric deformations within this category of surfaces. Next, we give a Weierstrass type representation formula for th ..."
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Our work studies immersions of arbitrary constant mean curvature in H 3 (−1), as integrable surfaces (i.e., associate families of surfaces). We first present new results related to isometric deformations within this category of surfaces. Next, we give a Weierstrass type representation formula for these immersions. Ultimately, we discuss correspondences between such immersions in Euclidean 3space E 3 and the ones in hyperbolic 3space H 3 (−1). In particular, we show that every nonzero constant mean curvature surface in E 3 has two hyperbolic ‘cousins ’ of constant mean curvature H  < 1 and H > 1 in H 3 (−1) respectively, where each ‘cousin ’ consists of two parallel surfaces, all having the same Weierstrass type data.
COVERINGS AND INTEGRABILITY OF THE GAUSS–MAINARDI–CODAZZI EQUATIONS
, 1998
"... Abstract. Using covering theory approach (zerocurvature representations with the gauge group SL2), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained. ..."
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Abstract. Using covering theory approach (zerocurvature representations with the gauge group SL2), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained.