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Complex surfaces of constant mean curvature fibered by minimal surfaces
 Hokkaido Math. J
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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]).
Actions of loop groups on the space of harmonic maps into reductive homogeneous spaces
 J. Math. Sci. Univ. Tokyo
, 1998
"... Abstract. In this paper we study special affine harmonic maps into reductive homogeneous spaces and prove that there exist loop group actions on such harmonic maps. ..."
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Abstract. In this paper we study special affine harmonic maps into reductive homogeneous spaces and prove that there exist loop group actions on such harmonic maps.
GRASSMANN GEOMETRIES IN INFINITE DIMENSIONAL HOMOGENEOUS SPACES AND AN APPLICATION TO REFLECTIVE SUBMANIFOLDS
, 2007
"... Abstract. Let U be a real form of a complex semisimple Lie group, and (τ, σ) a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. This giv ..."
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Abstract. Let U be a real form of a complex semisimple Lie group, and (τ, σ) a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. This gives many new examples of submanifolds as integrable systems. The solutions are shown to correspond to various special submanifolds, depending on which homogeneous space, U/L, one projects to. We apply the construction to a question which generalizes, to the context of reflective submanifolds of arbitrary symmetric spaces, the problem of isometric immersions of space forms with negative extrinsic curvature and flat normal bundle. For this problem, we prove that the only cases where local solutions exist are the previously known cases of space forms, in addition to our new example of constant curvature Lagrangian immersions into complex projective and complex hyperbolic spaces. We also prove nonexistence of global solutions in the compact case. For other reflective submanifolds, lower dimensional solutions exist, and can be described in terms of Grassmann geometries. We consider one example in detail, associated to the group G2, obtaining a special class of surfaces in S 6. 1.
Discretizing constant curvature surfaces via loop group factorization: the discrete sine and sinhGordon equations
 J. Geom. Phys
, 1995
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Goertsches, Generators for rational loop groups and geometric applications, arXiv:0803.0029v1 [math.DG
"... Abstract. Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n, C) that satisfy the U(n)reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simpl ..."
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Abstract. Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n, C) that satisfy the U(n)reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neoclassical groups and most of their symmetric spaces. In order to apply our theorems to submanifold geometry we also obtain explicit dressing and permutability formulae. We introduce a new submanifold geometry associated to G2/SO(4) to which our theory applies. Contents
Elliptic Integrable Systems: a Comprehensive Geometric Interpretation
, 2011
"... In this paper, we study all the elliptic integrable systems, in the sense of C.L. Terng [65]. That is to say the family of all the mth elliptic integrable systems associated to a k ′symmetric space N = G/G0. Here m ∈ N and k ′ ∈ N ∗ are integers. For example, it is known that the first elliptic i ..."
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In this paper, we study all the elliptic integrable systems, in the sense of C.L. Terng [65]. That is to say the family of all the mth elliptic integrable systems associated to a k ′symmetric space N = G/G0. Here m ∈ N and k ′ ∈ N ∗ are integers. For example, it is known that the first elliptic integrable system associated to a symmetric space (resp. to a Lie group) is the equation for harmonic maps into this symmetric space (resp. this Lie group). Indeed it is well known that this harmonic maps equation can be written as a zero curvature equation: dαλ+ 1 2 [αλ∧αλ] = 0, ∀λ ∈ C ∗ , where αλ = λ −1 α ′ 1+α0+λα ′′ 1 is a 1form on a Riemann surface L taking values in the Lie algebra g. This 1form αλ is obtained as follows. Let f: L → N = G/G0 be a map from the Riemann surface L into the symmetric space G/G0. Then let F: L → G be a lift of f, and consider α = F −1.dF its MaurerCartan form. Then decompose α according to the symmetric decomposition g = g0 ⊕ g1 of g: α = α0 + α1. Finally, we define αλ: = λ −1 α ′ 1 + α0 + λα ′′ 1, ∀λ ∈ C ∗ , where α ′ 1,α ′′ 1 are the resp. the (1,0) and (0,1) parts of
Geometric Interpretation of Second Elliptic Integrable Systems
, 2009
"... In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4symmetric spaces. We first show that a 4symmetric space G/G0 can be embedded into the twistor space of the corresponding symmetric space G/H. Then we prove that the second elliptic syste ..."
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In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4symmetric spaces. We first show that a 4symmetric space G/G0 can be embedded into the twistor space of the corresponding symmetric space G/H. Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift J taking values in G/G0 → Σ(G/H). We begin the paper with an example: G/H = R 4. We also study the structure of 4symmetric bundles over Riemannian symmetric spaces.
ON INFINITESIMAL DEFORMATIONS OF CMC SURFACES OF FINITE TYPE IN THE 3SPHERE
"... Abstract. We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3sphere. We use BakerAkhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral curve to distinguish between isospectral and noniso ..."
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Abstract. We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3sphere. We use BakerAkhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral curve to distinguish between isospectral and nonisospectral deformations.