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Geometries and symmetries of soliton equations and integrable elliptic systems
 IN SURVEYS ON GEOMETRY AND INTEGRABLE SYSTEMS, ADVANCED STUDIES IN PURE MATHEMATICS, MATHEMATICAL SOCIETY OF JAPAN NORTHEASTERN UNIVERSITY AND UC IRVINE EMAIL ADDRESS: TERNG@NEU.EDU MSRI, BERKELEY, CA 94720 EMAIL ADDRESS: EWANG@MRSI.ORG
, 2002
"... We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semisimple Lie algebra and finite order automorphisms. For example, the nonlinear Schrödinger equation, the nwave equation, and the sigmamodel are soliton flow ..."
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Cited by 26 (4 self)
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We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semisimple Lie algebra and finite order automorphisms. For example, the nonlinear Schrödinger equation, the nwave equation, and the sigmamodel are soliton flows; and the equation for harmonic maps from the plane to a compact Lie group, for primitive maps from the plane to a ksymmetric space, and constant mean curvature surfaces and isothermic surfaces in space forms are integrable elliptic systems. We also give a survey of • construction of solutions using loop group factorizations, • PDEs in differential geometry that are soliton equations or elliptic integrable systems, • similarities and differences of soliton equations and integrable elliptic
Curved flats into symmetric spaces
 In preparation
"... Over the past years many problems in classical surface theory and also more generally, submanifold theory, have been linked to certain types of completely integrable nonlinear PDE (soliton equations). In certain cases this has already been known to geometers in the last century but the explicit con ..."
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Cited by 26 (3 self)
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Over the past years many problems in classical surface theory and also more generally, submanifold theory, have been linked to certain types of completely integrable nonlinear PDE (soliton equations). In certain cases this has already been known to geometers in the last century but the explicit construction of solutions by the finite gap integration scheme has only been achieved recently. The crucial ingredient is to rewrite
Curved flats, pluriharmonic maps and constant curvature immersions into pseudoRiemannian space forms
, 2006
"... We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among ..."
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Cited by 9 (7 self)
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We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudoRiemannian space forms. As a corollary, we obtain a nonimmersibility theorem for spheres into certain pseudoRiemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.
Curved flats, exterior differential systems, and conservation laws, Complex, contact and symmetric manifolds
 235–254, Progr. Math., 234, Birkhauser
, 2005
"... Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. Th ..."
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Cited by 8 (3 self)
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Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. They noted that the equation for curved flats is an integrable system. Bryant used the involution σ to construct an involutive exterior differential system Iσ such that integral submanifolds of Iσ are curved flats. Terng used r first flows in the U/U0hierarchy of commuting soliton equations to construct the U/U0system. She showed that the U/U0system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many these systems occur as the GaussCodazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system Iσ. 1.
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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Cited by 5 (2 self)
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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.
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
GRASSMANN GEOMETRIES AND INTEGRABLE SYSTEMS
, 804
"... Abstract. We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann subma ..."
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Abstract. We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given from the recent article [2]. 1.
APPLICATIONS OF LOOP GROUP FACTORIZATION TO GEOMETRIC SOLITON EQUATIONS
, 2006
"... Abstract. The 1d Schrödinger flow on S 2, the GaussCodazzi equation for flat Lagrangian submanifolds in R 2n, and the spacetime monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the real ..."
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Abstract. The 1d Schrödinger flow on S 2, the GaussCodazzi equation for flat Lagrangian submanifolds in R 2n, and the spacetime monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated to SU(n). In this article, we explain the method developed jointly with K. Uhlenbeck, that uses various loop group factorizations to construct inverse scattering transforms, Bäcklund transformations, and solutions to Cauchy problems for these equations. 1.