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Hamiltonian stationary tori in complex projective plane
, 310
"... We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature equation on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest o ..."
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Cited by 11 (1 self)
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We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature equation on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.
The curvature of a Hessian metric
 Internat. J. Math
"... Given a smooth function f on an open subset of a real vector space, one can define the associated “Hessian metric ” using the second derivatives of f, gij: = ∂ 2 f/∂xi∂xj. In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case wher ..."
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Cited by 9 (0 self)
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Given a smooth function f on an open subset of a real vector space, one can define the associated “Hessian metric ” using the second derivatives of f, gij: = ∂ 2 f/∂xi∂xj. In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case where f is a homogeneous polynomial (also called a “form”) of degree d at least 2. Following Okonek and van de Ven [23], Wilson considers the “index cone, ” the open subset where the Hessian matrix of f is Lorentzian (that is, of signature (1, ∗)) and f is positive. He restricts the indefinite metric −1/d(d − 1)∂2f/∂xi∂xj to the hypersurface M: = {f = 1} in the index cone, where it is a Riemannian metric, which he calls the Hodge metric. (In affine differential geometry, this metric is known as the “centroaffine metric ” of the hypersurface M, up to a constant factor.) Wilson considers two main questions about the Riemannian manifold M. First, when does M have nonpositive sectional curvature? (It does have nonpositive sectional curvature in many examples.) Next, when does M have constant negative curvature?
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.
Transformations of flat Lagrangian immersions and Egoroff nets, accepted
 Asian Journal of Mathematics
"... Abstract. We associate a natural λfamily (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with nondegenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space U( ..."
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Cited by 7 (2 self)
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Abstract. We associate a natural λfamily (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with nondegenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space U(n)⋉Cn O(n)⋉Rn. An interesting observation is that the family degenerates to an Egoroff net on R n when λ → 0. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The action of the generator with one simple pole gives the geometric Ribaucour transformation and we provide the permutability formula for such transformations. The action of the generator with two poles and the action of a rational loop in the translation subgroup produce new transformations. The corresponding results for flat Lagrangian submanifolds in CP n−1 and ∂invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration. 1.
Survey on affine spheres
 Handbook of geometric analysis No. 2, International Press Beijing, Advanced Lectures in Mathematics, 13 (2010), 161C191; arXiv: math.DG/0809.1186v1, Zblpre05831739
"... We give a survey on the theory of affine spheres, emphasizing the convex cases and relationships to MongeAmpère equations. We discuss the twodimensional case, as originally investigated by Ţiţeica, and also the global classification theory of affine spheres, due to Calabi and ChengYau and othe ..."
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Cited by 6 (0 self)
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We give a survey on the theory of affine spheres, emphasizing the convex cases and relationships to MongeAmpère equations. We discuss the twodimensional case, as originally investigated by Ţiţeica, and also the global classification theory of affine spheres, due to Calabi and ChengYau and others. We also recount the relationship between hyperbolic affine spheres and convex RPn structures on manifolds, and briefly discuss the relationship of affine spheres to mirror symmetry via the StromingerYauZaslow conjecture. Affine maximal surfaces and the affine normal flow, which generalize affine spheres, are also discussed. MSC Classification: 53A15
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.
Tzitzéica transformation is a dressing action
 J. Math. Phys
, 2006
"... Abstract. We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzi ..."
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Cited by 4 (1 self)
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Abstract. We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitzéica transformations, and discuss the group structure for these transformations. 1 2 ERXIAO WANG 1.
TABLEAUX OVER LIE ALGEBRAS, INTEGRABLE SYSTEMS, AND CLASSICAL SURFACE THEORY
, 2006
"... Abstract. Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfa ..."
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Cited by 4 (2 self)
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Abstract. Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G0system of Terng and the curved flat system of Ferus–Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss–Codazzi equations are associated with tableaux over sl(4, R) are discussed. 1.
and A Quintino. Dressing transformations of constrained Willmore surfaces
 Comm. Anal. Geom
, 2014
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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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Cited by 3 (2 self)
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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.