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122
Hamiltonian stationary Lagrangian surfaces in C²
, 1999
"... We study Hamiltonian stationary Lagrangian surfaces in C², i.e. Lagrangian surfaces in C² which are stationary points of the area functional under smooth Hamiltonian variations. Using ..."
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Cited by 47 (11 self)
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We study Hamiltonian stationary Lagrangian surfaces in C², i.e. Lagrangian surfaces in C² which are stationary points of the area functional under smooth Hamiltonian variations. Using
Dressing orbits of harmonic maps
 Duke Math. J
, 1995
"... At the heart of the modern theory of harmonic maps from a Riemann surface to a Riemannian symmetric space is the observation that, in this setting, the harmonic map equations have a zero curvature representation [19, 24, 28] and so correspond to loops of flat connections. This fact was first exploit ..."
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Cited by 33 (6 self)
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At the heart of the modern theory of harmonic maps from a Riemann surface to a Riemannian symmetric space is the observation that, in this setting, the harmonic map equations have a zero curvature representation [19, 24, 28] and so correspond to loops of flat connections. This fact was first exploited in the mathematical literature by Uhlenbeck in her
Harmonic twospheres in compact symmetric spaces, revisited
, 1997
"... The purpose of this article is to give a new description of harmonic maps from the twosphere S 2 to a compact symmetric space G/K, using a method suggested by Morse theory. The method leads to surprisingly short proofs of most of the known results, and also to new results. ..."
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Cited by 29 (4 self)
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The purpose of this article is to give a new description of harmonic maps from the twosphere S 2 to a compact symmetric space G/K, using a method suggested by Morse theory. The method leads to surprisingly short proofs of most of the known results, and also to new results.
Poisson Actions and Scattering Theory for Integrable Systems
 J. Differential Geometry
, 1998
"... Conservat#se laws, heirarchies,scat#G556D t#cat# and Backlundt#ndPD9575PL54G are known t# bet#I building blocks of int#GG849P part#49 di#erent##e equat#PDDG We ident#en t#enP asfacet# of at#II46 of Poisson groupact#pPGG and applyt#p t#plyP t# t#p ZSAKNS nxn heirarchy (which includest#c nonlinear S ..."
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Cited by 28 (10 self)
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Conservat#se laws, heirarchies,scat#G556D t#cat# and Backlundt#ndPD9575PL54G are known t# bet#I building blocks of int#GG849P part#49 di#erent##e equat#PDDG We ident#en t#enP asfacet# of at#II46 of Poisson groupact#pPGG and applyt#p t#plyP t# t#p ZSAKNS nxn heirarchy (which includest#c nonlinear Schrodingerequat#rPG modified KdV, and t#d nwaveequat#7I4P Wefirst find a simple model Poisson group act#pP t#t# cont#nPD flows forsyst#G4 wit# a Lax pair whose t#ose all decay on R.Backlundt#ndPG9zD4PL47G and flows arise from subgroups of t#fP single Poisson group. Fort#G ZSAKNS nxn heirarchy defined by aconst#G t a # u(n), t#P simple model is no longercorrect# The adet#86PL44 t woseparat# Poissonst#sonP44G9 The flows come fromt#o Poissonact#so of t#P cent#nPDzID H a of a in t#P dual Poisson group (due t# t#u behavior of e a#xat infinit y). When a hasdist#DzG eigenvalues, H a is abelian and it act# symplect#L4I65 . The phase space oft#PG9 flows is t#P space S a ofleft coset# of t#P cent#PIII58 of a in D , where D is acert#87 loop group. The group D cont#nP4 bot# a Poisson subgroup correspondingt# t#d cont#n uousscat#GGG8P dat#t and arat#5z7P loop group correspondingt# t#d discret# scat#t#587 dat#t The H aact#95 is t#P right dressingact#si on S a .Backlundt#ndPI59I8PLD46 arise fromt#o act#PG of t#P simplerat#lePD loops on S a by right mult#7PLDDG84Pt Variousgeomet#I9 equat##I9 arise from appropriat# choice of a andrest#G94PLDD oft#I phase space and flows. Inpart#DzPLD we discussapplicat#z4G t# t#p sineGordonequat#Gor harmonic maps, Schrodinger flows onsymmet#7P spaces, Darboux ort#8794PL coordinat#LD and isomet#5G immersions of one spaceform inanot#D4z 1 Research supported in part by NSF Grant DMS 9626130 2 Research supported in part by Sid Rich...
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
Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero
, 2003
"... We construct the entire threeparameter family of embedded constant mean curvature surfaces with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the necksizes of the three ends. Because our surfaces are transcendental, and are not described by an ..."
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Cited by 22 (1 self)
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We construct the entire threeparameter family of embedded constant mean curvature surfaces with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the necksizes of the three ends. Because our surfaces are transcendental, and are not described by any ordinary differential equation, it is remarkable to obtain such an explicit determination of their moduli space.
Embedded Constant Mean Curvature Surfaces with Special Symmetry
, 1998
"... We give sharp, necessary conditions on complete embedded cmc surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a twodimensional variety in the moduli space of general cmc surfaces. Fundamental domains of our cmc surfaces are characterized by associated grea ..."
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Cited by 21 (8 self)
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We give sharp, necessary conditions on complete embedded cmc surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a twodimensional variety in the moduli space of general cmc surfaces. Fundamental domains of our cmc surfaces are characterized by associated great circle polygons in the threesphere.
Exploring Surfaces through Methods from the Theory of Integrable Systems
 Lectures on the Bonnet Problem, preprint SFB 288, N 403, TUBerlin
, 1999
"... A generic surface in Euclidean 3space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known as the Bonnet problem. Regarding the Bonnet problem, we sho ..."
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Cited by 15 (1 self)
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A generic surface in Euclidean 3space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known as the Bonnet problem. Regarding the Bonnet problem, we show how analytic methods of the theory of integrable systems – such as finitegap integration, isomonodromic deformation, and loop group description – can be applied for studying global properties of special surfaces.
Unitarization of monodromy representations and constant mean curvature trinoids in 3dimensional space forms
 J. London Math. Soc
"... Abstract. We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thricepunctured sphere in the simplyconnected 3dimensional space forms R 3, S 3 and H 3. ..."
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Cited by 15 (11 self)
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Abstract. We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thricepunctured sphere in the simplyconnected 3dimensional space forms R 3, S 3 and H 3. Additionally, we compute the extended frame for any associated family of Delaunay surfaces. 1. Introduction. Surfaces that minimize area under a volume constraint have constant mean curvature (cmc). The generalized Weierstraß representation [3] for nonminimal cmc surfaces involves solving a holomorphic complex linear 2 × 2 system of ordinary differential equations (ode) on a Riemann surface with values in a loop group. A
WILLMORE IMMERSIONS and Loop Groups
, 1998
"... We propose a characterisation of Willmore immersions inspired from the works of R. Bryant on Willmore surfaces and J. Dorfmeister, F. Pedit, H.Y. Wu on harmonic maps between a surface and a compact homogeneous manifold using moving frames and loop groups. We use that formulation in order to constru ..."
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Cited by 14 (1 self)
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We propose a characterisation of Willmore immersions inspired from the works of R. Bryant on Willmore surfaces and J. Dorfmeister, F. Pedit, H.Y. Wu on harmonic maps between a surface and a compact homogeneous manifold using moving frames and loop groups. We use that formulation in order to construct a Weierstrass type representation of all conformal Willmore immersions in terms of closed oneforms.