Results 1  10
of
34
Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems
, 2000
"... ..."
(Show Context)
Minimal surfaces and the affine Toda field model
"... Introduction Minimal immersions or, more generally, harmonic maps of a Riemann surface S into S n ; CP n and other Riemannian symmetric spaces have been intensively studied over the past twenty five years. The subject has been given considerable impetus by the interest of mathematical physicist ..."
Abstract

Cited by 27 (10 self)
 Add to MetaCart
Introduction Minimal immersions or, more generally, harmonic maps of a Riemann surface S into S n ; CP n and other Riemannian symmetric spaces have been intensively studied over the past twenty five years. The subject has been given considerable impetus by the interest of mathematical physicists in nonlinear oemodels, which are harmonic maps of S 2 into CP n , and in related problems which may be handled using twistor theory and methods of complex geometry. For example, [EW83], harmonic maps of S 2 into CP n may be characterised as being elements of Frenet frames of holomorphic curves in CP n or, equivalently, in twistorial terms as projections of suitable holomorphic horizontal curves in the twistor space SU(n
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 ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
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
On constant mean curvature surfaces with periodic metric, Pac.J.Math
 Pacific J. Math
, 1996
"... We investigate CMCsurfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMCsurface, we develop an alternative approach to the classification of CMCtori given by Pin ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
We investigate CMCsurfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMCsurface, we develop an alternative approach to the classification of CMCtori given by Pinkall and Sterling. 1. Introduction. In recent years two independent approaches to the construction of CMCimmersions in R 3 were developed. One of them, using finite type solutions of the sinhGordon equation, leads to the classification of CMCtori by Pinkall and Sterling [20].
Investigation and Application of the Dressing Action on Surfaces of Constant Mean Curvature
, 1997
"... ..."
(Show Context)
Equivariant harmonic cylinders
"... Abstract. We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries. ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
Delaunay ends of constant mean curvature surfaces
"... Abstract. The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay su ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
Abstract. The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.
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. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
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.
Constant Mean Curvature Surfaces in 3Dimensional Space Forms
"... We apply the method of Dorfmeister, Pedit and Wu [8] for constructing harmonic maps from simply connected domains to symmetric spaces to obtain a procedure for generating constant mean curvature (CMC) surfaces with nontrivial topology in all simplyconnected 3dimensional space forms. We emphasiz ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
We apply the method of Dorfmeister, Pedit and Wu [8] for constructing harmonic maps from simply connected domains to symmetric spaces to obtain a procedure for generating constant mean curvature (CMC) surfaces with nontrivial topology in all simplyconnected 3dimensional space forms. We emphasize how to solve period problems and in the process discuss new examples of nonsimply connected CMC surfaces.