Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

Preface.- 1. 1+1 Dimensional Integrable Systems.- 1.1 KdV equation, MKdV equation and their Darboux transformations. 1.1.1 Original Darboux transformation. 1.1.2 Darboux transformation for KdV equation. 1.1.3 Darboux transformation for MKdV equation. 1.1.4 Examples: single and double soliton solutions. 1.1.5 Relation between Darboux transformations for KdV equation and MKdV equation. 1.2 AKNS system. 1.2.1 2 x 2 AKNS system. 1.2.2 N x N AKNS system. 1.3 Darboux transformation. 1.3.1 Darboux transformation for AKNS system. 1.3.2 Invariance of equations under Darboux transformations. 1.3.3 Darboux transformations of higher degree and the theorem of permutability. 1.3.4 More results on the Darboux matrices of degree one. 1.4 KdV hierarchy, MKdV-SG hierarchy, NLS hierarchy and AKNS system with u(N) reduction. 1.4.1 KdV hierarchy. 1.4.2 MKdV-SG hierarchy. 1.4.3 NLS hierarchy. 1.4.4 AKNS system with u(N) reduction. 1.5 Darboux transformation and scattering, inverse scattering theory. 1.5.1 Outline of the scattering and inverse scattering theory for the 2 x 2 AKNS system . 1.5.2 Change of scattering data under Darboux transformations for su(2) AKNS system. 2. 2+1 Dimensional Integrable Systems.- 2.1 KP equation and its Darboux transformation. 2.2 2+1 dimensional AKNS system and DS equation. 2.3 Darboux transformation. 2.3.1 General Lax pair. 2.3.2 Darboux transformation of degree one. 2.3.3 Darboux transformation of higher degree and the theorem of permutability. 2.4 Darboux transformation and binary Darboux transformation for DS equation. 2.4.1 Darboux transformation for DSII equation. 2.4.2 Darboux transformation and binary Darboux transformation for DSI equation. 2.5 Application to 1+1 dimensional Gelfand-Dickey system. 2.6 Nonlinear constraints and Darboux transformation in 2+1 dimensions. 3. N + 1 Dimensional Integrable Systems.- 3.1 n + 1 dimensional AKNS system. 3.1.1 n + 1 dimensional AKNS system. 3.1.2Examples. 3.2 Darboux transformation and soliton solutions. 3.2.1 Darboux transformation. 3.2.2 u(N) case. 3.2.3 Soliton solutions. 3.3 A reduced system on Rn. 4. Surfaces of Constant Curvature, Backlund Congruences.- 4.1 Theory of surfaces in the Euclidean space R3. 4.2 Surfaces of constant negative Gauss curvature, sine-Gordon equation and Backlund transformations. 4.2.1 Relation between sine-Gordon equation and surface of constant negative Gauss curvature in R3. 4.2.2 Pseudo-spherical congruence. 4.2.3 Backlund transformation. 4.2.4 Darboux transformation. 4.2.5 Example. 4.3 Surface of constant Gauss curvature in the Minkowski space R2,1 and pseudo-spherical congruence. 4.3.1 Theory of surfaces in the Minkowski space R2,1. 4.3.2 Chebyshev coordinates for surfaces of constant Gauss curvature. 4.3.3 Pseudo-spherical congruence in R2,1. 4.3.4 Backlund transformation and Darboux transformation for surfaces of constant Gauss curvature in R2,1. 4.4 Orthogonal frame and Lax pair. 4.5 Surface of constant mean curvature. 4.5.1 Parallel surface in Euclidean space. 4.5.2 Construction of surfaces. 4.5.3 The case in Minkowski space. 5. Darboux Transformation and Harmonic Map.- 5.1 Definition of harmonic map and basic equations. 5.2 Harmonic maps from R2 or R1,1 to S2, H2 or S1,1. 5.3 Harmonic maps from R1,1 to U(N). 5.3.1 Riemannian metric on U(N). 5.3.2 Harmonic maps from R1,1 to U(N). 5.3.3 Single soliton solutions. 5.3.4 Multi-soliton solutions. 5.4 Harmonic maps from R2 to U(N). 5.4.1 Harmonic maps from R2 to U(N) and their Darboux transformations. 5.4.2 Soliton solutions. 5.4.3 Uniton. 5.4.4 Darboux transformation and singular Darboux transformation for unitons. 6. Generalized Self-Dual Yang-Mills and Yang-Mills-Higgs Equations.- 6.1 Generalized self-dual Yang-Mills flow. 6.1.1 Generalized self-dual Yang-Mills flow. 6.1.2 Darboux transformation. 6.1.3 Example. 6.1.4 Relation with AKNS system. 6.2 Yang-Mills-Higgs

Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry

Confluent Hypergeometric Functions By Dr L J Slater Pp ix + 247 (Cambridge: At the University Press, 1960) 65s net

Confluent Hypergeometric Functions

and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

A short survey on biharmonic maps between Riemannian manifolds

We review part of the classical theory of curves and surfaces in 3-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

/pdf/differential-geometry-of-curves-and-surfaces-in-lorentz-5f7aizouyb.pdf

Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space

Since the discovery of constant mean curvature tori in Euclidean 3-space, methods in the theory of integrable systems (or soliton theory) have been applied frequently in differential geometry. In particular, surfaces of constant mean or Gaussian curvature in Euclidean 3-space have been studied extensively [2], [3]. On the other hand, the geometry of surfaces in Minkowski 3-space has been a subject ofwide interest [9], [10]. For example, a Kenmotsu-type representation formula for spacelike surfaces with prescribed mean curvature has been obtained by K. Akutagawa and S. Nishikawa, and M. A. Magid [9] has obtained such a representation formula for timelike surfaces. L. McNertney studied spacelike maximal surfaces, timelike extremal surfaces and timelike surfaces of constant positive Gaussian curvature by classical methods [10]. In our previous paper [5], we have studied spacelike surfaces with constant mean or Gaussian curvature in Minkowski 3-space via the theory of finite-type harmonic maps. To adapt the methods in the theory of integrable systems for our purposes, namely to study the geometry of surfaces, we reformulated the fundamental equations of spacelike surfaces. This allowed us to considerably simplify the computations in the study of such spacelike surfaces. The main tool of this reformulation was our use of split-quaternion numbers. In addition we obtained representation formulae for immersions in terms of loop group theory [5]. The purpose of this paper is to develop a corresponding setting for timelike surfaces of constant mean curvature in Minkowski 3-space. In contrast to the case of spacelike surfaces or timelike extremal surfaces, no systematic theory exists for timelike constant (nonzero) mean curvature surfaces. For this reason, we shall devote Section 1 to preliminary materials. The reformulation of the fundamental equations will be carried out in Section 2. As a result we obtain a

/pdf/timelike-surfaces-of-constant-mean-curvature-in-minkowski-3-15ebt6g76r.pdf

Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space

A classical theorem by Lancret says that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant. In this paper we study Lancret type problems for curves in Sasakian 3-manifolds.

/pdf/on-slant-curves-in-sasakian-3-manifolds-3wscdv5cmo.pdf

On slant curves in sasakian 3-manifolds

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated. Introduction. This paper concerns curves and surfaces in 3-dimen- sional contact manifolds whose mean curvature vector eld is in the kernel of certain elliptic dieren tial operators. First we study submanifolds whose mean curvature vector eld is in the kernel of the Laplacian (submanifolds with harmonic mean curvature vector elds). The study of such submanifolds is inspired by a conjecture of Bang-yen Chen (14):

/pdf/submanifolds-with-harmonic-mean-curvature-vector-field-in-507y8ndefu.pdf

Submanifolds with harmonic mean curvature vector field in contact 3-manifolds

In this paper, we present a unified geometric interpretation of the second AKNS-hierarchies via the geometric concept of Schrodinger flows in the category of symplectic manifolds and binormal motion for curves in the Minkowski 3-space.

Schrödinger flows, binormal motion for curves and the second AKNS-hierarchies

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces. In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.

Jun-ichi Inoguchi

Papers

Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space

On slant curves in sasakian 3-manifolds

Submanifolds with harmonic mean curvature vector field in contact 3-manifolds

Schrödinger flows, binormal motion for curves and the second AKNS-hierarchies

Timelike Minimal Surfaces via Loop Groups