Abstract
In this article we consider geometrical interpretations of the two-dimensional affine Toda equations for a compact simple Lie group G. These equations originated from the work of Toda [33],[34] over 25 years ago on vibrations of lattices, and they have received considerable attention from both pure and applied mathematicians particularly over the last 15 years. (For the original context of the ideas the reader is referred to [35], [27] and [1] and for a survey of recent work to [29] and [21]).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves, Adv. Math. 38 (1980), 267–317.
M. Black, Harmonic maps into homogeneous spaces, Longman, Harlow, 1991.
A.I. Bobenko, All constant mean curvature tori in R3, S3, H3 in terms of theta-functions, Math. Ann. 290 (1991), 209–245.
A.I. Bobenko, Surfaces in terms of 2 by 2 matrices. Old and new integrable cases,this volume.
J. Bolton, G.R. Jensen, M. Rigoli and L.M. Woodward, On conformal minimal immersions of S 2 into CP“, Math. Ann. 279 (1988), 599–620.
J. Bolton, F. Pedit and L.M. Woodward, Minimal surfaces and the Toda field model,preprint, Universities of Durham and Massachussetts, Amherst.
J. Bolton, L.Vrancken and L.M. Woodward, On almost complex curves in the nearly Kiihler 6-sphere,Quarterly J. Math. (Oxford), to appear.
J. Bolton and L.M. Woodward, On immersions of surfaces into space forms, Soochow J. Math. 14 (1988), 11–31.
J. Bolton and L.M. Woodward, Congruence theorems for harmonic maps from a Riemann surface into CP“ and S”, J. London Math. Soc. 45 (2) (1992), 363–376.
J. Bolton and L.M. Woodward, Minimal surfaces in CP“ with constant curvature and Kähler angle, Math. Proc. Camb. Phil. Soc. 112 (1992), 287–296.
R.L. Bryant, Submanifolds and special structures on the octonians, J. Diff. Geom. 17 (1982), 185–232.
F.E. Burstall. Harmonic tori in spheres and complex projective spaces,in preparation.
F.E. Burstall, D. Ferus, F. Pedit and U. Pinkall, Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras, Ann. of Math. 138 (1993), 173–212.
F.E. Burstall and F. Pedit, Harmonic maps via Adler-Kostant-Symes theory,this volume.
F.E. Burstall and J.C. Wood, The construction of harmonic maps into complex Grassmannians„ J. Diff. Geom. 23 (1986), 255–297.
E. Calabi, Minimal immersions of surfaces into Euclidean spheres, J. Diff. Geom. 1 (1967), 111–125.
A.M. Din and W.J. Zakrzewski, General classical solutions in the CPn -1 model, Nuclear Phys. B 174 (1980), 397–406.
A. Doliwa and A. Sym, The non-linear a-model in spheres, preprint, Warsaw University, 1992.
J. Eells and J.C. Wood, Harmonic maps from surfaces to complex projective spaces, Adv. in Math. 49 (1983), 217–263.
D. Ferus, F. Pedit, U. Pinkall and I. Sterling, Minimal tori in S4, J. Reine Angew. Math. 429 (1992), 1–47.
A.P. Fordy, Integrable equations associated with simple Lie algebras and symmetric spaces, in: Soliton Theory: A Survey of Results, ed. A.P. Fordy, Manchester Univ. Press 1990, 315–337.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, San Francisco, London 1978.
N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Diff. Geom. 31 (1990), 627–710.
A. Jimenez, Addendum to “Existence of Hermitian n-symmetric spaces and non-commutative naturally reductive spaces”, Math. Z. 197 (1988), 455–456.
K. Kenmotsu, On minimal immersions of R 2 into P“(C), J. Math. Soc. Japan 37 (1985), 665–682.
B. Kostant, The principal three dimensional subgroups and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. Math. 34 (1979), 195–338.
H.B. Lawson, Complete minimal surfaces in S 3, Ann. of Math. 92 (1970), 335–374.
A.N. Leznov and M.V. Saveliev. Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems, Birkhäuser, Basel 1992.
I. McIntosh, Infinte dimensional Lie groups and the two-dimensional Toda lattice,this volume.
M. Melko and I. Sterling, Integrable systems, harmonic maps and the classical theory of surfaces,this volume.
U. Pinkall and I. Sterling, On the classification of constant mean curvature tori, Ann. of Math. 130 (1989), 407–451.
M. Toda, Vibrations of a chain with nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431–436.
M. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. Japan 23 (1967), 501–506.
M. Toda, Theory of Nonlinear Lattices, Springer, Berlin, Heidelberg, New York 1981.
J.G. Wolfson, On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature, Trans. Amer. Math. Soc. 290 (1985), 627–646.
J.C. Wood, Harmonic maps into symmetric spaces and integrable systems,this volume.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Bolton, J., Woodward, L.M. (1994). The affine Toda equations and minimal surfaces. In: Fordy, A.P., Wood, J.C. (eds) Harmonic Maps and Integrable Systems. Aspects of Mathematics, vol E 23. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14092-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-663-14092-4_4
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-06554-6
Online ISBN: 978-3-663-14092-4
eBook Packages: Springer Book Archive