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Book Review

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Book Information

Author(s): M. A. Guest
Title: Harmonic maps, loop groups, and integrable systems
Additional book information: London Mathematical Society Student Texts 38, Cambridge Univ. Press, Cambridge, UK, 1997, xiii + 194, $14.95 (paperback), 0-521-58932-0

References:

[1]
A. Bobenko, All constant mean curvature tori in ${R}\sp 3,\;S\sp 3,\;H\sp 3$ in terms of theta-functions. Math. Ann. 290 (1991), no. 2, 209-245. MR 92h:53072

[2]
F. Burstall, D. Ferus, F. Pedit and U. Pinkall, Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras. Ann. of Math. (2) 138 (1993), no. 1, 173-212. MR 94m:58057

[3]
F. Burstall and F. Pedit, Harmonic maps via Adler-Kostant-Symes theory. Harmonic maps and integrable systems, 221-272, Aspects Math., E23, Vieweg, Braunschweig, 1994. CMP 94:09

[4]
F. Burstall and F. Pedit, Dressing orbits of harmonic maps. Duke Math. J. 80 (1995), no. 2, 353-382. MR 97e:58052

[5]
F. Burstall and M. Guest, Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309 (1997), no. 4, 541-572. MR 99f:58046

[6]
J. Dorfmeister and H. Wu, Constant mean curvature surfaces and loop groups. J. Reine Angew. Math. 440 (1993), 43-76. MR 94j:53005

[7]
J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), no. 4, 633-668. MR 2000d:53099

[8]
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 82b:58033

[9]
J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524. MR 89i:58027

[10]
Ch. Jaggy, On the classification of constant mean curvature tori in ${R}\sp 3$. Comment. Math. Helv. 69 (1994), no. 4, 640-658. MR 95h:53010

[11]
M. Jimbo and T. Miwa, Monodromy, solitons and infinite-dimensional Lie algebras. Vertex operators in mathematics and physics (Berkeley, Calif., 1983), 275-290, Math. Sci. Res. Inst. Publ., 3, Springer, New York, 1985. CMP 17:10

[12]
M. Jimbo and T. Miwa, Integrable systems and infinite-dimensional Lie algebras. Integrable systems in statistical mechanics, 65-127, Ser. Adv. Statist. Mech., 1, World Sci. Publishing, Singapore, 1985. CMP 18:08

[13]
U. Pinkall and I. Sterling, On the classification of constant mean curvature tori. Ann. of Math. (2) 130 (1989), no. 2, 407-451. MR 91b:53009

[14]
K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207-221. MR 53:12299

[15]
A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University, New York, 1986. MR 88i:22049

[16]
M. Sato and Y. Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. Nonlinear partial differential equations in applied science (Tokyo, 1982), 259-271, North-Holland Math. Stud., 81, North-Holland, Amsterdam-New York, 1983. MR 86m:58072

[17]
G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. No. 61, (1985), 5-65. MR 87b:58039

[18]
G. Segal, Loop groups and harmonic maps. Advances in homotopy theory (Cortona, 1988), 153-164, London Math. Soc. Lecture Note Ser., 139, Cambridge Univ. Press, Cambridge, 1989. MR 91m:58043

[19]
K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1-50. MR 90g:58028

Additional Information:

Reviewer(s):
Josef Dorfmeister
Affiliation: TU-Muenchen
Email: dorfm@mathematik.tu-muenchen.de

Review Information:
Journal: Bull. Amer. Math. Soc. 38 (2001), 251-254.

MSC (2000): Primary 58E20, 22E67, 37K10
DOI: 10.1090/S0273-0979-00-00900-9
PII: S 0273-0979(00)00900-9
Posted: December 27, 2000
Copyright of article: Copyright 2000, American Mathematical Society

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