American Mathematical Society

Book Review

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Book Information:

Author: 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 pp., ISBN 0-521-58932-0, $14.95$, paperback

[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 1109632

[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 1230929

[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 1369397

[5]

F. Burstall and M. Guest, Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309 (1997), no. 4, 541-572. MR 1483823

[6]

J. Dorfmeister and H. Wu, Constant mean curvature surfaces and loop groups. J. Reine Angew. Math. 440 (1993), 43-76. MR 1225957

[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 1664887

[8]

J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 0495450

[9]

J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524. MR 0956352

[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 1303230

[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 1014929

[14]

K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207-221. MR 0408535

[15]

A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University, New York, 1986. MR 0900587

[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 0730247

[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 0783348

[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 1055875

[19]

K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1-50. MR 1001271

Review Information:

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