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 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 . 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**
**[1]**- A. Bobenko, All constant mean curvature tori in 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 . 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

Journal:
Bull. Amer. Math. Soc.

**38** (2001), 251-254

Published electronically:
December 27, 2000

Review copyright:
© Copyright 2000
American Mathematical Society