Isothermic Surfaces
by
Emilio Musso
Università di L'Aquila, Italy

Plenary Lecture

In recent years the theory of isothermic surfaces has received a renewed attention due to its connection with the theory of integrable systems [BHPP], [Bu], [CGS], [FP]. Our aim is to survey the classical theory of isothermic surfaces and illustrate some recent progress from the view-point of integrable systems. We first present the basic transformations of isothermic surfaces such as the Chrisfoffel transformation, the Bianchi-Darboux transformation and the T-transformation; we analyze their permutability properties and discuss in some details a class of Darboux's special isothermic surfaces in relation with constant mean curvature surfaces in 3-dimensional space forms [HP], [HMN], [MN]. We then shortly present the quaternionic formalism for Möbius differential geometry [HJ], and the interrelations between Christoffel pairs and Bonnet mates [Bo], [KPP]. Finally we discuss the loop group approach to the transformation theory of isothermic surfaces as developed by F. Burstall [Bu] within the general theory of Bäcklund transformations due to C. L. Terng and K. Uhlenbeck [TU].

References

[BHPP] F. Burstall, U. Hertrich-Jeromin, F. Pedit, U. Pinkall, Curved flats and isothermic surfaces, Math. Z. 225 (1997), 199-209.

[Bo] A. Bobenko, Exploring Surfaces through Methods from the Theory of Integrable Systems. Lectures on the Bonnet Problem, e-print math.DG/9909003.

[Bu] F. Burstall, Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, e-print math.DG/0003096.

[CGS] J. Cie\'sli\'nski, P. Goldstein, A. Sym, Isothermic surfaces in E³ as soliton surfaces, Physics Letters A 205 (1995), 37-43.

[FP] D. Ferus, F. Pedit, Curved flats in symmetric spaces, Manuscripta Math. 91 (1996), 445-454

[HJ] U. Hertrich-Jeromin, Supplement on curved flats in the space of point pairs and isothermic surfaces: a quaternionic calculus, Documenta Math. J. DMV 2 (1997), 335-350.

[HP] U. Hertrich-Jeromin, F. Pedit, Remarks on the Darboux transform of isothermic surfaces, Documenta Math. J. DMV 2 (1997), 313-333.

[HMN] U. Hertrich-Jeromin, E. Musso, L. Nicolodi, Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space, e-print math.DG/9810157, to appear in Ann. Golobal Analysis and Geom..

[KPP] G. Kamberov, F. Pedit, U. Pinkal, Bonnet pairs and isothermic surfaces, Duke Math. J. 92 (1998), 637-644.

[MN] E. Musso, L. Nicolodi, Special Isothermic Surfaces and Solitons, in preparation (2000).

[TU] C.L. Terng, K. Uhlenbeck, Bäcklund transformations and loop group actions, Comm. Pure Appl. Math. 53 (2000), 1-75.

Date received: May 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-70.