Generalized Weierstrass representation for surfaces in multi-dimensional Riemann spaces

Journal of Geometry and Physics

Volumn 29, Issue 4, 1999, Pages 319-333

  Konopelchenko, B G ^a,b   Landolfi, G ^a,b  

^a UNIVERSITY OF SALENTO   (Italy)

^b INFN SEZIONE DI LECCE   (Italy)

Author keywords

Generalized Weierstrass representation; Integrable deformation of surfaces; Riemann spaces

Indexed keywords

EID: 0033096350     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(98)00046-1     Document Type: Article

References (32)

1. G. Darboux, Lecons sur la Théorie des Surfaces et les Applications Geometriques du Calcul Infinitesimal, t. 1-4, Gauthier-Villars, Paris, 1877-1896.
2. L. Bianchi, Lezioni di Geometria Differenziale, 2nd Ed., Spoerri, Pisa, 1902.
3. L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1909.
4. S.T. Yau (Ed.), Seminar on Differential Geometry, Princeton University Press, Princeton, NJ, 1992.
5. S.S. Chern, in: I. Chavel, H.M. Farkas (Eds.), Differential Geometry and Complex Analysis, Springer, Berlin, 1985.
6. D. Nelson, T. Piran, S. Weinberg (EdS.), Statistical Mechanics of Membranes and Surfaces, World Scientific, Singapore, 1989.
7. D.J. Gross, T. Piran, S. Weinberg (EdS.), Two Dimensional Quantum Gravity and Random Surfaces, World Scientific, Singapore, 1992.
8. F. David, P. Ginsparg, Y. Zinn-Justin (Eds.), Fluctuating Geometries in Statistical Mechanics and Field Theory, Elsevier, Amsterdam, 1996.
9. K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979) 89-99.
10. B.G. Konopelchenko, Multidimensional Integrable Systems and Dynamics of Surfaces in Space, Preprint of Institute of Mathematics, Taipei, August 1993.
11. B.G. Konopelchenko, Induced surfaces and their integrable dynamics, Stud. Appl. Math. 96 (1996) 9-51; Preprint, Institute of Nuclear Physics, N 93-114, Novosibirsk, 1993.
12. B.G. Konopelchenko, Induced surfaces and their integrable dynamics, Stud. Appl. Math. 96 (1996) 9-51; Preprint, Institute of Nuclear Physics, N 93-114, Novosibirsk, 1993.
13. N, J. Diff. Geom. 18 (1983) 733-754.
14. 4, Proc. London Math. Soc. 50 (3) (1985) 27-56.
15. R. Carroll, B.G. Konopelchenko, Generalized Weierstrass-Enneper inducing, conformai immersion and gravity, Int. J. Mod. Phys. A11 (1996) 1183-1216.
16. B.G. Konopelchenko, I. Taimanov, Generalized Weierstrass Formulae, Soliton Equations and Willmore Surfaces, preprint N 187, University of Bochum, 1995.
17. B.G. Konopelchenko, I. Taimanov, Constant mean curvature surfaces via an integrable dynamical system, J. Phys. A 29 (1996) 1261-1265.
18. I. Taimanov, Modified Novikov-Veselov equation and differential geometry of surfaces, Trans. Am. Math. Soc., Ser. 2 179 (1997) 133-159.
19. 3, J. Geom. Phys. to appear; preprint SFB288 N 291, TU-Berlin, 1997.
20. J. Richter, Conformal Maps of a Riemann Surface into Space of Quaternions, Ph.D. Thesis, TU-Berlin, 1997.
21. I. Taimanov, Global Weierstrass representation and its spectrum, Uspechi Mat. Nauk 52 (6) (1997) 187-188.
22. 3, the Willmore numbers and soliton spheres, preprint SFB 288, N 302, TU-Berlin, 1998.
23. T.J. Willmore, Total Curvature in Riemannian Geometry, Ellis Horwood, New York, 1982.
24. K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge University Press, Cambridge, 1994.
25. U. Abresch, Spinor Representation of CMC Surfaces, Lecture at Luminy, 1989.
26. A.I. Bobenko, Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, in: A. Fordy, J. Wood (Eds.), Harmonic Maps and Integrable Systems, Vieweg, Braunschweig, 1994.
27. R. Kusner, N. Schmitt, The Spinor Representation of Surfaces in Space, preprint dg-ga/9610005, 1996.
28. T. Friedrich, On the Spinor Representation of Surfaces in Euclidean 3-spaces, preprint SFB 288, N 295, TU-Berlin, 1997.
29. 3 in terms of theta-furictions, Math. Ann. 290 (1991) 209-245.
30. L.P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, NJ, 1926.
31. J. Barrett, G.W. Gibbons, M.J. Perry, P. Ruback, Kleinian Geometry and the N = 2 superstring, Int. J. Mod. Phys. A 9 (1994) 1457-1493.
32. A.S. Fokas, I.M. Gelfand, Surfaces on Lie groups, on Lie algebras and their integrability, Comm. Math. Phys. 177 (1996) 203-220.