Abstract
In this paper we consider a method for obtaining isothermic surfaces based on the Ribaucour transformations. By applying the theory to the cylinder, we obtain two families of complete isothermic surfaces. In the first family we have bubbletons surfaces and surfaces with planar ends. In second family, we get Dupin surfaces. As an application, we provide new explicit solutions to the Calapso equation.
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References
L. Bianchi, Lezioni di Geometria Differenziale (N. Zanichelli, Bologna, 1927), Vol. II.
A. M. V. Corro, W. P. Ferreira, and K. Tenenblat, “On Ribaucour transformations for hypersurfaces,” Mat. Contemp. 17, 137–160 (1999).
A. M. V. Corro, W. P. Ferreira, and K. Tenenblat, “Ribaucour transformations for constant mean curvature and linear Weingarten surfaces,” Pac. J. Math. 212 (2), 265–296 (2004).
A. M. V. Corro, W. P. Ferreira, and K. Tenenblat, “Minimal surfaces obtained by Ribaucour transformations,” Geom. Dedicata 96, 117–150 (2003).
A. M. V. Corro and K. Tenenblat, “Ribaucour transformation revisited,” Commun. Anal. Geom. 12 (5), 1055–1082 (2004).
M. V. Lemes and K. Tenenblat, “On Ribaucour transformations and minimal surfaces,” Mat. Contemp. 29, 13–40 (2005).
K. Tenenblat and Q. Wang, “Ribaucour transformations for hypersurfaces in space forms,” Ann. Global Anal. Geom. 29, 157–185 (2006).
A. M. V. Corro, “Generalized Weingarten surfaces of Bryant type in hyperbolic 3-space,” Mat. Contemp. 30, 71–89 (2006).
W. P. Ferreira and K. Tenenblat, “Hypersurfaces with flat \(r\)-mean curvature and Ribaucour transformations,” Int. J. Appl. Math. Stat. 11, 38–51 (2007).
A. M. V. Corro, A. Martínez, and F. Milán, “Complete flat surfaces with two isolated singularities in hyperbolic \(3\)-space,” J. Math. Anal. Appl. 366 (2), 582–592 (2010).
E. B. Christoffel, “Über einige allgemeine Eigenschaften der Minimumsflächen,” J. Reine Angew. Math. 67, 218–228 (1867).
G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitesimal (Chelsea, New York, 1972).
G. Darboux, “Sur les surfaces isothermiques,” C. R. Acad. Sci., Paris 128, 1299–1305 (1899).
J. Cieśliński, “The Darboux–Bianchi transformation for isothermic surfaces. Classical results the solitons approach,” Differ. Geom. Appl. 7 (1), 1–28 (1997).
J. Cieśliński and Z. Hasiewicz, “Iterated Darboux transformation for isothermic surfaces in terms of Clifford numbers,” Symmetry 13 (1), 1–148 (2021).
J. Cho, K. Leschke, and Y. Ogata, “Generalised Bianchi permutability for isothermic surfaces,” Ann. Global Anal. Geom. 61 (4), 799–829 (2022).
A. Bobenko, U. Eitner, and A. Kitaev, “Surfaces with harmonic inverse mean curvature and Painlevé equations,” Geom. Dedicata 68 (2), 187–227 (1997).
J. Cieśliński, P. Goldstein, and A. Sym, “Isothermic surfaces in \(\mathbb{E}^3\) as solitons surfaces,” Phys. Lett. A 205 (1), 37–43 (1995).
F. Burstall, J. U. Hertrich, and U. Pinkall, “Curved flats and isothermic surfaces,” Math. Z. 225, 199–209 (1997).
J. U. Hertrich, “Supplement on curved flats in the space of point pairs and isothermic surfaces: A quaternionic calculus,” Doc. Math. 2, 335–350 (1997).
J. U. Hertrich and F. Pedit, “Remarks on the Darboux transform of isothermic surfaces,” Doc. Math. 2, 313–333 (1997).
P. Calapso, “Sulle superficie a linee di curvatura isoterme,” Rend. Circ. Mat. Palermo 17, 275–286 (1903).
R. Rogers and W. K. Schief, Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory (Cambridge University Press, Cambridge, 2002).
A. Davey and K. Stewartson, “On Three-dimensional packets of surface waves,” Proc. R. Soc. Lond., Ser. A 338, 101–110 (1974).
A. M. V. Corro, K. V. Fernandes, and C. M. C. Riveros, “Isothermic surfaces and solutions of the Calapso equation,” Serdica Math. J. 44 (3–4), 341–364 (2018).
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The authors are grateful to the referees for valuable comments.
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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
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Corro, A.M.V., Ferro, M.L. Isothermic Surfaces Associated with the Cylinder Obtained by Ribaucour Transformations. Math Notes 114, 728–747 (2023). https://doi.org/10.1134/S0001434623110093
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DOI: https://doi.org/10.1134/S0001434623110093