Non-special, non-canal isothermic tori with spherical lines of curvature (Q2706611)

This paper is based on the author's thesis [PhD thesis, Washington Univ. St. Louis (1999)]. In the reviewer's understanding, the Darboux transformation (for isothermic surfaces, cf. [\textit{G.~Darboux}, C. R. Acad. Sci., Paris 128, 1299-1305 (1899; JFM 30.0555.02)] or [\textit{W.~Blaschke}, ``Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. III: Differentialgeometrie der Kreise und Kugeln'', Springer, Berlin (1929; JFM 55.0422.01)]) would ``add'' soliton-like bubbles to an isothermic surface along a given curvature line while keeping the rest of the surface merely unchanged, that is, going away from such a bubble along a curvature line that intersects the first curvature line orthogonally the surface and its Darboux transform would approach asymptotically. This behaviour can be observed when looking at Darboux transformations of a circular cylinder, for example, and the reviewer always observed this behaviour when trying to obtain computer graphics of Darboux transforms -- also of Darboux transforms of tori of revolution. As a consequence such Darboux transformations of an isothermic torus could never close both periods.NEWLINENEWLINENEWLINEThis understanding dramatically changed with the results presented in the paper under review: here, the author uses explicit solutions of Darboux's linear system for Clifford tori to obtain (non-trivially: of course, the central sphere congruence of the quadratic Clifford torus for example also provides a closed Darboux transform, but a rather trivial one) isothermic tori as Darboux transforms of the original Clifford tori.NEWLINENEWLINENEWLINEThe paper starts with a comprehensive introduction to the employed formalism: after introducing an invariant frame, as in [\textit{R.~Bryant}, J. Differ. Geom. 20, 23-53 (1984; Zbl 0555.53002)], for surfaces in the classical (projective) model of Möbius differential geometry [cf. \textit{W.~Blaschke}, loc. cit.], isothermic surfaces and their transformations are discussed.NEWLINENEWLINENEWLINEHere the Calapso potential of a polarized isothermic surface is assigned a key role. This potential is then used to characterize speciality of isothermic surfaces as well as spherical curvature lines: an isothermic surface is called ``special'' if the Calapso equation reduces to a second order equation and the surface becomes conformally equivalent to a surface of constant mean curvature in a space form (cf.~[\textit{L.~Bianchi}, Ann. Mat. III 11, 93-157 (1905; JFM 36.0674.01)]), and it has one family of spherical curvature lines if the Calapso potential satisfies a certain \(3\)rd order differential equation. NEWLINENEWLINENEWLINEIn the second part of the paper the geometry of Darboux transformations of Clifford tori (non-singular Dupin cyclides in the \(3\)-sphere) is analyzed. A priori the Darboux transformation is only locally defined -- using an explicit solution of Darboux's linear system, the author determines those transforms that are defined on the universal cover of the original torus as well as those that become doubly periodic. Moreover it is shown that the investigated Darboux transforms have spherical curvature lines, and may carry (isolated or lines of) umbilics. The discussion is complemented by two examples and nice computer graphics thereof.