The depicted net is a cmc Darboux transform of a discrete cylinder.

The model is a discrete bubbleton, which is a discrete constant mean curvature surface obtained from a (discrete) cylinder by a Darboux or Bianchi-Baecklund transformation, cf. [8].

A smooth constant mean curvature surface in Euclidean 3-space has (away from umbilics) a parallel constant mean curvature surface (Bonnet's theorem). On the one hand, this parallel surface carries a conformally equivalent metric, has corresponding curvature directions and also parallel tangent planes and, therefore, is a Christoffel transform of the original surface. On the other hand, besides carrying a conformally equivalent metric and corresponding curvature directions, it is the second envelope of a 2-parameter family of spheres (of constant radius) and, therefore, it is also a Darboux transform of the original surface. It turns out that these facts provide a characterization for surfaces of constant mean curvature among all isothermic surfaces [5].

Half way between two parallel cmc surfaces there is a (possibly degenerate) surface of constant (positive) Gauss curvature. For such surfaces, Bianchi defined a Baecklund transformation [1] which can be arranged, by aligning parameters appropriately, to provide a real surface after two transformations, see [10]. This provides a notion of Bianchi-Baecklund transformation for surfaces of constant mean curvature. The smooth (single-) bubbletons are obtained by this procedure from the cylinder: they are parallel to Sievert's surfaces [9], see [10] and [8]. Multiple application of the transformation provides the multibubbletons of [10].

On the other hand, the Darboux transformation transforms cmc surfaces (considered as isothermic surfaces) to cmc surfaces when the involved parameters are chosen appropriately, cf. [2]. It turns out that these Darboux transformations are the aforementioned Bianchi-Baecklund transformations, see [5] and [8], so that (single- and multi-) bubbletons can be obtained by (one resp. mutliple) Darboux transformations from a cylinder.

Using the transformation theory for discrete isothermic nets then yields a definition for discrete cmc nets by requiring the existence of a simultaneous Christoffel and Darboux transform of an isothermic net, see [6]. These discrete cmc nets can as well be defined in an analytic way by discretizing the frame equations, which yields a unit normal field that is defined on the vertices of the net and provides another way of defining the parallel surface, see [3]. Note that this definition can be extended to discrete isothermic nets in the broad sense, that is, for discrete nets with factorizing cross ratios rather than just cross ratio -1, cf. [7].

Also, by using the appropriate Bianchi permutability theorems, one can show that discrete cmc nets in this sense allow a (1+2)-parameter family of Darboux transformations into discrete cmc nets, see [6]. The described model is obtained as such a Darboux transform of a discrete cylinder.

Model produced with: Mathematica and JavaView v3.00

Keywords | discrete net; cmc surface; constant mean curvature; Darboux transformation; Baecklund transformation; Bubbleton | |

MSC-2000 Classification | 53A10 (37K25, 37K35) | |

Zentralblatt No. | 05264902 |

- L. Bianchi: Vorlesungen ueber Differentialgeometrie, Teubner, Berlin (1899), .
- L. Bianchi:
*Ricerche sulle superficie isoterme e sulla deformazione delle quadriche*, Ann. Mat.**11**(1905), 93-157, . - A. Bobenko, U. Pinkall:
*Discretization of surfaces and integrable systems*, in A. Bobenko, R. Seiler (Eds.): Discrete integrable geometry and physics, Clarendon Press, Oxford (1999), 3-58, . - G. Darboux:
*Sur les surfaces isothermiques*, Comptes Rendus**128**(1899), 1299-1305, 1538, . - U. Hertrich-Jeromin, F. Pedit:
*Remarks on the Darboux transform of isothermic surfaces*, Doc. Math. J. DMV**2**(1997), 313-333, . - U. Hertrich-Jeromin, T. Hoffmann, U. Pinkall:
*A discrete version of the Darboux transform for isothermic surfaces*, in A. Bobenko, R. Seiler (Eds.): Discrete integrable geometry and physics, Clarendon Press, Oxford (1999), 59-81, . - U. Hertrich-Jeromin: Introduction to Moebius differential geometry, Cambridge Univ. Press (2003), .
- S. Kobayashi:
*Bubbletons and their parallel surfaces in Euclidean 3-space*(2002), Electronic Geometry Model No. 2002.03.001, http://www.eg-models.de/2002.03.001. - H. Sievert:
*Ueber die Zentralflaechen der Enneper'schen Flaechen konstanten Kruemmungsmasses*(1886), Inaugural-Dissertation, Tuebingen, . - I. Sterling, H.C. Wente:
*Existence and classifiation of constant mean curvature multibubbletons of finite and infinite type*, Indiana Univ. Math. J.**42**(1993), 1239-1266, .

- Master File: DiscreteBubbleton_Master.jvx
- Applet File: DiscreteBubbleton_Master.jvx
- Preview: DiscreteBubbleton_Preview.jpg

Submitted: Fri Oct 29 12:05:34 MET DST 2004.

Revised: Wed Oct 5 10:27:48 MET DST 2005.

Accepted: Mon Oct 31 10:29:20 CET 2005.

University of BathTim Hoffmann

Department of Mathematical Sciences

University of Bath

Bath, BA2 7AY

United Kingdom

u.hertrich-jeromin@maths.bath.ac.uk

http://www.maths.bath.ac.uk/~masuh/

Universitaet Muenchen

Mathematisches Institut

Theresienstr. 39

D-80333 Muenchen

Germany

hoffmann@mathematik.uni-muenchen.de

http://www.mathematik.uni-muenchen.de/personen/hoffmann.php