Elastic strips

Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Moebius strip, but did not give an integrable solution. We derive two conservation laws, which describe the equilibrium equations of elastic strips. In applying these laws we find two new classes of integrable elastic strips which correspond to spherical elastic curves. We establish a connection between Hopf tori and force--free strips, which are defined by one of the integrable strips, we have found. We introduce the P--functional and relate it to elastic strips.Comment: 21 pages, 2 figure

Holomorphic vector fields and quadratic differentials on planar triangular meshes

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a M\"obius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gau{\ss} map and prescribed Hopf differential.Comment: 17 pages; final version, to appear in "Advances in Discrete Differential Geometry", ed. A. I. Bobenko, Springer, 2016; references adde

Isothermic triangulated surfaces

We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is M\"{o}bius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes. A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.Comment: 29 pages, 7 figures; v2: references added, typos corrected and minor changes; v3: section 1-2,8-11 revised and references added; v4: minor change

A discrete version of the Darboux transform for isothermic surfaces

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all isothermic surfaces: they can be characterized by the fact that their parallel constant mean curvature surfaces are Christoffel and Darboux transforms at the same time. This characterization is used to define discrete nets of constant mean curvature. Basic properties of discrete nets of constant mean curvature are derived.Comment: 30 pages, LaTeX, a version with high quality figures is available at http://www-sfb288.math.tu-berlin.de/preprints.htm

Constrained Willmore Surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint.Comment: 17 pages, 8 figures; v2: Hopf tori added as an example, minor changes in presentation, numbering changed; v3: new abstract and appendix, several changes in presentatio