Udo Hertrich-Jeromin
Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Bryant's Weierstrass type representation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Links between the integrable systems of CMC surfaces, isothermic surfaces and constrained Willmore surfaces
本文统一刻画了CMC曲面、等温曲面与约束威尔莫尔曲面所对应的三个可积系统，通过平坦联络族的平行截面揭示其代数联系，并阐明各类简单因子dressing与Darboux变换之间的关系及相关族的极限结构。
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points o…
This paper studies the discrete differential geometry of the checkerboard pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns out to be a versatile tool which allows us to consistently define principal nets, Koenigs nets and eventually isothermic nets as a combination of both. Principal nets are based on the notions of orthogonality and conjugacy and can be identified with sphere congruences that are entities of Moebius geometry. Discrete Koenigs nets are defined via the existence of the so called conic of Koenigs. We find several interesting properties of Koenigs n…
We give an elaborated treatment of discrete isothermic surfaces and their analogs in different geometries (projective, Möbius, Laguerre, Lie). We find the core of the theory to be a novel projective characterization of discrete isothermic nets as Moutard nets. The latter belong to projective geometry and are nets with planar faces defined through a five-point property: a vertex and its four diagonal neighbors span a three dimensional space. Analytically this property is equivalent to the existence of representatives in the space of homogeneous coordinates satisfying the discrete Moutard equati…
Isothermic nets with spherical parameter lines from discrete holomorphic maps
本文证明了具有平面或球面曲率线的离散等温网可通过提升折叠从特殊离散全纯映射获得，该新方法推广并离散化了经典光滑曲面理论，并为构造离散等温拓扑环面提供了有效途径。
The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.
Isothermic surfaces and conservation laws
本文研究了3维空间形式中CMC曲面的矩类与等温曲面可积系统理论的上同调类之间的联系，并证明二者均源于变分的Noether电流。
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients introduced previously by Bobenko and Suris for isothermic nets. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case.
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 net…
Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality for cmc-1 surfaces are interpreted in terms of transformations of isothermic surfaces. A new Weierstrass type representation is introduced and a Moebius geometric characterization of cmc-1 surfaces in hyperbolic space and minimal surfaces in Euclidean space is given.
We consider the generalization of classical Blaschke's Problem to higher codimension case, characterizing Darboux pair of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere congruence and conformally correspond to each other. When the sphere congruence is the mean curvature spheres of one envelop surface, it must be a cmc-1 surface in hyperbolic 3-space, or a S-Willmore surface. A study of conformally immersed surface pairs is indicated with discussion on the geometric meaning of new invariants.
We provide a convincing discretisation of Demoulin's $Ω$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.
We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interestin…
We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. Bäcklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a construction for discrete cmc surfaces of revolution in space forms is provided.
Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory better, it is described how to give the classical characterization a rigorous meaning in the sense of modern differential geometry.
Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces
本文肯定回答了离散层面s-嵌入提升为极大曲面的问题，通过引入s-嵌入与相切Lorentz球面汇的对应关系，识别出提升为S-等温曲面的等温s-嵌入子类，为后续构造离散极大曲面奠定基础。
Discrete Koenigs nets, inscribed quadrics and autoconjugate curves
本研究推广离散Koenigs网的内接圆锥曲线结论，证明其存在高维内接二次曲面，研究Koenigs d网格的全局切触性质，并建立其与离散自共轭曲线对的双射关系，丰富离散微分几何理论。
Constant mean curvature surfaces from ring patterns: Geometry from combinatorics
本文利用正交相交圆的球填充定义了三维欧几里得和洛伦兹空间中的离散常平均曲率曲面，提出变分原理求解边界值问题，并基于纯组合数据构造经典曲面的离散类比，推广了Koebe多面体的离散极小曲面理论。
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting orthogonal Koenigs nets). Our construction is based on an integrable discretization of the Euler-Poisson-Darboux equation and leads to discrete nets with the separability property, with all two-dimensional…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。