arXiv 0803.07162008-03-05math.DG

Discrete holomorphic geometry I. Darboux transformations and spectral curves

Christoph Bohle, Franz Pedit, Ulrich Pinkall

发表信息：: J. Reine Angew. Math. 637 (2009), 99-139
DOI：: 10.1515/CRELLE.2009.092

英文摘要：

Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over "discrete surfaces", by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Moebius invariant and based on linear equations. In the special case of maps into the 2-sphere we obtain a reinterpretation of the theory of complex holomorphic functions on discrete surfaces introduced by Dynnikov and Novikov. As an application of our theory we introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. This Darboux transformation can be interpreted as the space- and time-discrete Davey-Stewartson flow of Konopelchenko and Schief.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

Darboux charts around holomorphic Legendrian curves and applications

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We…

2017-06-16math.CVmath.DG

Backlund Transformations for Darboux Integrable Differential Systems

We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.

2014-07-10math.DG

Minimal Darboux transformations

We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.

2016-02-22math.DG

Infinitesimal Darboux transformation and semi-discrete mKdV equation

We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg-de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux transformations that include the aforementioned motions, and also give an alternate geometric interpretation for establishing the semi-discrete potential mKdV equation.

2022-05-29math.DG

Lorentz-Minkowski平面中类空曲线的Darboux变换

Darboux transformations of spacelike curves in the Lorentz-Minkowski plane

该研究借助Penrose图共形紧化，考察Lorentz-Minkowski平面中类空曲线的Lorentz-Darboux变换，揭示其奇点与爆破等独特性质，并讨论类型变化曲线。

2023-12-06math.DG

Darboux transforms and simple factor dressing of constant mean curvature surfaces

We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N is the Gauss map of a classical Darboux transform f. More generally, for all complex parameter the transformation on the harmonic Gauss map of f is induced by a (generalized) Darboux transformation on f. We show that this operation on harmonic maps coincides with simple factor dre…

2010-09-27math.DG

更多相关论文收起更多论文

The Bianchi-Darboux transform of L-isothermic surfaces

We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutability theorem for iterated Bianchi-Darboux transforms.

2000-01-11math.DG

Backlund Transformations for Darboux Integrable Differential Systems: Examples and Applications

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of Bäcklund transformations. We then showed how this construction can be applied to construct Bäcklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new applications which demonstrate the theory. In particular our final example demonstrates how our group theoretical approach produces all the Bäcklund transformations in arXiv:0707.4408. We also prove, using group methods, a non-existence theorem for Bäcklund transformatons which d…

2015-09-01math.DG

Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations

We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of the wave equation for the invariants of the G-structure associated to the Backlund transformation. The other direction constructs Backlund transformations for Darboux integrable equations as solutions of an involutive exterior differential system. Explicit transformations are give…

2008-06-27math.APmath.DG

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, …

2017-02-24math-phmath.DGmath.MP

Darboux transformations for a 6-point scheme

We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6-point difference scheme. The scheme is appropriate to solving a hyperbolic type initial-boundary value problem. We discuss several reductions and specifications of the transformations as well as construction of other Darboux covariant schemes by means of existing ones. In particular we introduce a 10-point scheme which can be regarded as the discretization of self-adjoint hyperbolic equation.

2006-06-08math.DGnlin.SI

Darboux可积椭圆系统的几何学

The Geometry of Darboux Integrable Elliptic Systems

Utah State University（犹他州立大学）；College of Charleston（查尔斯顿学院）

本文研究了通过复Pfaffian系统H的全纯解构造实椭圆微分系统，并证明其具有Darboux可积性。

2026-02-12math.APmath.DG

Discrete mKdV equation via Darboux transformation

We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.

2022-03-05math.DGnlin.SI

Invertible Darboux Transformations

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an in…

2013-01-04math-phmath.DGmath.MP

Harmonic maps, Backlund-Darboux transformations and "line soliton" analogs

Harmonic maps from $\BR^2$ or one-connected domain $Ø\subset \BR^2$ into $GL(m, \BC)$ and $U(m)$ are treated. The GBDT version of the Bäcklund-Darboux transformation is applied to the case of the harmonic maps. A new general formula on the GBDT transformations of the Sym-Tafel immersions is derived. A class of the harmonic maps similar in certain ways to line-solitons is obtained explicitly and studied.

2006-06-20math.DGnlin.SI

Darboux transformations for differential operators on the superline

We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary line.

2015-05-19math-phmath.CAmath.DG

Proof of the Completeness of Darboux Wronskian Formulae for Order Two

Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux Transformations of total order two.

2012-06-02math-phmath.APmath.DG

Sequences of Levy Transformations and Multi-Wroński Determinant Solutions of the Darboux System

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wroński determinants. These formulae are for the tangent vectors, Lamè coefficients, rotation coefficients and points of the surface.

1997-07-21dg-gamath.DGnlin.SI

Reduced Vectorial Ribaucour Transformation for the Darboux-Egoroff Equations

The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff reduction. We also show that a permutability property holds for all these transformations. Finally, as an example, we apply these transformations to the Cartesian background.

1998-05-11math-phmath.DGmath.MP

The Multidimensional Darboux Transformation

A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n greater than one. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.

1996-12-10dg-gahep-thmath-ph