首页/论文/1010.1978

arXiv 1010.19782010-10-11math.DG

Discrete constant mean curvature surfaces via conserved quantities

Wayne Rossman

备注：: survey article, 130 pages
发表信息：: MI Lecture Note Series 25, 2010

英文摘要：

This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way.

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

Polynomial Conserved Quantities of Lie Applicable Surfaces

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and $L$-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.

2017-07-06math.DG

Polynomial conserved quantities for constrained Willmore surfaces

We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature surfaces. We show that this hierarchy is preserved under both spectral deformation and Baecklund transformation, for special choices of parameters, defining, in particular, transformations of constant mean curvature surfaces into new ones, with preservation of the mean curvature, i…

2019-03-28math.DG

Formal conserved quantities for isothermic surfaces

Isothermic surfaces in $S^n$ are characterised by the existence of a pencil $\nabla^t$ of flat connections. Such a surface is special of type $d$ if there is a family $p(t)$ of $\nabla^t$-parallel sections whose dependence on the spectral parameter $t$ is polynomial of degree $d$. We prove that any isothermic surface admits a family of $\nabla^t$-parallel sections which is a formal Laurent series in $t$. As an application, we give conformally invariant conditions for an isothermic surface in $S^3$ to be special.

2013-01-03math.DG

共形球面上特殊曲线的守恒量

Conserved quantities of distinguished curves on conformal sphere

本文研究了共形球面上共形圆的两种推广，分别基于特殊平行tractor和共形Mercator方程，通过Lagrangian形式主义计算守恒量，并揭示了两者守恒量之间的关系。

2025-04-26math-phmath.DGmath.MP

Conserved quantities on multisymplectic manifolds

Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved under transgression to spaces of maps into M. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. We show that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family o…

2016-10-18math-phmath.DGmath.DS

Conserved quantities on asymptotically hyperbolic initial data sets

In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previous ones to validate a Bondi-type mass loss formula. The Lorentz group acts on the asymptotic infinity through the exchange of foliations by coordinate spheres. For foliations aligning with the total energy-momentum vector, we prove that the limits of quasi-local center of mass an…

2014-09-05gr-qcmath.DG

更多相关论文收起更多论文

On symmetries and conserved quantities in Nambu mechanics

In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number. The origin of the difference may be traced back to a shift in degrees of relevant forms present in equations of motion, or, alternatively, to a corresponding shift in degrees of relevant objects in action integral for Nambu mechanics.

2013-06-25math-phmath.DGmath.DS

Conserved Quantities of harmonic asymptotic initial data sets

In the first half of this article, we survey the new quasi-local and total angular momentum and center of mass defined in [9] and summarize the important properties of these definitions. To compute these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is rather difficult in general. We found a large family of initial data sets on which such a calculation can be carried out effectively. These are initial data sets of harmonic asymptotics, first proposed by Corvino and Schoen to solve the full vacuum constraint equation. In the secon…

2014-09-17gr-qcmath.DG

Conserved quantities in General Relativity: the case of initial data sets with a noncompact boundary

It is well-known that considerations of symmetry lead to the definition of a host of conserved quantities (energy, linear momentum, center of mass, etc.) for an asymptotically flat initial data set, and a great deal of progress in Mathematical Relativity in recent decades essentially amounts to establishing fundamental properties for such quantities (positive mass theorems, Penrose inequalities, geometric representation of the center of mass by means of isoperimetric foliations at infinity, etc.) under suitable energy conditions. In this article I first review certain aspects of this classical…

2021-03-10gr-qcmath.DG

Conserved quantities in general relativity -- the view from null infinity

In general relativity, an idealized distant observer is situated at future null infinity where light rays emitted from the source approach. This article concerns conserved quantities such as mass, energy-momentum, angular momentum, and center of mass at future null infinity. The classical definitions of Bondi mass at future null infinity ascertains the mass radiated away in gravitational waves distinctively. However, the same question for other conserved quantities such as angular momentum has been a subtle issue since the discovery of "supertranslation ambiguity" in the 1960's. Recently, new …

2022-04-08gr-qcmath-phmath.DG

非保守 Herglotz 变分问题的不变 Euler-Lagrange 方程与守恒量

Invariantised Euler-Lagrange equations and conserved quantities for nonconservative Herglotz variational problems

Federal University of Catalao（卡塔劳联邦大学）；University of Aveiro（阿威罗大学）；Federal University of Ceara（塞阿拉联邦大学）

本文利用活动标架和不变量变分法，导出了一维 n 阶 Herglotz 变分问题的广义 Euler-Lagrange 方程及其守恒量的结构，并应用于受限 Lorentz 群 SO^+(1,2) 不变的问题。

2026-05-27math-phmath.DGmath.MP

Conserved quantities and Hamiltonization of nonholonomic systems

This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic dynamics and carrying an almost symplectic foliation (determined by the common level sets of the first integrals). In appropriate coordinates, this 2-form is shown to agree with the one previously introduced locally in [34]. We use our coordinate-free viewpoint to study various g…

2019-03-30math-phmath.DGmath.MP