F Burstall, U Hertrich-Jeromin, W Rossman, S Santos
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.
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We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
Discrete constant mean curvature cylinders and isothermic tori
本文利用离散极化曲线的可积理论解决离散等温曲面达布变换的单值化问题，并首次给出离散等温圆柱、离散常平均曲率圆柱及离散等温环面的闭式离散参数化。
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.
Weierstrass representations of discrete constant mean curvature surfaces in isotropic space
该研究获得三维迷向空间中离散常平均曲率曲面的Weierstrass表示，还借助该表示构造出带离散闭式参数化的相关实例。
Ricci flow of discrete surfaces of revolution, and relation to constant Gaussian curvature
该研究通过可积系统方法给出离散常高斯曲率旋转曲面的显式参数化，探究离散旋转曲面的Ricci流几何实现与该类常高斯曲率曲面的趋近关系。
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…
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.
We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.
This article is an application of the author's paper about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d'Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition. As an application, we draw figures of discrete constant negative Gaussian curvature surfaces given by this method.
In this paper we define the class of constant affine mean curvature (CAMC) discrete asymptotic nets, which contains the well-known classes of affine spheres and affine minimal asymptotic nets. This class is defined by considering fields of compatible interpolating quadrics, i.e., quadrics that have common tangent planes at the edges of the net. We show that, for CAMC asymptotic nets, ruled discrete asymptotic nets is equivalent to ruled compatible interpolating quadrics. Moreover, we prove discrete counterparts of some known properties of the Demoulin transform of a smooth CAMC surface.
Nonexistence of stable discrete maps into some homogeneous spaces of nonnegative curvature
本文研究了从有限加权图到光滑黎曼流形的离散映射的加权长度或能量泛函的稳定性，并证明了到特定齐性空间（如正全纯截面曲率的Kähler C-空间及部分单连通紧致黎曼对称空间）不存在稳定离散极小浸入或非常数稳定离散调和映射。
Perpetual cutoff method and discrete Ricci curvature bounds with exceptions
针对图上Bakry-Emery曲率研究需全局一致下曲率界、不允许例外集的障碍，提出永久截断方法，证明了局部曲率界下的梯度估计，并给出带例外集的一致正曲率图到例外集距离的严格上界。
Using Banchoff's discrete Morse Theory, in tandem with Bloch's result on the strong connection between the former and Forman's Morse Theory, and our own previous algorithm based on the later, we show that there exists a curvature-based, efficient Persistent Homology scheme for networks and hypernetworks. We also broaden the proposed method to include more general types of networks, by using Bloch's extension of Banchoff's work. Moreover, we show the connection between defect and Forman's Ricci curvature that exists in the combinatorial setting, thus explaining previous empirical results showin…
We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and *-scalar curvature.
Singular minimal surfaces with constant curvature
本文证明了常高斯曲率的奇异极小曲面仅为平面、球面和圆柱面，并进一步分类了常主曲率和常平均曲率的奇异极小曲面。
Constant Mean Curvature n-noids with Platonic Symmetries
Positive curvature and discrete abelian symmetry
本研究用初等阿贝尔2-群替代环面，推广了Grove、Searle的最大对称性结果与Wilking关于带等距环面作用正曲率流形的半最大对称性结果，拓展了正曲率流形对称作用的研究范畴。
Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic
本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。
Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $π$ is a spacelike 2 plane, let $R(π)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. We also classify the Ivanov-Petrova algebraic curvature tensors of rank 2; these are the algebraic curvature tensors of constant rank 2 such that the complex Jordan normal form of R(-) is constant.
In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterize…
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients.
It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.
The aim of this paper is to describe Kahler surfaces with quasi-constant holomorphic curvature
We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of $M$.
In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is $0$ or $1$ and that the surface is ruled. If the generating curve is a Lorentzian circle, we show that the only possibility is that the axis is spacelike and the center of the circle lies in the axis.
Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, t\in\mathbb{R}$. In particular, there are no topological obstructions for metrics with $\varepsilon$-pinched Weyl curvature and negative scalar curvature.
In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.
In this paper paraSasakian manifolds with a constant paraholomorphic section curvature are considered.
This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/
Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.
In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.
In this paper conditions for a Kropina structure to be of constant flag curvature are obtained.
We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.
Constant curvature rotational nets and periodic Bäcklund transforms
本文研究了常负高斯曲率旋转网的Bäcklund变换，构造了具有显式参数化的非旋转离散伪球面新例子，并给出了保持周期性的条件。
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when the initial energy is small enough. The proof relies on a careful analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix of the curvature map. As a consequence, combinatorial curvature flow provides an algorithm to compute discrete sphere packing metrics wit…
We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler metrics that have constant flag curvature.
In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.
We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.
Biconservative hypersurfaces are hypersurfaces which have conservative stress-energy tensor with respect to the bienergy, containing all minimal and constant mean curvature hypersurfaces. The purpose of this paper is to study biconservative hypersurfaces $M^n$ with constant scalar curvature in a space form $N^{n+1}(c)$. We prove that every biconservative hypersurface with constant scalar curvature in $N^4(c)$ has constant mean curvature. Moreover, we prove that any biconservative hypersurface with constant scalar curvature in $N^5(c)$ is ether an open part of a certain rotational hypersurface …
The classification of Finsler spaces of constant curvature is an interesting and important topic of research in differential geometry. In this paper we obtain necessary and sufficient conditions for generalized Kropina space to be of constant flag curvature.
We explain how the current knowledge on the set of complete noncompact constant mean curvature surfaces can be exploited to produce new examples of compact constant mean curvature surfaces of genus greater than or equal to 3.
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons w…
We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.
We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.
We show that closed starshaped hypersurfaces of space forms with almost constant mean curvature or almost constant higher order mean curvature are closed to geodesic spheres.
A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces
本文研究了曲面上高斯曲率的自然离散化，建立了离散单值化定理，并引入带手术的组合$\alpha$-Ricci流和组合$\alpha$-Calabi流，证明了其长期存在性与收敛性，从而解决了预定组合$\alpha$-曲率的带装饰分片双曲度量的存在性问题。
This paper constructs a family of constant mean curvature immersions of the thrice-punctured Riemann sphere into Euclidean 3-space with asymptotically Delaunay ends via loop group methods.
We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.
Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation (after Chen and Wu)
针对常标量曲率与常边界平均曲率方程，本文将低能量解集失去紧性的光滑反例维数阈值由 Chen 和 Wu 的不小于62维改进至不小于35维。
We investigate projective spherically symmetric Finsler metrics with constant flag curvature in $R^n$ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk are found to have constant negative flag curvature.
A logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector
该研究在微分几何领域证明了非负截面曲率流形中具常长度平均曲率向量的闭子流形满足对数Sobolev不等式，丰富了子流形几何的分析工具。
In this paper we study the behavior of the scalar curvature $S$ of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of $S$. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli and Setti \cite{PRS}.
We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.
Geodesics in Randers spaces of constant curvature are classified.
In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.
The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.
We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.
In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.
We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.
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