Keisuke Teramoto
We show relation between sign of Gaussian curvature of cuspidal edge and geometric invariants through types of singularities of Gauss map. Moreover, we define and characterize positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and show relation between sign of cusps and of the Gaussian curvature.
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We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularities types in terms of differential geometry (Theorem 3.4 and 3.5).
Cuspidal crosscaps and folded singularities on a maxface and a minface
该研究针对洛伦兹闵可夫斯基空间中带折叠奇点的零平均曲率曲面，构造了含递增尖点交叉帽且收敛于该曲面的极大面与极小面族，并做了相关一般性讨论。
Geometric deformations of cuspidal $S_1$ singularities
本研究基于作者前期得到的$S_1$奇点几何形变表示形式，给出其为frontal映射的充要条件，进而得到尖点型$S_1$奇点的形变表示，并借此研究该类奇点及形变中尖点交叉帽的几何性质。
We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.
We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the di…
Singularities of generalized timelike minimal surfaces in Lorentz-Minkowski 3-space
本文研究Minkowski 3维空间中类时极小曲面的奇点，证明其存在性与不存在性定理，给出多类奇点的判别准则、对偶与不变性定理及例子。
We study the geometry of cuspidal $S_k$ singularities in $\mathbb R^3$ obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We study geometrical invariants associated to $M$ and show that they determine it up to order 5. We then study the flat geometry (contact with planes) of a generic cuspidal cross-cap by classifying submersions which preserve it and relate the singularities of the resulting height functions with the geometric invariants.
Parallel Surfaces of Cuspidal Cross Caps
本文研究尖点交叉帽（基本非前额正面奇点）的平行曲面的几何与奇点，通过距离平方函数的退化准则描述奇点配置，并证明在特定距离下平行曲面退化为退化尖点S1奇点，该距离类比于主曲率半径。
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometric properties of initial cuspidal edges.
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges and types of singularities of Gauss maps of cuspidal edges. Further, we consider extended height functions on fronts with non-degenerate singular points.
Cuspidal edges on focal surfaces of regular surfaces
本文研究了正则曲面焦点曲面上尖点边的几何不变量，通过分析满足特定条件的平行曲面的奇异性来明确尖点边处奇异性曲率的符号。
Singularities on timelike minimal surfaces in Lorentzian Heisenberg group
研究了洛伦兹希格纳群中时状极小曲面的构造及其奇点特性，通过调和映射和具体例子探讨了不同奇点类型。
We give necessary and sufficient conditions on the singular Björling data to the singular Björling problem's solution has a prescribed nature of singularity. As an application, we prove that near a maxface with a particular type of singularity, there is another maxface having a cuspidal-edge. An example of a maxface having various types of singularities is also given.
It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we shall prove that generically flat fronts in the hyperbolic 3-space admit only cuspidal edges and swallowtails. Moreover, we will show that every complete flat front (which is not rotationally symmetric) has associated …
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the geometric Cauchy data for the generic singularities, as well as for the cuspidal beaks and cuspidal butterfly singularities. We consider the bifurcations of generic one parameter families of spherical fronts and provide evidence that suggests that these are the cuspidal beaks, cu…
In this paper, we study the singularities of spacelike constant mean curvature one (CMC 1) surfaces in the de Sitter 3-space. We prove the duality between generalized conelike singular points and 5/2-cuspidal edges on spacelike CMC 1 surfaces. To describe the duality between $A_{k+3}$ singularities and cuspidal $S_k$ singularities, we introduce two invariants, called the $α$-invariant and $σ$-invariant, of spacelike CMC 1 surfaces at their singular points. Moreover, we give a classification of non-degenerate singular points on spacelike CMC 1 surfaces.
We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized su…
Bour's theorem for helicoidal surfaces with singularities
该研究推广Bour定理的技术，证明欧氏3维空间中螺旋运动不变的一般尖点棱、n型棱存在非平凡等距形变，并得出多个相关几何不变量为外蕴不变量的结论。
Maximal surfaces in the Lorentzian Heisenberg group
本文研究洛伦兹海森堡群中平均曲率为零的类空曲面，通过调和映射和圈群方法刻画其一般奇点类型，并证明类光边界正则极大圆盘至少含两个尖交叉帽奇点。
Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field $ν$ is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regular space curve) is not orthogonal to $ν$. This genericity is equivalent to the condition that its limiting normal curvature $κ_ν$ takes a non-zero value. In this paper, we show that a given generic (real analytic) cuspidal edge can be isometrically deformed preserving $κ_ν$ into …
In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.
The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.
We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curvature (resp. $μ_Π$ called normalized product curvature) of $κ_c$ (resp. $μ_c$) and the limiting normal curvature $κ_ν$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $κ_Π$ whe…
We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails.
We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvatureand for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.
Singularities of Discrete Improper Indefinite Affine Spheres
本文研究基于渐近网的离散非正常仿射球面，区分了其中的奇异边与奇异顶点并给出对应奇点类型，为离散网奇点分类提供初步进展，同时证明了直纹离散非正常仿射球面的若干与光滑情形类似的刻画。
Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in $\boldsymbol{R}^N$
本文提出构建有限乘数曲线奇点准则的系统方法，给出乘数二、三、四的显式准则，包括仅出现在三维及以上空间的特定尖点，并推广标准化曲率函数和尖点曲率到R^N中的奇异曲线，重新解释Fukui在R^N中有限乘数曲线的存在唯一性定理。
We study singularities and geometric properties of surfaces given by the singular loci of normal congruence of frontals with pure-frontal singular points. These surfaces consist of the normal ruled surface and focal surfaces of the initial frontal. For the normal ruled surface, we give characterizations of singularities in terms of geometric invariants of the initial frontal defined on the set of singular points. For focal surfaces, we show relation between certain singularities of them and geometric property of the given frontal. Moreover, we consider behavior of Gaussian curvature of focal s…
Gauss-Bonnet formula for metrics with logarithmic singularities on compact Riemann surfaces
该研究在高斯曲率关于度量面积形式Lebesgue可积的条件下，推广了紧黎曼曲面上带对数奇异性度量的Gauss-Bonnet公式，并将其应用于关联三次微分亚纯的特殊Kähler度量。
It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,ν)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this pape…
In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface which admits some kind of singular points.
Geometry of $D_4^-$-front singularities and a Gauss-Bonnet type formula
通过源坐标变换和目标等距构造$\R^3$中前缘的$D_4^-$奇点形式，计算其微分几何不变量，并给出单参数一般前缘的高斯-博内型定理。
On singularities of the Gauss map components of surfaces in ${\mathbb R}^4$
本文研究$\mathbb R}^4$中一般浸入曲面的 Gauss 映射两球面分量的奇点，证明其一般稳定性，并联系曲面接触类型、$\mathcal J$-全纯曲线及 Gauss-Bonnet 型拓扑几何公式。
We propose a new approach to the question of prescribing Gaussian curvature on the 2-sphere with at least three conical singularities and angles less than $2π$, the main result being sufficient conditions for a positive function of class at least $C^2$ to be the Gaussian curvature of such a conformal conical metric on the round sphere. Our methods particularly differ from the variational approach in that they don't rely on the Moser-Trudinger inequality. Along the way, we also prove a general precompactness theorem for compact Riemann surfaces with at least three conical singularities and angl…
Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surface…
We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the partial derivatives $f_x$ and $f_y$ are given by $\frac{\dd f_0}{\dd x}$ and $v$ along the curve. Within the clas…
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2π$.
Deformations of swallowtails in a 3-dimensional space form
本文通过建立欧氏三维空间中燕尾奇点的表示公式，研究了一般燕尾奇点映射芽的性质，并证明任意燕尾奇点可保号形变为常高斯曲率燕尾奇点。
In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discrete minimal and maximal surfaces, and discrete constant mean curvature $1$ surfaces in de Sitter $3$-space, including comparisons with different previously known definitions of such singularities.
We investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities on fronts. We also study the convergence to infinite limits of these geometrical invariants and show how this is tightly related to a particular property of uniform approximation of fronts by parallel surfaces.
We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a new Chern-Gauss-Bonnet formula with error terms that can be expressed as isoperimetric deficits. This is the first such formula in a dimension higher than two which allows the underlying manifold to have isolated branch points or conical singularities.
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…
Shapes of light: singularities of slant functions and the Gauss map of a surface
本文研究三维空间中正则曲面上倾斜函数的奇点，通过抛物集及其在高斯映射下的球面像的微分几何特征来刻画奇点类型。
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solutio…
On the boundedness of the geodesic curvature measure of a regular curve passing through a cross cap singularity
该研究针对内蕴交叉帽奇点相关几何问题，证明了穿过该奇点的正则曲线测地曲率测度的有界性，推广了已有Gauss-Bonnet型公式至带边界曲面情形，并给出奇点处测地曲率有界的条件。
We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A-determined map-germs from the plane to the plane can be represented by harmonic maps.
The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect in terms of singularities and tangencies at infinity.
Prescribing curvatures on surfaces with conical singularities and corners
本文研究了带锥奇点和角点的带边界紧致曲面上的高斯曲率与测地曲率预定问题，通过将新变分公式推广至奇异情形求得解，是该奇异曲面预定曲率问题的首个结果。
In this paper we first prove that every differential character can be represented by differential form with singularities. Then we lift the Gauss-Bonnet-Chern theorem for vector bundles to differential characters.
We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles less than $π$ along time-like curves), the complement of the convex core in $N$ admits a unique foliation by constant Gauss curvature surfaces. This extends, and provides a new proof of, a result of \cite{BBZ2}. We also describe a parametrization of the space of convex GHM AdS metrics on a given manifold, with particles of given angles, by the product of two copies of the Teichmüller space of hyperbolic metrics with cone singularities of …
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a…
We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with c…
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