不定非正常仿射球面的一个表示公式

凸平面曲线与广义仿射球面的面积距离

Area distances of Convex Plane Curves and Improper Affine Spheres

到凸平面曲线的面积距离是计算机视觉中的一个重要概念。在本文中，我们描述了一个强有力的区域距离和不适当的仿射球之间的联系。这种联系使得更好地理解这两种理论成为可能。仿射球面理论的概念导致凸平面弧的外部部分上的面积距离的新定义。同时，基于离散仿射球理论，我们提出了计算区域距离的快速算法。另一方面，面积距离提供了一个很好的几何理解不适当的仿射球。

反常仿射球面和常高斯曲率曲面的奇异性

Singularities of improper affine spheres and surfaces of constant Gaussian curvature

本文从切触点几何的角度研究了反常（抛物）仿射球方程，给出了方程几何解中出现的奇异性的一般分类及其奇异性的一般形式。我们也给出了常高斯曲率曲面和可展曲面的结果。特别是，我们确认，一般奇点出现在这样的表面只是尖形的边缘和燕尾。

具有低总曲率奇异的完备非固有仿射球及新例子的分类

The classification of complete improper affine spheres with singularities of low total curvature and new examples

本文研究了幺模仿射三维空间中带奇异性的完备非固有仿射球，给出了总曲率大于等于$-6π$及$-8π$情形的分类，并提供了亏格0和1的新例子及嵌入端的渐近行为分析。

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in $n$-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the $n$th power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood-Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the …

具有不定度量的离散仿射极小曲面

Discrete Affine Minimal Surfaces with Indefinite Metric

受不确定度量的光滑仿射极小曲面的Weierstrass表示的启发，我们提出了一个构造过程，产生了一大类离散曲面，我们称之为离散仿射极小曲面.我们证明了它们是定义在四边形离散曲面空间上的仿射面积泛函的临界点。该构造利用渐近坐标，并允许定义某些微分几何对象的离散模拟，如法向量场和余法向量场、三次型和相容性方程。

Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric

Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.

离散不定仿射极小曲面的奇点

Singularities of Discrete Indefinite Affine Minimal Surfaces

本文研究离散不定仿射极小曲面的奇点，通过离散Lelieuvre公式定义离散渐近网的奇异边和顶点，使光滑情形中奇异集的关键性质在离散情形下仍然保持。

广义仿射球面高斯映射的Value distribution

Value distribution of the Gauss map of improper affine spheres

给出了仿射三维空间中完全非正常仿射阵面的拉格朗日高斯映射的例外值个数的最佳可能上界。对于弱完备情形，我们也得到了尖锐的估计.作为这一结果的应用，我们给出了非正常仿射球面的参数仿射Bernstein问题的一个新的简单证明。此外，我们还得到了三维双曲空间中弱完备平坦锋的标准形之比的相同估计。

允许点态对称的不定仿射超球面。第2部分

Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

一个仿射超曲面$M$被称为允许点态对称，如果对于所有的$p\in M$，存在${\rm Aut}(T_p M)$的子群$G$，它保持（点态）仿射度量$h$，差分张量$K$和仿射形状算子$S$。这里，我们考虑三维不定仿射超球面，即$S= HId$（因此$S$是平凡保持的）。在第一部分中，我们找到了可能的对称群$G$，并对每个$G$给出了$K$的标准型。我们开始一个分类表明，超球体承认逐点${\mathbb Z}_2\times {\mathbb Z}_2$ resp。${\mathbb R}$ -对称性是众所周知的，它们分别具有恒定的截面曲率和Pick不变量$J<0$。J=0。在这里，我们继续研究具有逐点${\mathbb Z}_3$ -或SO（2）-对称性的仿射超球面。它们是仿射球面（${\mathbb Z}_3$）或二次曲面（SO（2））与曲线的翘曲积。

A variational setting for an indefinite Lagrangian with an affine Noether charge

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold. Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function. Our setting includes different types of Lorentz-Finsler Lagrangians admitting a timelike Killing vector field.

允许点态对称的不定仿射超球面

Indefinite affine hyperspheres admitting a pointwise symmetry

一个仿射超曲面M称为点态对称，如果对M中的所有p，存在Aut（T_p M）的子群G，它（点态）保持仿射度量h，差张量K和仿射形状算子S.在这里，我们考虑三维不定仿射超球面，即S= H Id（因此S是平凡保持的）。首先我们解决一个代数问题。在等距群SO（1，2）的作用下，我们确定了Lorentz-Minkowski空间R^3_1上无迹三次型的非平凡稳定子G，并找到了每个SO（1，2）/G-轨道的代表。由于仿射三次型是由h和K定义的，这给了我们可能的对称群G和每个G的K的标准型。其次，我们对所有非平凡稳定子G（除Z_2外）具有点态G-对称性的超球面进行了分类。除了著名的超球（对于Z_2 x Z_2 resp. R的超球面具有常截面曲率和Pick不变量J<0。J=0，我们得到了丰富的新的例子，例如，二维仿射球的翘曲积结构（分别）。二次曲面）和曲线。此外，我们还找到了一种由二维二次曲面或正定仿射球面构造不定仿射超球面的方法。

Singular Improper Affine Spheres from a given Lagrangian Submanifold

Given a Lagrangian submanifold $L$ of the affine symplectic $2n$-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension $2n$, both of whose sets of singularities contain $L$. Although these improper affine spheres (IAS) always present other singularities away from $L$ (the off-shell singularities studied in our previous paper), they may also present singularities other than $L$ which are arbitrarily close to $L$, the so called singularities "on shell". These on-shell singularities possess a hidden $\mathbb Z_2$ symmetry that is absent fr…

偶数维反常仿射球面

Even Dimensional Improper Affine Spheres

二维广义仿射球面有两种类型：非凸球面可以用平面曲线对的中心弦变换来表示，而凸球面可以用全纯映射来表示。本文证明了这两种构造都可以推广到任意偶数维：前者对应于一对拉格朗日子流形的中心弦变换，而后者则与特殊的Kähler流形有关.此外，我们还证明了用这种方法得到的广义仿射球面是某些外微分系统的解。最后，我们还讨论了简单稳定的勒让德奇点作为这些反常仿射球面奇点的实现问题。

一类带奇点的仿射极大曲面及其与极小曲面理论的关系

A class of affine maximal surfaces with singularities and its relationship with minimal surface theory

本文定义了带奇点的仿射极大曲面子类 affine maxfaces，证明其满足 Osserman 型不等式，并揭示了其与 Euclidean 极小曲面理论的关系。

Discrete Affine Surfaces based on Quadrangular Meshes

In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-norma…

Crofton Formulas and Indefinite Signature

We study the $O(p,q)$-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an $O(p,q)$-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the $(p-1)$ homogeneous case of $O(p,p)$, yielding a Crofton formula for the centro-affine surface area when $p\not\equiv 3\mod 4$. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly…