Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.07738

Envelopes created by pseudo-circle families in the Minkowski plane

Yongqiao Wang·Lin Yang·Yuan Chang·Pengcheng Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we address the topic of envelopes created by pseudo-circle families in the Minkowski plane, which exhibit some different properties when compared with the Euclidean case. We provide solutions to all four basic problems associated with these envelopes, namely the existence problem, representation problem, problem on the number of envelopes, and problem on the relationships of definitions.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yongqiao Wang, Lin Yang, Yuan Chang, Pengcheng Li. 2024-07-10. Envelopes created by pseudo-circle families in the Minkowski plane. https://arxiv.org/abs/2407.07738

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Curvature equations coupling symmetric tensors with a metric

There are described hierarchies of equations coupling a metric with a trace-free tensor having prescribed symmetries and in the kernel of certain generalized gradients. These specialize, when the tensor vanishes identically, to the usual hierarchy of constant sectional curvature (projectively flat), Einstein, and constant scalar curvature. At the Ricci curvature level these equations are formal analogues of the Einstein-Maxwell and supergravity equations that couple differential forms with a metric. The particular cases coupling a metric with trace-free symmetric tensors satisfying the Codazzi or conformal Killing equations are studied in detail. Examples of solutions are obtained from mean curvature zero immersions, affine spheres, isoparametric hypersurfaces, and related algebraic constructions. The formalism yields a hierarchy of curvature equations for statistical structures. There are deduced constraints on the scalar curvature of the metric occurring in a solution that generalize classical results of Simons, for mean curvature zero hypersurfaces in spheres, and of Calabi, for hyperbolic affine spheres.

math.DG↗
arXiv2026

The signature of geometrically decomposable aspherical 4-manifolds

We construct examples of geometrically decomposable aspherical 4-manifolds with non-zero signature. We show that all such 4-manifolds satisfy the inequality (of Bogomolov--Miyaoka--Yau type) $χ\geq 3|σ|$. We also construct examples attaining the equality that are non-geometric and have non-zero signature. Finally, we prove that for higher graph 4-manifolds, with complex-hyperbolic vertices, the strict inequality always holds. Moreover, we construct infinitely many examples of higher graph 4-manifolds with non-zero signature and prove that the inequality is strict and sharp in this class.

math.DG↗
arXiv2026

Proper affine deformations of positive representations

We define for every positive Anosov representation of a nonabelian free group into $\mathrm{SO}(2n,2n-1)$ a family of $\mathbb{R}^{4n-1}$-valued cocycles which induce proper affine actions on $\mathbb{R}^{4n-1}$. We construct fundamental domains in $\mathbb{R}^{4n-1}$ bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.

math.DG↗