Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.17116

Jordan curves inscribe a positive measure of rectangles

Joshua Evan Greene·Andrew Lobb
View original source ↗Read source document ↗Export metadata ↓

Abstract

Suppose that $γ\subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,π)$ of measure at least $A/R^2$ such that if $θ\in I$, then there exist four points on $γ$ at the vertices of a rectangle whose diagonals meet at angle $θ$.

Explore related subjects

math.GTmath.COmath.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joshua Evan Greene, Andrew Lobb. 2026-04-18. Jordan curves inscribe a positive measure of rectangles. https://arxiv.org/abs/2604.17116

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

KEEP EXPLORING

Related papers

arXiv2026

Bending parameterization of one-sided degenerate Kleinian surface groups

It was recently proved that quasi-Fuchsian manifolds are uniquely determined by their bending laminations. This paper concerns a similar result for certain non-quasi-Fuchsian manifolds~: those obtained by degenerating one end but not the other, i.e. those appearing in boundaries of Bers slices. More precisely, we show that such hyperbolic manifolds are uniquely determined by the end structure of the degenerated end and the bending lamination of the other. The end structure consists of the parabolic locus, which is a multicurve, together with ending laminations or conformal structures on each components of its complement.

math.GT↗
arXiv2026

Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact $3$-Manifolds with Boundary

In this paper, we study a combinatorial Ricci flow on closed pseudo $3$-manifolds $(M,\mathcal{T})$. We prove that if every edge in the triangulation $\mathcal{T}$ has valence at least $9$, then the combinatorial Ricci flow converges exponentially fast to the unique zero-curvature hyper-ideal metric. As a consequence, for any compact $3$-manifold $N$ with boundary admitting an ideal triangulation $\mathcal{T}_N$ whose edges all have valence at least $9$, there exists a unique complete hyperbolic metric with totally geodesic boundary on $N$ such that $\mathcal{T}_N$ is isotopic to a geometric decomposition of $N$. This provides a partial solution to the conjecture of Costantino, Frigerio, Martelli and Petronio, and hence an affirmative answer to Thurston's geometric ideal triangulation conjecture for such manifolds. Moreover, we obtain explicit upper and lower bounds for the resulting hyperbolic metric.

math.GT↗