arXiv subjects

Friedrich Bauermeister

Publications and source records attributed to Friedrich Bauermeister.

4 recordsLinked to original sources

arXiv2026

Gordian split links in the Gehring ropelength problem

A thick link is a link in $\mathbb{R}^3$ such that each component of the link lies at distance at least $1$ from every other component. Strengthening the notion of thickness, a thickly embedded link is a thick link whose open radius-$\tfrac{1}{2}$ normal disk bundles of all components are embedded. A thick homotopy is a link homotopy of a thick link that preserves thickness and total length throughout. A thick isotopy is a link isotopy of a thickly embedded link that preserves thick-embeddedness and total length throughout. We construct an isotopically Gordian split link, that is, a thickly embedded 4-component link which is topologically split but which cannot be split by a thick isotopy. This is the first time a Gordian split link is shown to exist in this most permissive setting where length trading between components is allowed. We then prove for the first time that local, non-global minima for Gehring ropelength exist. In particular, we construct a 2-component homotopically Gordian unlink, that is, a link in the link homotopy class of the unlink which cannot be split by any thick homotopy.

math.GT↗

arXiv2026

Refocusing spacetimes need not be strongly refocusing

We prove that there are globally hyperbolic spacetimes which are refocusing but not strongly refocusing. In fact, every globally hyperbolic strongly refocusing spacetime of dimension at least $3$ admits globally hyperbolic metrics which are refocusing but not strongly refocusing. This answers a question by Chernov, Kinlaw, and Sadykov. We then prove that globally hyperbolic spacetimes which are Legendrian refocusing (a notion introduced in this paper) admit globally hyperbolic strongly refocusing metrics. As a corollary, a contact Bott-Samelson type result by Frauenfelder, Labrousse, and Schlenk can be applied to Legendrian refocusing spacetimes to show that the Cauchy surface of a globally hyperbolic Legendrian refocusing spacetime of dimension at least $3$ is compact, that its fundamental group is finite, and that its universal cover has the integral cohomology ring of a compact rank one symmetric space (CROSS).

math.DG↗

arXiv2026

Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds

Let $(M,h)$ be a connected, complete Riemannian manifold, $x\in M$, and $l>0$. Then $M$ is called a $Z^x$ manifold if all geodesics starting at $x$ return to $x$, and it is called a $Y^x_l$ manifold if every unit-speed geodesic starting at $x$ returns to $x$ at time $l$. It is unknown whether there are $Z^x$ manifolds that are not $Y^x_l$ manifolds for any $l>0$. By the Bérard-Bergery theorem, any $Y^x_l$ manifold of dimension at least $2$ is compact with finite fundamental group. We prove the same result for $Z^x$ manifolds $M$ for which all unit-speed geodesics starting at $x$ return to $x$ in uniformly bounded time. We also prove that any $Z^x$ manifold $(M,h)$ with $h$ analytic is a $Y^x_l$ manifold for some $l>0$. We start by defining a class of globally hyperbolic spacetimes (called observer-refocusing) such that any $Z^x$ manifold is the Cauchy surface of some observer-refocusing spacetime. We then prove that under suitable conditions the Cauchy surfaces of observer-refocusing spacetimes are compact with finite fundamental group, and we show that analytic observer-refocusing spacetimes of dimension at least $3$ are strongly refocusing. We end by stating a contact-theoretic conjecture analogous to our results in Riemannian and Lorentzian geometry.

math.DG↗

arXiv2025

Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection

We show that strata of oriented toric hyperplane arrangements are in bijection with a collection of lattice points in a zonotope. Moreover, we relate the dimension of the stratum and the dimension of the minimal face of the zonotope containing the corresponding lattice point. We discuss how this correspondence is related to toric varieties and the Bondal-Thomsen generators of their derived categories.

math.CO↗