arXiv subjects

Maria Trnkova

Publications and source records attributed to Maria Trnkova.

7 recordsLinked to original sources

arXiv2026

Verified Length Spectrum and Margulis number for Hyperbolic 3-Manifolds

We present a new algorithm to compute a stream for the length spectrum of an oriented, finite-volume hyperbolic 3-manifold $M$. The advantage over the existing algorithm by Hodgson-Weeks is that our algorithm does not require finding a Dirichlet domain and thus can use interval arithmetic to verify the result. Our algorithm is also significantly faster in some cases. We also present a new algorithm to compute the optimal Margulis number of $M$. The authors have made these algorithms available in SnapPy.

math.GT↗

arXiv2020

Quadratic differentials, measured foliations and metric graphs on punctured surfaces

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic direction at each pole, and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue of a theorem of Gardiner-Masur, for meromorphic quadratic differentials. We also prove an analogue of the Hubbard-Masur theorem, namely, for a fixed punctured Riemann surface there exists a meromorphic quadratic differential with any prescribed horizontal foliation, and such a differential is unique provided we prescribe the singular-flat geometry at the poles.

math.GT↗

arXiv2020

Approximating Surfaces in $R^3$ by Meshes with Guaranteed Regularity

We study the problem of approximating a surface $F$ in $R^3$ by a high quality mesh, a piecewise-flat triangulated surface whose triangles are as close as possible to equilateral. The MidNormal algorithm generates a triangular mesh that is guaranteed to have angles in the interval $[49.1^o, 81.8^o]$. As the mesh size $e\rightarrow 0$, the mesh converges pointwise to $F$ through surfaces that are isotopic to $F$. The GradNormal algorithm gives a piecewise-$C^1$ approximation of $F$, with angles in the interval $[35.2^o, 101.5^o]$ as $e\rightarrow 0$. Previously achieved angle bounds were in the interval $[30^o, 120^o]$.

cs.CG↗

arXiv2019

Hyperbolic Flowers

Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This approach helps to understand various properties of hyperbolic geometry that are demonstrated in the paper: a sum of angles and a relation between edges and angles in a hyperbolic triangle, tiling of a hyperbolic plane, ratio of the circumference to the radius of a hyperbolic disc and even Nash-Kuiper embedding theorem. Oriented on students learning basics of Riemannian geometry.

math.HO↗

arXiv2013

Exceptional hyperbolic 3-manifolds

We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of radius log(3)/2.

math.GT↗

arXiv2009

On models of non-Eucludian spaces generated by associative algebras

We present the non-trivial example how to generate non-Euclidean geometries from associative unital algebras. We consider bundles of the sphere of the degenerate non-Eucleadian space and its two models. The first (conformal) model is obtained by the mapping S onto a plane pass through the origin. It is analogous to the stereographic mapping. The second model (projective) is con- structed by the Norden normalization method, where we project the sphere onto a plane of normalization defining the metric and Christoffel symbols which allow us to find geodesic curves.

math.DG↗

arXiv2008

On principal fibrations associated with one algebra

In this paper we study two types of fibrations associated with a 3-dimensional unital associative irreducible algebra and their basic properties. We investigate trivial principal fibrations of degenerate semi-Euclidean sphere and their semi-conformal and projective models. We use Norden normalization method for constructing second model.

math.DG↗