arXiv research

Sharp bounds on distortion of surfaces in 3D space.

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.

New bounds on knot distortion and Seifert surface properties.

New metric defines surface shapes, minimizing area and angle distortions.

Let $g \colon S \looparrowright N$ be a properly immersed $π_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of $π_1(S)$ in $π_1(N)$ and show that how it is related to separability of $π_1(S)$ in $π_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, an…

We prove that the distortion function of the Gauss map of a harmonic surface coincides with the distortion function of the surface. Consequently, Gauss map of a harmonic surface is ${\mathcal{K}}$ quasiregular if and only if the surface is ${\mathcal{K}}$ quasiregular, provided that the Gauss map is regular or what is …

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute low…

Our main result is a nontrivial lower bound for the distortion of some specific knots. In particular, we show that the distortion of the torus knot $T_{p,q}$ satisfies $δ(T_{p,q})>\frac 1{160}\min(p,q)$. This answers a 1983 question of Gromov.

Triangulates surfaces with bounded energy using diffeomorphisms.

New method corrects complex distortions in single view images.

The study finds minimal distortion embeddings of surfaces into small domains.

In this work, we are concerned with the spherical quasiconformal parameterization of genus-0 closed surfaces. Given a genus-0 closed triangulated surface and an arbitrary user-defined quasiconformal distortion, we propose a fast algorithm for computing a spherical parameterization of the surface that satisfies the pres…

QC-SPHARM detects Alzheimer's Disease early using hippocampal surface geometry.

New approach to nematic fields on surfaces, relaxing uniformity to quasi-uniformity.

We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level $d$ mapping class group is the same as that o…

Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are theoretically guaranteed [1]. Rece…

We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at least two marked points or spots.

Linear progress observed in fibered 3-manifold embeddings.

Method flattens complex surfaces with consistent density and shape.

In this article we explore some finer properties of equi-areal mirrors and introduce techniques for developing new mirror surfaces that simultaneously minimize angular and areal distortion.

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…

The paper uses thermodynamics to improve machine learning representation quality.

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orienta…

We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing …

Vertex distortion detects if a knot is unknot.

Algorithm finds optimal affine transformation to minimize overall distortion.

We study the Teichmüller metric on the Teichmüller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichmüller metric is approximated up to bounded additive distortion by the sup metric on a product of lower dimensional spac…

Vertex distortion measures how far lattice knots deviate from straight lines.

We consider the problem of distortion minimal morphing of $n$-dimensional compact connected oriented smooth manifolds without boundary embedded in $\R^{n+1}$. Distortion involves bending and stretching. In this paper, minimal distortion (with respect to stretching) is defined as the infinitesimal relative change in vol…

The paper proposes methods for volumetric parameterization of 3D solid manifolds.

This paper shows how to calculate risk measures for sums of two counter-monotonic risks.

A new family of stochastic dominance orders based on distortion functions.

Study distortion risk measures for step-weighted distributions.

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil kno…

The paper calculates subgroup distortions in 3-manifold groups.

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the…

Study on curvatures of surfaces in specific Lie groups.

Computed distortion coefficients for the α-Grushin plane.

Study on risk measures using distorted Choquet integrals with random distortions.

Sequence distortion measures large-scale distances in metric spaces.

We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, ha…

The distortion of a curve is the supremum, taken over distinct pairs of points of the curve, of the ratio of arclength to spatial distance between the points. Gromov asked in 1981 whether a curve in every knot type can be constructed with distortion less than a universal constant C. Answering Gromov's question seems to…

The study shows exponential distortion in virtually special groups containing free subgroups.

Estimates rate-distortion function for large datasets using neural networks.

Introduces new performance criteria for investment under distorted probabilities.

New coding theorem shows achievable rate matches theoretical limit.

We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.

Paper proposes a new black-box attack approach to minimize visual distortion.