arXiv research

Study on Gauss images of specific minimal surfaces with finite curvature.

The study constructs new minimal surfaces with more ramified values than previously known.

Unified approach to totally ramified values in various surface theories.

In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean four-space and give a kind of unicity theorem.

In this thesis, we study value distribution theoretical properties of the Gauss map of pseudo-algebraic minimal surfaces in n-dimensional Euclidean space. After reviewing basic facts, we give estimates for the number of exceptional values and the totally ramified value numbers and the corresponding unicity theorems for…

We give an effective estimate for the totally ramified value number of the hyperbolic Gauss maps of complete flat fronts in the hyperbolic three-space. As a corollary, we give the upper bound of the number of exceptional values of them for some topological cases. Moreover, we obtain some new examples for this class.

We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the …

We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a …

The paper studies Gauss maps of space-like stationary surfaces in Lorentz-Minkowski space, focusing on ramification and unicity.

New monoids tied to symmetric group and Jones/Brauer monoids discovered.

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…

In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed four-manifold X. We find that in order to have sensible solutions to these equations, only …

The study finds conditions for Kaehler-Einstein and cscK metrics on certain manifold coverings.

New algorithm for Coxeter connections with maximally ramified singularities.

Three great theorems of Thurston read: Haken manifolds are hyperbolic; big ramified coverings are hyperbolic; big surgeries are hyperbolic. Recent developments indicate that the later two theorems are essentially a corollary of the first, that is there are much more Haken manifolds than expected by Thurston. In fact Fr…

The study analyzes Dirac operators twisted by specific bundles, revealing their geometric and regularity properties.

Solves Deligne-Simpson problem for special connections on Gm.

For all $k\in]0,1[$, we construct a canonical bijection between the space of ramified coverings of the sphere and the space of complete immersed surfaces in $3$-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to $k$. We show, furthermore, that this bijection restricts to a homeomor…

We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also …

Let S be a K3 surface that admits a non-symplectic automorphism $ρ$ of order 3. We divide $S\times \mathbb{P}^1$ by $ρ\timesψ$ where $ψ$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the …

Topology of non-orientable spaces without boundary is studied.

We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X. Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witte…

The paper proves local isometric embeddings for singular metrics near a point.

New examples contradict a conjecture about knot surgeries.

We demonstrate that the question whether or not a given postcritically finite topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvat…

We discuss how the fractional Dehn twist coefficient behaves under a fully ramified branched covering of an open book, and give applications to both topological and contact 3-manifolds. Among them, we show that non-right-veering closed braids represent virtually loose transverse links.

The Birman-Hilden theory is extended to infinite type surfaces and branched covers.

There is a natural sequence of CY manifolds that are double covers of the projective g dimensional spaces ramified over 2g+2 hyperplanes. We observe that some of them are obtained as quotient of the action of the semi-direct product of g-1 copies of cyclic Z/2Z groups with the symmetric g group on the product of g-copi…

We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

Estimates curvature for holomorphic maps on Riemann surfaces.

We consider the universal family $E_n^d$ of superelliptic curves: each curve $Σ_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$ distinct points. We fin…

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the ma…

We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed…

We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gord…

Study optimality conditions for interval-valued optimization problems on Riemannian manifolds.

Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over…

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified coveri…

Study finds TVL doesn't predict cryptocurrency returns.

We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the exis…

We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $μ$ be the Euler-Poincaré measure on $PSL_2(\mathbb R)^n$ and $χ=μ/2^n$. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49…

The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagran…

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent…

This paper investigates the scaling dependencies between measures of "activity" and of "size" for companies included in the FTSE 100. The "size" of companies is measured by the total market capitalization. The "activity" is measured with several quantities related to trades (transaction value per trade, transaction val…

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…

Study on TVL computation in DeFi protocols, proposing verifiable metrics.

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a …