arXiv research

The 3D Index is extended to meromorphic functions on triangulated 3-manifolds.

Study rationality of meromorphic functions between real algebraic sets in the plane.

Constructs a function to prove meromorphic differential strata don't have complete subvarieties.

Researchers compute the index of meromorphic functions on tori.

Pólya's theorem extended to meromorphic functions on Riemann surfaces.

Transforms study meromorphically isothermic surfaces near singular points.

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We …

One-parameter smooth families of circles in the complex plane with the following property are described: a function is polyanalytic if and only if it has meromorphic extension inside any circle from the family, with the only singularity-a pole at the center.

For any connected component $H_0$ of the space of real meromorphic functions we build a compactification $N(H_0)$ of the space $H_0$. Then we express the Euler characteristics of the spaces $H_0$ and $N(H_0)$ in terms of topological invariants of functions from $H_0$.

We give a necessary condition for a meromorphic function in several variables to give rise to a Milnor fibration of the local link (respectively of the link at infinity). In the case of two variables we give some necessary and sufficient conditions for the local link (respectively the link at infinity) to be fibred.

In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.

Study of holomorphic correspondences combining entire maps and Fuchsian groups.

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on t…

New insights into knot homology growth, extending previous results.

Study uses graph techniques to understand meromorphic quadratic differential strata.

We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of $2π$, determined by…

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces,…

The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $|ω|^2$, where $ω$…

Study describes how to realize periods of meromorphic differentials with specific properties.

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D …

Paper analyzes meromorphic (1,0)-forms on compact Riemann surfaces.

Study of meromorphic connections and their spectral duals in $\mathfrak{gl}_3(\mathbb{C})$.

Study geodesics of meromorphic connections on Riemann surfaces.

Study the determinant of Laplacians on Riemann surfaces with conical metrics.

We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.

The Brylinski beta function is extended for coaxial layers on submanifolds.

A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a me…

This paper classifies components of meromorphic differential strata.

The chaotic geodesic flow on a jet space is non-integrable.

Classifies meromorphic affine connections on complex surfaces.

Classifies connected components of meromorphic differentials with residue conditions.

Proves theorem about meromorphic projective structures with complex poles.

The paper proves a grafting theorem for meromorphic projective structures and shows the monodromy map is a local homeomorphism.

Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that $a(M)\leq k$, where $a(M)$ is the algebraic dimension $a(M)$ (i.e. the transcendence degre…

The paper studies complex affine structures near irregular singularities.

Paper generalizes Bloch-Ros principle to various surface classes.

The paper explores meromorphic connections over Frobenius manifolds.

For a Riemann surface with cusps we define a theta function using the eigenvalues of the Laplacian and the singularities of the scattering determinant. We provide its meromorphic continuation and discuss its singularities.

The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.

An analogue of Brylinski's knot beta function is defined for a submanifold of d-dimensional Euclidean space. This is a meromorphic function on the complex plane. The first few residues are computed for a surface in three dimensional space.

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

Study of energy functional on Deligne-Hitchin moduli space sections.

New Riesz distributions for differential forms on Euclidean space.

This work is based on the approach developed by J.~Dorfmeister, F.~Pedit and H.~Wu [GANG and KITCS preprint, Report KITCS94-4-1] to construct maps $Φ:D\rightarrow R^3$, $D$ being the unit disk in $C$, whose images are surfaces of constant mean curvature. They start from certain meromorphic one forms, so called meromorp…

Study of asymptotics of meromorphic 3D-index as q approaches 1.

We investigate the Lawson genus $2$ surface by methods from integrable system theory. We prove that the associated family of flat connections comes from a family of flat connections on a $4-$punctured sphere. We describe the symmetries of the holonomy and show that it is already determined by the holonomy around one of…

The paper studies the geometric structure of modular curves $X_{1}(N)$ using meromorphic differentials.