arXiv research

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…

Formula derived for mean curvature of a special surface with boundary curve.

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

Extends Teichmüller space parametrization using poles of higher order.

The paper constructs harmonic maps from punctured surfaces to the hyperbolic plane.

Transforms study meromorphically isothermic surfaces near singular points.

This paper generalizes a result about bounded differentials to higher-order differentials and studies their geometric implications.

Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.

Survey on quadratic differentials in Teichmüller theory.

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manif…

Finite intersection numbers between horizontal foliations of quadratic differentials.

Abstract compares two norms in holomorphic quadratic differentials.

It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products $\mathbb{S}^2\times\Rr$ and $\mathbb{H}^2\times \Rr$ discovered by U. Abresch and H. Rosenberg could be obtained as a linear combination of usual Hopf differentials. Using this fact, we are able to extend it for L…

The Heights Theorem is extended to all Riemann surfaces with a first kind fundamental group.

The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.

Holomorphic QDs dual to Fenchel-Nielsen coords for hyperbolic metrics.

The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.

Unified theory of discrete minimal surfaces using integrable systems.

Thurston's boundary to the universal Teichmüller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichmüller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prio…

We prove a uniform estimate, valid for every closed Riemann surface of genus at least two, that bounds the distance of any quadratic differential to the finite dimensional space of holomorphic quadratic differentials in terms of its antiholomorphic derivative.

Develops second order infinitesimal structures on Teichmüller space.

Study on surfaces with same mean curvature in 4D space forms.

We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

The paper extends the Hopf differential concept to associative submanifolds in G2-manifolds.

New proofs for complex Hopf manifolds using geometric structures.

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with di…

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify th…

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in parti…

Entire minimal graphs in Heisenberg space have negative Gauss curvature.

Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.

Holomorphic vector bundles on Hopf manifolds admit flat connections.

The study describes how quadratic differentials influence foliations on Riemann surfaces.

Study of hyperkähler reduction on Riemann surfaces, finding more solutions.

The study constructs differential systems on Riemann surfaces and explores their monodromy properties.

We prove that the Teichmueller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov diffeomorphism contains at least two closed Teichmueller geodesics. This proves that the corresponding flat surface does not have a cyclic Veech group. In addition, we prove that this Teichmueller disc is dense inside the hyperelliptic lo…

Classifies surfaces in hyperbolic space with constant Gaussian curvature.

The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.

Non-negativity of curvature lost on Chern-Ricci flow on Hopf manifolds.

The study connects triangulated surfaces to complex projective structures and circle patterns.

We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil.

Study surfaces with constant mean curvature in 4D spaces.

Quadratic differentials on Riemann surfaces uniquely determine foliations.

Holomorphic structures on quantum flag manifolds uniquely defined.

New Hopf surfaces found in LCK manifolds with potential.

Researchers confirm a conjecture about metrics on a specific Teichmüller space.

Quadratic differentials induce spiralling foliations on Riemann surfaces.

Study leafwise holomorphic automorphisms of Reeb components.