arXiv research

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

Finite intersection numbers between horizontal foliations of quadratic differentials.

Abstract compares two norms in holomorphic quadratic differentials.

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which as…

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…

This expository survey describes how holomorphic quadratic differentials arise in several aspects of Teichmüller theory, highlighting their relation with various geometric structures on surfaces. The final section summarizes results for non-compact surfaces of finite type, when the quadratic differential has poles of f…

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

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.

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…

Develops second order infinitesimal structures on Teichmüller space.

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The …

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

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.

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the principal parts of $\sqrt q$ at the poles. This generalizes a theorem of Hubbard and …

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.

Quadratic differentials on Riemann surfaces uniquely determine foliations.

In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps f…

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

The paper studies how hyperbolic surfaces degenerate along harmonic map rays.

Study of non-Archimedean Hitchin map for SL2(F) characters.

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

A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that f…

We consider those simply connected isothermic surfaces for which their Hopf differential factorizes into a real function and a meromorphic quadratic differential that has a zero or pole at some point, but is nowhere zero and holomorphic otherwise. Upon restriction to a simply connected patch that does not contain the z…

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isotherm…

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf's pa…

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(κ,τ)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(κ,τ)$, and obtain some consequence…

Proof of wall-crossing formula using spectral networks.

Affine vector fields on pseudo-Kähler manifolds are symplectic.

Bounds projective structure norms by bending lamination lengths.

We prove quantitative recurrence and large deviations results for the Teichmuller geodesci flow on a connected component of a stratum of the moduli space $Q_g$ of holomorphic unit-area quadratic differentials on a compact genus $g \geq 2$ surface.

A short proof of the Caratheodory conjecture about index of an isolated umbilic on the convex 2-dimensional sphere is suggested. The argument is based on the study of geodesic lines near cone-type singularity of a metric induced by holomorphic quadratic differentials.

Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an ass…

The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We prov…

New theory connects string theory to swampland distance conjecture.

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riem…

We study an infinite-dimensional hyperkähler reduction introduced by Donaldson and associated with the constant scalar curvature equation on a Riemann surface. It is known that the corresponding moment map equations admit special solutions constructed from holomorphic quadratic differentials. Here we obtain a more gene…

We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension $2n+1$). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surface…

Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential $ω$ on a Riemann surface of genus $g \geq 2$ the smallest gap between saddle connecti…

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

The Bers embebbing realizes the Teichmüller space of a Fuchsian group $G$ as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for $G$. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective holomorphic function on the di…

The paper introduces new metrics on complex domains with specific geometric properties.

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent …

Classifies components of strata of k-differentials on Riemann surfaces.

Let X be some Riemann surface, and let omega be a meromorphic quadratic differential form on X, that is, omega can be written in local coordinates as f(z) dz^2, for some meromorphic function f. We say that a curve gamma is part of a horizontal leaf of omega if for each t in I, we have that f(gamma(t)) (gamma'(t))^2 is …