arXiv research

Paper finds surfaces where KVol is close to the surface's genus.

Study identifies specific subvarieties in translation surfaces with quadratic field.

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varyi…

Uniqueness of circle packings on certain translation surfaces is proven.

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maxima…

The Gauss-Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more. We study w…

Classifies Teichmüller curves in genus three from genus two.

Circle packings on translation surfaces are consistent across different surfaces.

Random translation surfaces converge to a Poisson plane as genus grows.

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the …

We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.

Study finds saddle connections on random surfaces follow Poisson distribution.

There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one.

The study finds large systoles in translation surfaces and hyperelliptic ones.

We complete the classification of rank two affine manifolds in the moduli space of translation surfaces in genus three. Combined with a recent result of Mirzakhani and Wright, this completes the classification of higher rank affine manifolds in genus three.

As main result we show that for each g > 1 there is some translation surface of genus g whose Veech group is a non congruence subgroup of SL(2,Z). We use origamis/square-tiled surfaces to produce our examples. The article is divided into two parts: In the first part we introduce translation surfaces, origamis, Veech gr…

We extend asymptotic formulas for saddle connections on translation surfaces.

Algorithm constructs algebraic curves from translation surfaces.

Study confirms non-injective monodromy for even genus 4 translation surfaces.

Researchers compute gap distributions for saddle connection directions on specific translation surfaces.

This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard sp…

New families of translation surfaces with multiple oblivious points discovered.

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient c…

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $ω$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,ω)$ and, moreover, that this bound is realized by a unique translation surface up to homo…

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.

A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known classification results for properly embedded minimal surfaces with genus zero in $\mathbb{R…

Finite groups can be automorphism groups of translation surfaces with poles.

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of p…

The study constructs minimal surfaces in a product space with specific properties.

Geometrically boundary of surface moduli space defined.

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingr…

We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surfa…

In this paper, we investigate the closure of a large class of Teichmüller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The …

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular $pq$-fold covering…

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank of their set of dir…

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…

We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.

A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with a translation structure with a topology, which we call the immersive topology be…

We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also show that when the genus is fixed and the number of punctures varies the behavio…

New findings on translation lengths in Teichmüller and curve graphs for pseudo-Anosovs.

The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1 < a_2$ such that the minimal translation length is bounded below and above by $a…

Maps with a single face converge to hyperbolic surfaces in large genus.

We show that every countable subgroup $G<\rm GL_+(2,\mathbb{R})$ without contracting elements is the Veech group of a tame translation surface $S$ of infinite genus, for infinitely many different topological types of $S$. Moreover, we prove that as long as every end has genus, there are no restrictions on the topologic…

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant subman…

In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus greater than or equal to two, one has up to three components with a similar descri…

This note provides an easy construction of fake octagons.