arXiv research

Study finds saddle connections on random surfaces follow Poisson distribution.

Study saddle connections on hyperelliptic surfaces, finding growth rates.

Study precise rates of horizontal gap shrinkage on generic translation surfaces.

Golden L surface has unbounded bunching of saddle connections

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the …

We show that area minimizing polyhedral surfaces are saddle.

The study shows how to measure translation surfaces with short saddle connections.

We extend asymptotic formulas for saddle connections on translation surfaces.

For a half-translation surface (S,q), the associated saddle connection complex A(S,q) is the simplicial complex where vertices are the saddle connections on (S,q), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. …

Researchers create new triply periodic minimal surfaces by gluing saddle towers.

Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such structures have different geometrical and dynamical properties than usual translatio…

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k…

Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.

Constructs minimal surfaces by gluing saddle towers with Scherk ends.

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

Researchers provide counterexamples to Pogorelov and Toponogov's questions about saddle surfaces.

We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.

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…

Study shows saddle connection graph's geometry and quasi-isometry properties.

Analytic saddle spheres in S^3 are equators.

In this paper we study flows $\varphi:M\times\mathbb{R}\longrightarrow M$ having an isolated non-saddle set. We see that the complexity of the region of influence of an isolated non-saddle set $K$ depends on the way in which $K$ sits on the phase space at the cohomological level. We construct flows in surfaces having i…

Revisit Fenn's table theorem from a differential-topological perspective.

The paper proves geometric properties of square tables and saddle surfaces.

Bounds on saddle connections on flat spheres with conical singularities.

Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the $SL(2,\mathbb{R})$-action is ergodic in co…

Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of …

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…

Researchers calculate complexity of billiard paths in regular polygons.

In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the quotient, called {\it saddle towers}. They have been recently classified by Pér…

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…

Study dynamics and topology of flows near non-saddle sets or W-sets.

We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension) behave as they are expected to: they have no conjugate points and thus minimize len…

Study calculates slope gaps on polygon surfaces, finding non-unimodal distributions.

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré se…

The paper classifies surfaces formed by quadrilateral gluings.

The paper calculates gap distributions for translation surfaces, focusing on the double heptagon.

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desing…

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…

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…

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

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $ξ_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ξ_{m,k}$ in terms of their genus $g=m k \gg1$.

DEO uses gradient information to escape saddle points in neural networks.

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.

We construct most symmetric Saddle towers in Heisenberg space i.e. periodic minimal surfaces that can be seen as the desingularization of vertical planes intersecting equiangularly. The key point is the construction of a suitable barrier to ensure the convergence of a family of bounded minimal disks. Such a barrier is …

This paper calculates interaction strength for translation surfaces with multiple singularities.

We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.

We prove the existence of nonperiodic, properly embedded minimal surfaces in $\mathbb{R}^2\times\mathbb{S}^1$ with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).

We give an explicit formula for the limiting gap distribution of slopes of saddle connections on the golden L, or any translation surface in its SL(2, R)-orbit, in particular the double pentagon. This is the first explicit computation of the distribution of gaps for a flat surface that is not a torus cover.