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David Aulicino

Publications and source records attributed to David Aulicino.

16 recordsLinked to original sources
arXiv2026

Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.

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arXiv2025

Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces

We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is always equal to the reciprocal of the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.

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arXiv2024

Algebraically primitive invariant subvarieties with quadratic field of definition

We show that the only algebraically primitive invariant subvarieties of strata of translation surfaces with quadratic field of definition are the decagon, Weierstrass curves, and eigenform loci in genus two and the rank two example in the minimal stratum of genus four translation surfaces discovered by Eskin-McMullen-Mukamel-Wright.

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arXiv2023

A Zero Lyapunov Exponent in Genus $3$ Implies the Eierlegende Wollmilchsau

We prove that the closed orbit of the Eierlegende Wollmilchsau is the only $\text{SL}(2,\mathbb{R})$-orbit closure in genus three with a zero Lyapunov exponent in its Kontsevich-Zorich spectrum. The result recovers previous partial results in this direction by Bainbridge-Habegger-Möller and the first named author. The main new contribution is an understanding of the Forni subspace along a degeneration toward the boundary of the moduli space of curves. This results in a simple geometric criterion that excludes the existence of a Forni subspace. Another key ingredient is the solution to the jump problem from the work of Hu and the third named author.

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arXiv2021

Counting Tripods on the Torus

Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in $\mathbb C^2$, and we give an asymptotic counting result using lattice point counting techniques.

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arXiv2020

Rank 2 Affine Manifolds in Genus 3

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.

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arXiv2019

Shimura-Teichmüller Curves in Genus 5

We prove that there are no Shimura-Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura-Teichmüller curves. Moreover, the property of being a Shimura-Teichmüller curve is equivalent to having completely degenerate Kontsevich-Zorich spectrum. The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.

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arXiv2019

Platonic solids and high genus covers of lattice surfaces

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 primitive lattice surfaces, we show that the Teichmüller curve of the unfolded dodecahedron has genus 131 with 19 cone singularities and 362 cusps. We provide both theoretical and rigorous computer-assisted proofs that there are no closed saddle connections on the surfaces associated to the tetrahedron, octahedron, cube, and icosahedron. We show that there are exactly 31 equivalence classes of closed saddle connections on the dodecahedron, where equivalence is defined up to affine automorphisms of the translation cover. Techniques established here apply more generally to Platonic surfaces and even more generally to translation covers of primitive lattice surfaces and their Euclidean cone surface and billiard table quotients.

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arXiv2019

A New Approach to the Automorphism Group of a Platonic Surface

We borrow a classical construction from the study of rational billiards in dynamical systems known as the "unfolding construction" and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular polygon or double polygon by the unfolded Platonic surface yields a normal subgroup of the rotation group of the Platonic surface. The quotient of this rotation group by the normal subgroup is always a cyclic group, where explicit bounds on the order of the cyclic group can be given entirely in terms of the Schläfli symbol of the Platonic surface. As a consequence, we provide a new derivation of the rotation groups of the dodecahedron and the Bolza surface.

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arXiv2016

The Cantor-Bendixson Rank of Certain Bridgeland-Smith Stability Conditions

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendixson rank and give a tight bound. Finally, we present a family of surfaces realizing all possible Cantor-Bendixson ranks. The techniques in the proof of this result exclusively concern Abelian differentials on Riemann surfaces, also known as translation surfaces. The concept of a "slit translation surface" is introduced as the primary tool for studying meromorphic quadratic differentials with higher order poles.

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arXiv2015

Rank two affine submanifolds in $\mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$

We classify all rank two affine manifolds in strata in genus three with two zeros. This confirms a conjecture of Maryam Mirzakhani in these cases. Several technical results are proven for all strata in genus three, with the hope that they may shed light on a complete classification of rank two manifolds in genus three.

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arXiv2015

Affine Manifolds and Zero Lyapunov Exponents in Genus 3

In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curve in genus three has two zero Lyapunov exponents in the Kontsevich-Zorich cocycle, then it lies in the principal stratum and has at most quadratic trace field. Moreover, there can be at most finitely many such Teichmüller curves.

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arXiv2015

Teichmüller Discs with Completely Degenerate Kontsevich-Zorich Spectrum

We reduce a question of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich, which asks for a classification of all $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum, to a conjecture of Möller's. Let $\mathcal{D}_g (1)$ be the subset of the moduli space of Abelian differentials $\mathcal{M}_g$ whose elements have period matrix derivative of rank one. There is an $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measure $ν$ with completely degenerate Kontsevich-Zorich spectrum, i.e. $λ_1 = 1 > λ_2 = \cdots = λ_g = 0$, if and only if $ν$ has support contained in $\mathcal{D}_g (1)$. We approach this problem by studying Teichmüller discs contained in $\mathcal{D}_g (1)$. We show that if $(X,ω)$ generates a Teichmüller disc in $\mathcal{D}_g (1)$, then $(X,ω)$ is completely periodic. Furthermore, we show that there are no Teichmüller discs in $\mathcal{D}_g (1)$, for $g = 2$, and the two known examples of Teichmüller discs in $\mathcal{D}_g (1)$, for $g = 3, 4$, are the only two such discs in those genera. Finally, we prove that if there are no genus five Veech surfaces generating Teichmüller discs in $\mathcal{D}_5(1)$, then there are no Teichmüller discs in $\mathcal{D}_g (1)$, for $g = 5,6$.

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arXiv2015

Affine Invariant Submanifolds with Completely Degenerate Kontsevich-Zorich Spectrum

We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL$(2,\mathbb{R})$-invariant submanifold is completely degenerate, i.e. $λ_2 = \cdots = λ_g = 0$, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there are at most finitely many such Teichmueller curves.

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arXiv2014

Classification of higher rank orbit closures in H^{odd}(4)

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). Together with work of Matheus-Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmuller curves) in H^{odd}(4) outside of the Prym locus.

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