arXiv research

New minimal surfaces found in 3D sphere with low genus.

Study on mapping class groups for low genus surfaces.

This study exhausts curve graphs of low-genus surfaces.

Flexible knot construction for low genus surfaces.

In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal surfaces.

In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…

The study finds arithmetic groups often in square-tiled surface monodromies.

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give sever…

Given a flow on a 3-dimensional integral homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. We illustrate the formula with several examples, in particular with surfaces of low genus. As an application, we show that for a right-handed flow with an …

Circle packings on translation surfaces are consistent across different surfaces.

Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of `low' genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homolo…

We show that every surface in H^hyp(4) is either a Veech surface or a generic surface, i.e. its GL^+(2,R)-orbit is either a closed or a dense subset of H^hyp(4) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Recent results of Eskin-Mirzak…

Study investigates minimal surfaces in spherical caps, extending previous findings.

A new construction is presented of scalar-flat Kaehler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new example…

We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …

New bounds on hyperbolic surfaces' properties using linear programming.

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(R)$ be the curve graph of $R$. We prove that if $g=0, n \geq 5$ or $g=1, n \geq 3$, and $λ: \mathcal{C}(R) \rightarrow\mathcal{C}(R)$ is an edge preserving map, then $λ$ is induced by a homeomorphism of $R$, …

We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmuell…

We construct, somewhat non-standard, Legendrian surgery diagrams for some Stein fillable contact structures on some plumbing trees of circle bundles over spheres. We then show how to put such a surgery diagram on the pages of an open book for $S^3,$ with relatively low genus. Thus we produce open books with low genus p…

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…

Characterizes braid group actions on R and mapping class group actions on S1.

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$.…

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency d…

We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabili…

Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2 by a homeomorphism φ. We show that when φis "sufficiently complicated" then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of M…

The paper proves conditions for minimal surfaces to be holomorphic and stable.

New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.

We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general in…

For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslashη(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critica…

The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a sign…

The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposit…

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

Improved lower bounds for faithful linear representations of mapping class groups.

We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.

Each Morita--Mumford--Miller (MMM) class e_n assigns to each genus g >= 2 surface bundle S_g -> E^{2n+2} -> M^{2n} an integer e_n^#(E -> M) := <e_n,[M]> in Z. We prove that when n is odd the number e_n^#(E -> M) depends only on the diffeomorphism type of E, not on g, M, or the map E -> M. More generally, we prove that …

In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we…

Minimal surfaces in lens spaces identified with specific counts.

This paper focuses on the interplay between the intersection theory and the Teichmueller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simp…

Researchers prove quantum invariants remain hard even when restricted.

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. M…

Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $φ:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map φis "sufficiently complicated." This a…

The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$-th power of the canonical line bund…

Machine learning predicts arithmetic curve invariants with high accuracy.

We study the class of holomorphic and isometric submersions between finite-type Teichmüller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map $\mathcal{T}_{g,n} \rightarrow \mathcal{T}_{g,m}$ obtained by filling in punctures. This generalizes a classical r…

Develops a method for random manifolds and submanifolds, focusing on 3-ball knots.

Nonorientable 4-manifolds can be fibred over 2-disks with nonorientable fibers.

Study of origamis in minimal stratum with single cylinders, calculating spin parities and monodromy groups.