arXiv research

Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.

Modernizes classical theory linking isothermic surfaces to Bonnet pairs.

Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

Study on the minimum length of curves on once-punctured hyperbolic surfaces.

Defines height pairing for differential forms on Riemann surface degenerations.

Let $S_g$ be a closed surface of genus $g$ and let $(α, β)$ be a filling pair on $S_g$; then $i(α, β) \geq 2g-1$, where $i$ is the (geometric) intersection number. Aougab and Huang demonstrated that (exponentially many) minimally-intersecting filling pairs exist on $S_g$ when $g > 2$ by a construction w…

This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.

The paper constructs minimal coherent filling pairs on surfaces.

Paper constructs infinitely many pairs of Seifert surfaces for each link.

In this note we classify all Bonnet pairs on a simply connected domain. Our main intent was to apply what we call a quaternionic function theory to a concrete problem in differential geometry. The ideas are simple: conformal immersions into quaternions or imaginary quaternions take the place of chart maps for a Riemann…

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvat…

Study trisections on rational elliptic surfaces to find new Zariski pairs.

Constructs projective moduli spaces for Calabi-Yau pairs.

We consider the generalization of classical Blaschke's Problem to higher codimension case, characterizing Darboux pair of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere congruence and conformally correspond to each other. When the sphere congruence is the …

New method connects veering triangulations to dynamic pairs.

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for all surfaces completing the work of Aougab-Huang and Aougab-Taylor.

4-manifolds can be broken down into pairs-of-pants and K3 surfaces.

The paper studies orbifold splice quotients and log covers of surface pairs.

The article finds conditions for separating filling pairs on surfaces and constructs a Morse function.

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and to…

We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of $G_2$-manifolds. We find a large class …

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. W…

Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.

Paper describes eigenvalues of genus 3 surfaces graphs.

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

We calculate a projective space of essential measured laminations in a surface pair, which will be used in another paper to help describe spaces of "finite height laminations."

We submit a new way to detect pairs of non-cobordant surface-links. We find a new example of a pair of non-cobordant surface-links with the following properties: Orr invariant, Cochran sequence, Sato-Levine invariant, the alinking number and one of Stallings's theorems cannot distinguish them. However our new way can d…

This paper studies properties of weak reducing pairs in critical Heegaard splittings.

Curved flats linked to pairs of Lie applicable surfaces.

We start with a disk with $2n$ vertices along its boundary where pairs of vertices are connected with $n$ strips with certain restrictions. This forms a {\it pairing}. To relate two pairings, we define an operator called a cut-and-glue operation. We show that this operation does not change an invariant of pairings know…

The study finds an upper limit for the number of minimal origami pairs on a surface.

Shortest non-simple closed geodesics on hyperbolic surfaces found.

We give an explicit construction of any simply-connected superconformal surface $φ\colon M^2\to \R^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h\colon M^2\to\R^4$. That $φ$ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs $(g,h)$ of …

The paper finds symplectic mapping class relations using pencil pairs.

The paper finds infinite pairs of CP1-structures sharing same holonomy.

We found unique tori with same curvatures using isometric transformations.

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S', K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.

Researchers found multiple surfaces with same topological and symmetry properties.

Study homotopy motions of surfaces in 3-manifolds.

Nous montrons que les équations du repère mobile des surfaces de Bonnet conduisent à une paire de Lax matricielle isomonodromique d'ordre deux pour la sixième équation de Painlevé. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixt…

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…

Extends Gordon-Litherland pairing to links in thickened surfaces, defining new invariants.

We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawa…

In this paper we first give a Bonnet theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two…

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice pol…

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.