arXiv research

Gradient flows for surface energies with tensor fields are derived and analyzed.

The study finds bounds on homologically independent loops on hyperelliptic hyperbolic surfaces.

In this expository article, we illustrate how two independent flat structures on minimal surfaces induce a harmonic function, which captures the uniqueness of Enneper's surface.

Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped surfaces have positivity of a certain curvature, related to surface gravity. Increas…

Yoshikawa moves were introduced at least quarter-century ago and are still actively used by researchers. For any marked graph diagram we will define its twisted diagram and its mirror cut surface. By using a surface-link group of a mirror cut surface of a twisted diagram we will prove the independence of Yoshikawa eigh…

Many extensions of General Relativity are based on considering metric and affine structures as independent properties of spacetime. This leads to the possibility of introducing torsion as an independent degree of freedom. In this article we examine the effects of torsion on the affine Killing vectors of two-dimensional…

Paper discovers simplicial complexes connecting trained models for improved ensembling.

We give an independent proof of a theorem of Danciger of Zhang: surface groups with Hitchin linear part cannot act properly on the affine space

In this note we clarify that the Rcci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

On the thick part of the moduli space of Riemann surfaces, where there is a positive lower bound of the systole of the surface, we show that all Weil-Petersson Riemannian curvatures are bounded, independent of the genus of the surface.

In this paper, we are interested in short homologically and homotopically independent loops based at the same point on Riemannian surfaces and metric graphs. First, we show that for every closed Riemannian surface of genus $g \geq 2$ and area normalized to $g$, there are at least $\ceil{\log(2g)+1}$ homotopically indep…

Roseman moves are seven types of local modification for surface-link diagrams in $3$-space which generate ambient isotopies of surface-links in $4$-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of an…

Study on singular points of translation surfaces under linearly dependent conditions.

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a conseq…

We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for s…

A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.

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.

Improved bounds on geodesic intersections on hyperbolic surfaces.

Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L2-bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.

In trying to generalize Bianchi's Bäcklund transformation of quadrics to Bäcklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the Bäcklund transformation with isometric correspondence of leaves of a general nature (…

The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…

Study null conformal Killing vector fields on complex surfaces.

A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and a…

We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.

We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying t…

Classifies special homogeneous surfaces with unique properties.

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a $(g,b)$-bridge surface for a…

Study geodesic curvature of logarithmic spirals on curved surfaces.

We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recent…

Study shows systole behavior changes significantly for large genus hyperbolic surfaces.

The study defines Finsler metrics on special surfaces and constructs geodesic currents.

Study on ordering geodesics on hyperbolic surfaces.

Overlay framework simplifies exotic derivative pricing.

Sharp inequalities and symmetries on Riemannian surfaces quantified.

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we comput…

The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.

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…

Paper develops streaming algorithms to estimate classifier accuracy on unlabeled data.

Study finds saddle connections on random surfaces follow Poisson distribution.

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitel…

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative …

Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual lin…

Generalizes existence of bending laminations for Kleinian groups.

Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic $\text{SL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree…

A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant of a non-Kähler complex surface, which was originally due to O. Biquard \cite{Biq} and S.R. Williams \cite{W} independentl…

This paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic bounds in g for the number of possible slopes, independent of the 3-manifold. We al…

Proves stability of geodesic flows on closed surfaces.