arXiv research

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

Study finds negatively curved spheres in elliptic surfaces and their modifications.

Characterizes totally elliptic surface group representations into Lie groups.

The paper studies elliptical surfaces in 3D affine space, classifying them based on curvature.

Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.

Elliptic surfaces without 1-handles proven for specific cases.

Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.

Study of 17 surface behaviors and singularities for elliptic Weingarten equations.

Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.

Estimates heights of special surfaces in warped products.

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respec…

Constructs a new type of metric for elliptic surfaces.

Study on ruled surfaces over elliptic curves with unique foliations and parallelizable 4-webs.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

Researchers classify cmc surfaces using Jacobi elliptic functions.

Surveying the signature theorem, researchers cancel anomalies on elliptic surfaces.

Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for $n\geq 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5).

This paper constructs explicit trisection diagrams for elliptic surfaces.

Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.

Study timelike meridian surfaces in Minkowski 4-space with specific properties.

We determine obstructedness or unobstructedness of (holomorphic) Poisson deformations of ruled surfaces over an elliptic curve.

Study asymptotic behavior of Weingarten surfaces at infinity.

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

Extends theory of minimal surfaces to elliptic special Weingarten surfaces.

Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.

Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.

We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $Γ$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\…

We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit cir…

The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kaehler surface. We consider the case of the elliptic surfaces E(n) and focus on a slightly weaker conjecture for the closure of the symplec…

The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

Study describes Abel-Jacobi map for elliptic surfaces, refining cubic-line arrangements topology.

Study topological hyperbolicity of moduli spaces of elliptic surfaces.

Karcher reimagined elliptic functions using geometry.

We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new examples of a minimal surface of general type with $p_g=0$, $π_1=\mathbb{Z}/2\mathbb{…

Study on K3 surfaces' collapsing and special Kähler structures.

Quadratic points on surfaces are studied in projective 3-space.

We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp $#(9-k)$ \cpb $$ with $k$ base points and twelve singular fibers. By blowing up the base points we get an el…

This paper establishes an interesting connection between the family of CMC surfaces of revolution in $\mathbb E_1^3$ and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the $\mathbb E_1^3$, only spacelike cylinders and stand…

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…

The paper studies elliptic surfaces and proves unique fibered structures.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypothes…

The study shows conditions for elliptic surfaces without 1-handles.

We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a K…