arXiv research

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

Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo 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 {\…

Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.

New solutions found for elliptic sinh-Gordon and sine-Gordon equations.

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.

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface ov…

We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an ``elliptic end'' (resp. a ``hyperbolic end'') if the monodromy matrix at the …

Characterizes totally elliptic surface group representations into Lie groups.

Paper classifies minimal graph transformations into new families of surfaces.

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic fu…

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.

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We al…

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.

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

Researchers classify cmc surfaces using Jacobi elliptic functions.

We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,μ}$ multivalued solutions to the minimal surface equation, to produce $C^{1,μ}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the La…

In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…

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).

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a min…

Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.

This paper constructs explicit trisection diagrams for elliptic surfaces.

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

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

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

Study asymptotic behavior of Weingarten surfaces at infinity.

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

The study classifies points on ruled surfaces in 4-space based on geometric properties.

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

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 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.

Study topological hyperbolicity of moduli spaces of elliptic surfaces.

The paper proves existence of special surfaces in 3D manifolds with constant curvature.

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{…

Karcher reimagined elliptic functions using geometry.

Study on K3 surfaces' collapsing and special Kähler structures.

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…

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is…