arXiv research

Mathematicians embed a Klein's quartic cover in hyperbolic space.

Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.

We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Bar…

We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function ${}_3F_2$. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.

Paper constructs an infinite 3-7 surface in 3D space.

We study the moduli space of null curves in Klein's quartic in the four-dimensional (complex) projective plane using methods developed by Robert Bryant. As a consequence, we show that minimal surfaces with $9$ embedded planar ends do not exist and formulate some conjectures about the previous moduli space.

Improved bound on the product of first Laplacian eigenvalue and area for genus three surfaces.

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S,f) to be conformally equivalent to (S',f') where S' is a bordered orientable Klein surface embedded in the Euclidean space and f' is the restrict…

We investigate the action of the automorphism group of a closed Riemann surface on its set of theta characteristics (or spin structures). We give criteria for when an automorphism fixes all spin structures, or when it fixes just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.

Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.

Minimal dimensions for Riemann surface embeddings computed for specific groups.

The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains Ω of R^4. When Ω is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which Ω is the complement of a parabola is st…

Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.

Classifies special quartic curves up to equivalence.

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.

Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.

Minimal surfaces with isothermal parameters admitting Bézier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of th…

Classifies branched Willmore spheres using conformal Gauss maps.

The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.

Researchers compute monodromy groups of surface families over quartic curves.

We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and…

Study Einstein-Weyl spaces from Segre quartic surfaces, finding unique geodesics and deformations.

High-order Klein geometries constructed using Lie algebras.

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y\rightarrow X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the …

Infinite Klein bottles with 4-fold meridians found.

This paper contains some results about Teichmüller spaces of non-orientable surfaces (Klein surfaces). We prove several theorems giving isomorphisms between deformation spaces of Klein surfaces. These results show the similarity between the deformation theory of Klein surfaces, and the theory of Riemann surfaces. We al…

In this paper, we consider a Finsler space with a Randers change of Quartic metric F = $\sqrt[4]{α^4 + β^4} + β$. The conditions for this space to be with reversible geodesics are obtained. Further, we study some geometrical properties of F with reversible geodesics and prove that the Finsler metric F induces a general…

The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.

The paper studies diffeomorphisms of a specific foliation on a Klein bottle.

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp$-functi…

First explicit isometric immersion of a flat Klein bottle in 3D space.

Holomorphic maps between configuration spaces are classified, resolving quartic and elliptic curve problems.

New homeomorphism found in Klein bottle group.

Study of zero-divisors in sedenions via determinant factorization.

We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S^3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S^3 bounds at most six Seifert Klein bottles with a given boundary slope. The Seifert Klein bottles in a minimal projection of hyperbolic p…

A new framework for hyperbolic neural networks using the Klein model is introduced.

Classifies compact Clifford-Klein forms for specific Lie algebras.

The study sets limits on the complexity of Klein geometries.

Given a hyperelliptic Klein surface, we construct companion Klein bottles. Bavard's short loops on companion bottles are studied in relation to the surface to improve an inequality of Gromov's in systolic geometry.

Solves a problem about deforming symplectic forms on a Klein bottle.

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin …

No Lagrangian Klein bottles found in $S^2 imes S^2$.

New Klein-Maskit theorems for Anosov subgroups.

We study the small perturbations of the $1+3$-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on …

We find all m-spin structures on Klein surfaces of genus larger than one. An m-spin structure on a Riemann surface P is a complex line bundle on P whose m-th tensor power is the cotangent bundle of P. A Klein surface can be described by a pair (P,tau), where P is a Riemann surface and tau is an anti-holomorphic involut…

We construct one-parameter families of solutions to the Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-p…

We investigate the structure of a variety of new Moishezon twistor spaces, by utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of these twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of…

Develops theory of d-holomorphic connections on Klein surfaces.