arXiv research

The study classifies hypersurfaces with constant principal curvatures in $\mathbb{S}^3 imes \mathbb{R}$ and $\mathbb{H}^3 imes \mathbb{R}$.

In this paper, by using the $G_2$-structure on Im$(\mathbb O)\cong\mathbb R^7$ from the octonions $\mathbb O$, the $G_2$-binormal motion of curves $γ(t,s)$ in $\mathbb R^7$ associated to the almost complex structure on $\mathbb S^6$ is studied. The motion is proved to be equivalent to Schrödinger flows from $\mathbb R^…

We define a notion of isotropic surfaces in $\mathbb{O}$, i.e. on which some canonical symplectic forms vanish. Using the cross-product in $\mathbb{O}$ we define a map $ρ\colon Gr\_2(\mathbb{O})\to S^6$ from the Grassmannian of $\mathbb{O}$ to $S^6$. This allows us to associate to each surface $Σ$ of $\mathbb{O}$ a fun…

Two families of genus one surfaces with $H$-curvature in $\mathbb{H}^2 imes\mathbb{R}$ are constructed.

Special orthogonal representations from octonions have geometric properties linked to binary cubics.

The paper constructs surfaces with constant mean curvature in a specific space and explores their properties.

Classifies quotients of $S^n imes S^n$ by $\mathbb Z_{/p} imes \mathbb Z_{/p}$ actions.

Let $(X,o)$ be a complex normal surface singularity with rational homology sphere link and let $\widetilde{X}$ be one of its good resolutions. Fix an effective cycle $Z$ supported on the exceptional curve and also a possible Chern class $l'\in H^2(\widetilde{X},\mathbb{Z})$. Define ${\rm Eca}^{l'}(Z)$ as the space of e…

The paper classifies hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with constant curvature.

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

Study CMC hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with a specific symmetry.

The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.

Complex pinning problem simplified for simple multiloops.

Uniform bounds on $SO(2) imes SO(3)$-invariant Ricci solitons on $\mathbb{S}^4$.

New approach to Carrollian geometry using $\mathbb{R}^ imes$-bundles.

Study $G_2$-instantons on $\mathbb{R}^4 imes S^3$ with invariant properties.

Study of a $G_2$-equivariant octonionic operator and its right spectrum.

The paper quantizes hybrid topological-holomorphic field theories on $\mathbb{R}^m imes \mathbb{C}^n$.

2D complexes can be almost-embedded in 4D space without self-intersections.

Alternative proof for links not smoothly slice in $S^2 imes S^2$.

The pinning ideal of multiloops is shown to be NP-complete.

Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.

Discuss knots in $S_{g} imes S^{1}$, introducing rotating information and invariants.

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

Study of singularities in mean curvature flow with focus on $\mathbb{S}^3 imes\mathbb{R}$.

We classify non-minimal biconservative surfaces with parallel mean curvature vector field in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$. When these surfaces do not lie in $\mathbb{S}^n$ or $\mathbb{H}^n$ and they are not vertical cylinders, we find their explicit (local) equation. We also prove a…

The paper constructs exotic complex projective surfaces using rational blowdowns.

Researchers found equations for special surfaces in curved spaces.

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ${\mathbb{H}}^3$ orthogonal to the geodesics of $Σ$. We prove that the induced metr…

This paper aims to define and study a notion of orientability in the Heisenberg sense ($\mathbb{H}$-orientability) for the Heisenberg group $\mathbb{H}^n$. In particular, we define such notion for $\mathbb{H}$-regular $1$-codimensional surfaces. Analysing the behaviour of a Möbius Strip in $\mathbb{H}^1$, we find a $1$…

We construct non-zero constant mean curvature H surfaces in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height and are invariant under a discrete group of horizontal translati…

We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\frac{1}{2}$ and arbitrary genus in $\mathbb{S}^2\times\mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\frac{1}{2}$ tangent along an equator. This is a particular case of a conjug…

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${…

Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a characterization of $δ$-parabolicity we prove that under additional conditions on $\mathbb{M}$,…

We prove new local inequality for divisors on surfaces and utilize it to compute $α$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6…

Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.

The study finds new constant mean curvature surfaces in curved spaces.

In this article, we consider compact surfaces $Σ$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $Γ=\partialΣ\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the warped product $\mathbb{M}\times_f\mathbb{R}$, here $\mathbb{M}$ denotes a Hadamard surface…

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the consta…

We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in ${\mathbb L}({\mathbb H}^3)$ is a maximal surface. We then classify Lagrangian maximal surfa…

Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.

Study defines hyper-dual spheres and ruled surfaces, proving geometric relationships.

We simplify and prove a splitting theorem for mapping class groups of connect sums of $S^2 imes S^1$.

In this article we study surfaces in $\mathbb{S}^3(1) \times \mathbb{R}$ for which the $\mathbb{R}$-direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature vector.

Two classification theorems for Willmore surfaces in S² × S².

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.

Introduces $\mathbb{K}$-framings for surfaces, generalizing quadratic forms.

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology.…