arXiv research

The study classifies surfaces in Berger spheres as Willmore and Hopf tori.

Characterizes conformal classes of tori using differential geometry.

We produce a new general family of flat tori in R^4, the first one since Bianchi's classical works in the 19th century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in S^3, we first present a global description of all isometric immersions of R^2 into R^4 with flat normal bundle.

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones att…

In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elas…

We show that the extrinsic diameter of immersed flat tori in the 3-sphere is $π$ under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration.

Classifies mapping tori of specific groups, generalizing known results.

This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 …

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori whic…

In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.

Synthetic construction of Hopf fibration in 4D space.

In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology.…

The study examines stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.

The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simp…

The paper classifies biharmonic immersions and submersions in specific spheres.

We survey generalisations of the Chang-Skjelbred Lemma for integral coefficients. Moreover, we construct examples of manifolds with actions of tori of rank > 2 whose equivariant cohomology is torsion-free, but not free. This answers a question of Allday's. The "mutants" we construct are obtained from compactified repre…

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we derive some consequences for strongly stable surfaces in such ambient spaces. M…

Infinitely many new examples of compact Lorentzian surfaces without conjugate points are given. Further, we study the existence and the stability of this property among Lorentzian metrics with a Killing field. We obtain a new obstruction and prove that the Clifton- Pohl torus and some of our examples are as stable as p…

In this note we prove that any minimal $2$-torus in $S^4$ has Morse index at least $6$, with equality if and only if it is congruent to the Clifford torus in some great $S^3\subset S^4$.For a minimal $2$-torus in $S^n$ with vanishing Hopf differential, we show that its index is at least $n+3$, and that this estimate is…

The paper explores triangulations of spheres and projective spaces, focusing on Hopf triangulations and equilibrium structures.

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira…

Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Mo…

This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operat…

We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statement…

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.

Analyses cohomology relations for moving frames and coframes.

Functor connects Lie groupoid algebras to bornological structures.

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…

Automorphism groups of Hopf manifolds are finite and have a bounded order.

Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.

The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.

Algorithm calculates Hopf invariant for simplicial mappings.

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

Analyzes properties of Hopf manifolds from analytic and metric perspectives.

The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.

We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…

Classifies Legendrian Hopf links in lens spaces.

The Hopf invariant is linked to null-homotopy properties of maps.

Study finds non-isotopic transverse tori in Engel manifolds.

In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on four-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hop…

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…

The study characterizes Sasakian manifolds from magnetic Hopf surfaces.

Generalizes Hopf degree theorem to nontrivial bundles.

We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

For a positive Hopf plumbed arborescent Seifert surface $S$, we study the set of Hopf bands $H\subset S$, up to homology and up to the action of the monodromy. The classification of Seifert surfaces for which this set is finite is closely related to the classification of finite Coxeter groups.