arXiv research

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…

Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.

Paper proves unbounded stabilization heights of fiber surfaces using Hopf invariant.

In a three-dimensional Riemannian manifold M that admits a unit Killing vector field $ξ$, we regard $ξ$ as a magnetic vector field. A magnetic Hopf surface is a surface obtained by Lie dragging the magnetic curve with $ξ$. Then we characterize Sasakian structure on M from magnetic Hopf surfaces. That is, we show that i…

Study links Hopf differentials to curvature line flows on time-like CMC surfaces.

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

We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.

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.

Finite-type surfaces have a topological Hopf property.

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

Classifies Real primary Hopf surfaces and their associated groups.

The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern-Ricci flow. We use a construction of Gauduchon-Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern-Ricci flow. These solutions reach a volu…

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-pl…

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

Researchers propose a new approach to the Hopf problem for aspherical varieties.

Hopf's theorem generalized to curved spaces.

Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk. In t…

We investigate slice-quaternionic Hopf surfaces. In particular, we construct new structures of slice-quaternionic manifold on $\mathbb{S}^1\times\mathbb{S}^7$, we study their group of automorphisms and their deformations.

Authors show that genus defects of Hopf arborescent links are decidable.

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

Find first (0,2) mirror symmetry examples on Hopf surfaces.

Study classifies triharmonic surfaces in 3D homogeneous spaces.

We give a classification of compact solitons for the pluriclosed flow on complex surfaces. First, by exploiting results from the Kodaira classification of surfaces, we show that the complex surface underlying a soliton must be Kähler except for the possibility of steady solitons on minimal Hopf surfaces. Then, we const…

We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …

Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This com…

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

Classifies surfaces of section for Seifert fibrations.

The paper extends Hopf's theorem to convex surfaces and discrete triangulations.

We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manif…

The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.

An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on $X$ by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomor…

Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in parti…

Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space $E^3$ which preserve the Hopf differential. We prove that, given a CMC $H$ surface $f$, either minimal or not, and a fixed basepoint $z_0$ on this surface, there is a naturally defined family $f_h$, …

A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 Jänich presented a Poincaré-Hopf th…

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…

Solves an Arnold trivium problem using calculus and topology.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.

Quantum invariants for fibered links determined by genus and Hopf invariant.

Study proves properties of compact Hermitian surfaces with specific curvature conditions.

Involutive Hopf monoids yield surface invariants.

The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.

Characterizes conformal classes of tori using differential geometry.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.