On a Class of Kato Manifolds

2020 ◽  
Author(s):  
Nicolina Istrati ◽  
Alexandra Otiman ◽  
Massimiliano Pontecorvo
Keyword(s):  
Spherical Shell ◽  
Vector Fields ◽  
Specific Class ◽  
Complex Dimension ◽  
Hirzebruch Surfaces ◽  
Higher Dimensional ◽  
Analytical Properties ◽  
Locally Conformally Kähler ◽  
Algebraic Reduction ◽  
Kato Surfaces

Abstract We revisit Brunella’s proof of the fact that Kato surfaces admit locally conformally Kähler metrics, and we show that it holds for a large class of higher-dimensional complex manifolds containing a global spherical shell. On the other hand, we construct manifolds containing a global spherical shell that admit no locally conformally Kähler metric. We consider a specific class of these manifolds, which can be seen as a higher-dimensional analogue of Inoue–Hirzebruch surfaces, and study several of their analytical properties. In particular, we give new examples, in any complex dimension $n \geq 3$, of compact non-exact locally conformally Kähler manifolds with algebraic dimension $n-2$, algebraic reduction bimeromorphic to $\mathbb{C}\mathbb{P}^{n-2}$, and admitting nontrivial holomorphic vector fields.

scholarly journals EMBEDDING FRACTIONAL QUANTUM HALL SOLITONS IN M-THEORY COMPACTIFICATIONS

2011 ◽  
Vol 08 (07) ◽  
pp. 1507-1518 ◽  
Author(s):  
A. BELHAJ ◽  
N.-E. FAHSSI ◽  
E. H. SAIDI ◽  
A. SEGUI
Keyword(s):  
Quantum Hall ◽  
Target Space ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Hirzebruch Surfaces ◽  
Dynkin Diagrams ◽  
Higher Dimensional ◽  
Dynkin Quivers ◽  
M Theory ◽  
Chern Simons

We engineer U(1)n Chern–Simons type theories describing fractional quantum Hall solitons (QHS) in 1 + 2 dimensions from M-theory compactified on eight-dimensional hyper-Kähler manifolds as target space of N = 4 sigma model. Based on M-theory/type IIA duality, the systems can be modeled by considering D6-branes wrapping intersecting Hirzebruch surfaces F0's arranged as ADE Dynkin Diagrams and interacting with higher-dimensional R-R gauge fields. In the case of finite Dynkin quivers, we recover well known values of the filling factor observed experimentally including Laughlin, Haldane and Jain series.

scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

scholarly journals Locally conformally Kähler metrics on Kato surfaces

2011 ◽  
Vol 202 ◽  
pp. 77-81 ◽  
Author(s):  
Marco Brunella
Keyword(s):  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Locally Conformally Kähler ◽  
Kahler Metric ◽  
Kato Surfaces

AbstractWe show that every Kato surface admits a locally conformally Kähler metric.

An existence theorem for a two-dimensional nonlinear shell model of Koiter’s type

2018 ◽  
Vol 28 (14) ◽  
pp. 2833-2861 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Shell Model ◽  
Energy Function ◽  
Vector Fields ◽  
Elastic Shell ◽  
Existence Theory ◽  
Specific Class ◽  
Two Dimensional ◽  
Stored Energy ◽  
Nonlinear Shell ◽  
Order Part

We propose a minimization problem with a stored energy function that is polyconvex and satisfies all the other assumptions of John Ball’s theorem, while being at the same time well adapted for modeling a nonlinearly elastic shell. By restricting the admissible deformations to be specific quadratic polynomials with respect to the transverse variable, we are able to define a new nonlinear shell model for which a satisfactory existence theory is available and that is still two-dimensional, in the sense that minimizing the corresponding total energy amounts to finding three vector fields defined on the closure of a bounded open subset of [Formula: see text]. The most noteworthy feature of our nonlinear shell model is that the “lowest order part” of its stored energy function coincides, at least formally, with the stored energy function found in Koiter’s model for a specific class of deformations that are to within the first-order identical to the Kirchhoff–Love deformations considered by W. T. Koiter.

scholarly journals Invariants of four-manifolds with flows via cohomological field theory

2014 ◽  
Author(s):  
Hugo Garcia-Compeân ◽  
Roberto Santos-Silva ◽  
Alberto Verjovsky
Keyword(s):  
Field Theory ◽  
Probability Measure ◽  
Vector Fields ◽  
Invariant Probability Measure ◽  
Yang Mills ◽  
Higher Dimensional ◽  
Coupling Dynamics ◽  
Four Manifolds ◽  
Cohomological Field Theory ◽  
Asymptotic Cycles

This chapter argues that the Jones–Witten invariants can be generalized for smooth, nonsingular vector fields with invariant probability measure on three-manifolds, thus giving rise to new invariants of dynamical systems. After a short survey of cohomological field theory for Yang–Mills fields, Donaldson–Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. The chapter studies the case of Kähler manifolds by using the Witten's consideration of the strong coupling dynamics of N = 1 supersymmetric Yang–Mills theories. The whole construction is performed by implementing the notion of higher-dimensional asymptotic cycles. In the process Seiberg–Witten invariants are also described within this context. Finally, the chapter gives an interpretation of the asymptotic observables of four-manifolds in the context of string theory with flows.

scholarly journals Compact lcK manifolds with parallel vector fields

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Andrei Moroianu
Keyword(s):  
Vector Fields ◽  
Kähler Manifold ◽  
Kahler Manifold ◽  
Locally Conformally Kähler

AbstractWe show that for n > 2 a compact locally conformally Kähler manifold (M

Sectional-hyperbolic systems

2008 ◽  
Vol 28 (5) ◽  
pp. 1587-1597 ◽  
Author(s):  
R. METZGER ◽  
C. MORALES
Keyword(s):  
Vector Fields ◽  
Hyperbolic Systems ◽  
Negative Answer ◽  
Lorenz Attractor ◽  
Closed Orbits ◽  
Higher Dimensional ◽  
Singular Sets ◽  
Uniform Hyperbolicity ◽  
Mathematical Sciences ◽  
Lorenz Attractors

AbstractWe introduce a class of vector fields onn-manifolds containing the hyperbolic systems, the singular-hyperbolic systems on 3-manifolds, the multidimensional Lorenz attractors and the robust transitive singular sets in Liet al[Robust transitive singular sets via approach of an extended linear Poincaré flow.Discrete Contin. Dyn. Syst.13(2) (2005), 239–269]. We prove that the closed orbits of a system in such a class are hyperbolic in a persistent way, a property which is false for higher-dimensional singular-hyperbolic systems. We also prove that the singularities in the robust transitive sets in Liet alare similar to those in the multidimensional Lorenz attractor. Our results will give a partial negative answer to Problem 9.26 in Bonattiet al[Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005].

scholarly journals Locally conformally Kähler metrics on Kato surfaces

2011 ◽  
Vol 202 ◽  
pp. 77-81 ◽  
Author(s):  
Marco Brunella
Keyword(s):  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Locally Conformally Kähler ◽  
Kahler Metric ◽  
Kato Surfaces

AbstractWe show that every Kato surface admits a locally conformally Kähler metric.

scholarly journals Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

2012 ◽  
Vol 33 (5) ◽  
pp. 1550-1583 ◽  
Author(s):  
STEFAN MÜLLER ◽  
PETER SPAETH
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Contact Dynamics ◽  
Dimensional Manifold ◽  
Contact Form ◽  
Hamiltonian Flows ◽  
Higher Dimensional ◽  
Calabi Invariant ◽  
Regular Contact ◽  
Volume Preserving

AbstractWe compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the isotopy. Based on these results, we provide positive answers to two questions posed by Arnold in [The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4) (1986), 327–345]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume-preserving homeomorphisms, and is invariant under conjugation by volume-preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed by Banyaga and Spaeth [On the uniqueness of generating Hamiltonians for topological strictly contact isotopies.Preprint, 2012], Buhovsky and Seyfaddini [Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symplectic Geom. to appear, arXiv:1003.2612v2], Müller [The group of Hamiltonian homeomorphisms in the$L^\infty $-norm. J. Korean Math. Soc.45(6) (2008), 1769–1784], Müller and Oh [The group of Hamiltonian homeomorphisms and$C^0$-symplectic topology. J. Symplectic Geom. 5(2) (2007), 167–219], Müller and Spaeth [Topological contact dynamics I: symplectization and applications of the energy-capacity inequality.Preprint, 2011, arXiv:1110.6705v2] and Viterbo [On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. (2006), 34028; Erratum,Int. Math. Res. Not.(2006), 38748]. Moreover, we generalize an example of Furstenberg [Strict ergodicity and transformation of the torus. Amer. J. Math. 83(1961), 573–601] on topologically conjugate but not$C^1$-conjugate area-preserving diffeomorphisms of the two-torus to trivial$T^2$-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not$C^1$-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.

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