scholarly journals Planarity in Higher-Dimensional Contact Manifolds

2020 ◽  
Author(s):  
Bahar Acu ◽  
Agustin Moreno
Keyword(s):  
Higher Dimensions ◽  
Symplectic Manifolds ◽  
Contact Form ◽  
Contact Manifolds ◽  
Contact Type ◽  
Weinstein Conjecture ◽  
Higher Dimensional ◽  
Alternative Approach ◽  
Weak Contact ◽  
Planar Contact

Abstract We obtain several results for (iterated) planar contact manifolds in higher dimensions. (1) Iterated planar contact manifolds are not weakly symplectically co-fillable. This generalizes a 3D result of Etnyre [ 14] to a higher-dimensional setting, where the notion of weak fillability is that due to Massot-Niederkrüger-Wendl [ 38]. (2) They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds. This generalizes a result by Albers-Bramham-Wendl [ 4]. (3) They satisfy the Weinstein conjecture, that is, every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [ 2] and is a higher-dimensional generalization of a result of Abbas-Cieliebak-Hofer [ 1]. The results follow as applications from a suitable symplectic handle attachment, which bears some independent interest.

scholarly journals Symplectic cobordisms and the strong Weinstein conjecture

2012 ◽  
Vol 153 (2) ◽  
pp. 261-279 ◽  
Author(s):  
HANSJÖRG GEIGES ◽  
KAI ZEHMISCH
Keyword(s):  
Moduli Spaces ◽  
Quantitative Character ◽  
Contact Manifolds ◽  
Symplectic Capacity ◽  
Contact Type ◽  
Weinstein Conjecture ◽  
Cotangent Bundles ◽  
Higher Dimensional ◽  
Definition Of ◽  
Symplectic Cobordisms

AbstractWe study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

scholarly journals THE WEINSTEIN CONJECTURE IN THE PRESENCE OF SUBMANIFOLDS HAVING A LEGENDRIAN FOLIATION

2011 ◽  
Vol 03 (04) ◽  
pp. 405-421 ◽  
Author(s):  
KLAUS NIEDERKRÜGER ◽  
ANA RECHTMAN
Keyword(s):  
Projective Space ◽  
Holomorphic Curves ◽  
Real Projective Space ◽  
Contact Manifolds ◽  
Connected Sum ◽  
Novel Technique ◽  
Weinstein Conjecture ◽  
Higher Dimensional

Helmut Hofer introduced in 1993 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3-manifolds (M, ξ) with π2(M) ≠ 0. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher-dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

scholarly journals Affine varieties, singularities and the growth rate of wrapped Floer cohomology

2018 ◽  
Vol 10 (03) ◽  
pp. 493-530
Author(s):  
Mark McLean
Keyword(s):  
Growth Rate ◽  
Cotangent Bundle ◽  
Betti Numbers ◽  
Symplectic Manifolds ◽  
Isolated Singularity ◽  
Contact Manifolds ◽  
Floer Cohomology ◽  
Complex Singularities ◽  
Smooth Affine ◽  
Simply Connected

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if [Formula: see text] is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.

scholarly journals Open books and exact symplectic cobordisms

2018 ◽  
Vol 29 (04) ◽  
pp. 1850026 ◽  
Author(s):  
Mirko Klukas
Keyword(s):  
Disjoint Union ◽  
Contact Manifold ◽  
Higher Dimensions ◽  
Fiber Bundles ◽  
Open Book ◽  
Contact Manifolds ◽  
Open Books ◽  
Symplectic Cobordism ◽  
The Given ◽  
Symplectic Cobordisms

Given two open books with equal pages, we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods, we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact [Formula: see text]-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by Dörner–Geiges–Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic [Formula: see text]-handle.

scholarly journals Periodic orbits in virtually contact structures

2018 ◽  
Vol 12 (02) ◽  
pp. 371-418
Author(s):  
Youngjin Bae ◽  
Kevin Wiegand ◽  
Kai Zehmisch
Keyword(s):  
Periodic Solutions ◽  
Potential Function ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Contact Structure ◽  
Critical Value ◽  
Holomorphic Curves ◽  
Contact Structures ◽  
Contact Form ◽  
Contact Manifolds

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.

scholarly journals Structural transformations in quasicrystals induced by higher dimensional lattice transitions

2012 ◽  
Vol 468 (2141) ◽  
pp. 1452-1471 ◽  
Author(s):  
Giuliana Indelicato ◽  
Tom Keef ◽  
Paolo Cermelli ◽  
David G. Salthouse ◽  
Reidun Twarock ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Higher Dimensions ◽  
Structural Transformations ◽  
Icosahedral Symmetry ◽  
Local Transformation ◽  
Project Method ◽  
Higher Dimensional ◽  
Transformation Mechanisms ◽  
Transition Paths ◽  
Penrose Tilings

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

The Pons Asinorum in higher dimensions

2009 ◽  
Vol 46 (2) ◽  
pp. 263-273 ◽  
Author(s):  
Mowaffaq Hajja
Keyword(s):  
Higher Dimensions ◽  
Point Of View ◽  
Higher Dimensional ◽  
Bridge Of Asses ◽  
Pons Asinorum ◽  
Two Sides

The Pons Asinorum , or the Bridge of Asses , refers to Proposition 5 of Book I of Euclid’s Elements . This proposition and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. Analogues of these propositions for higher dimensional d -simplices are considered in this paper, and satisfactory results are obtained for orthocentric d -simplices. These results do not hold for non-orthocentric d -simplices, thus supporting the point of view that orthocentric d -simplices and not arbitrary ones are the adequate generalization of triangles.

Asymptotic stability of heteroclinic cycles in systems with symmetry

1995 ◽  
Vol 15 (1) ◽  
pp. 121-147 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Ad Hoc ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Higher Dimensions ◽  
Systematic Investigation ◽  
Heteroclinic Cycles ◽  
General Sufficient Condition ◽  
Higher Dimensional ◽  
The Stability

AbstractSystems possessing symmetries often admit heteroclinic cycles that persist under perturbations that respect the symmetry. The asymptotic stability of such cycles has previously been studied on an ad hoc basis by many authors. Sufficient conditions, but usually not necessary conditions, for the stability of these cycles have been obtained via a variety of different techniques.We begin a systematic investigation into the asymptotic stability of such cycles. A general sufficient condition for asymptotic stability is obtained, together with algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied in ℝ3 and often satisfied in higher dimensions. We end by applying our results to several higher-dimensional examples that occur in mode interactions with O(2) symmetry.

MASS, PARTICLES AND WAVES IN HIGHER DIMENSIONS

2003 ◽  
Vol 12 (09) ◽  
pp. 1721-1727 ◽  
Author(s):  
PAUL S. WESSON
Keyword(s):  
Equations Of Motion ◽  
Higher Dimensions ◽  
Particle Mass ◽  
Dimensional Manifold ◽  
Higher Dimensional ◽  
Matter Theory ◽  
Induced Matter

Using 5D membrane/induced-matter theory as a basis, we derive the equations of motion for a novel gauge. The latter admits both particle and wave behaviour, as well as super-communication (wherein there is causal contact in the higher-dimensional manifold among points which are disjoint in spacetime). Possible ways to test this model are suggested, notably using particle mass.

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