Showing papers on "Reeb vector field published in 2016"

A Generalization of the Goldberg Conjecture for CoKähler Manifolds

Abstract: Let M be a compact almost coKahler manifold. If the metric g of M is a Ricci soliton and the potential vector field is pointwise collinear with the Reeb vector field, then we prove that M is Ricci-flat and coKahler and the soliton g is steady. This generalizes a Goldberg-like conjecture for coKahler manifolds obtained by Cappelletti-Montano and Pastore, namely any compact Einstein K-almost coKahler manifold is coKahler. Without the assumption of compactness, Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKahler manifolds are also studied. Moreover, we prove that there exist no gradient Ricci solitons on proper \({(\kappa, \mu)}\)-almost coKahler manifolds.

Ricci tensors on three-dimensional almost coKähler manifolds

Abstract: Let M3 be a three-dimensional almost coKahler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M3 for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.

From a Reeb orbit trap to a Hamiltonian plug

Abstract: We present a simple construction of a plug for Hamiltonian flows on hypersurfaces of dimension at least five by doubling a trap for Reeb orbits.

Gromov-Witten theory of a locally conformally symplectic manifold.

Abstract: We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new phenomenon (relative to the symplectic case) is the potential existence of holomorphic sky catastrophes, an analogue for pseudo-holomorphic curves of sky catastrophes in dynamical systems originally discovered by Fuller. We are able to rule these out in some situations, particularly for certain $\lcs$ 4-folds, and as one application we show that in dimension 4 the classical Gromov non-squeezing theorem has certain $C ^{0} $ rigidity or persistence with respect to $\lcs$ deformations, this is one version of $\lcs$ non-squeezing a first result of its kind. In a different direction we study Gromov-Witten theory of the $\lcsm$ $C \times S ^{1} $ induced by a contact manifold $(C, \lambda)$, and show that the Gromov-Witten invariant (as defined here) counting certain elliptic curves in $C \times S ^{1} $ is identified with the classical Fuller index of the Reeb vector field $R ^{\lambda} $. This has some non-classical applications, and based on the story we develop, we give a kind of `holomorphic Seifert/Weinstein conjecture' which is a direct extension for some types of $\lcsm$'s of the classical Seifert/Weinstein conjecture. This is proved for $\lcs$ structures $C ^{\infty} $ nearby to the Hopf $\lcs$ structure on $S ^{2k+1} \times S ^{1} $.

Gromov-Witten theory of a locally conformally symplectic manifold and the Fuller index

Abstract: We set up a Gromov-Witten type theory of locally conformally symplectic manifolds. The moduli spaces are non-compact due to unboundedness of energy, and to obtain (Gromov-Witten) invariants in this non compact setting, we propose that the energy function on the moduli space of holomorphic curves in a l.c.s.m. behaves in crucial examples like a bounded from below proper abstract moment map for an $T=S ^{1} $ action, in the sense of Karshon, the Gromov-Witten invariant must then be considered in a suitable $T $ equivariant sense. One basic example considered here, where this holds, is that of a locally conformally symplectic manifold $C \times S ^{1} $ coming from a contact manifold $(C, \xi)$. And using our theory we define a $\mathbb{Q}$ valued "quantum Euler characteristic of $(C, \xi)$" which "counts" Reeb orbits of $(C, \xi)$ with respect to a contact form $ \lambda$, in terms of certain Gromov-Witten invariants counting holomorphic tori in $C \times S ^{1} $. Using this we obtain certain new existence results for periodic orbits of a Reeb vector field.

The AdS/CFT correspondence and generalized geometry

Abstract: The most general AdS$_5 imes Y$ solutions of type IIB string theory that are AdS/CFT dual to superconformal field theories in four dimensions can be fruitfully described in the language of generalized geometry, a powerful hybrid of complex and symplectic geometry. We show that the cone over the compact five-manifold $Y$ is generalized Calabi-Yau and carries a generalized holomorphic Killing vector field $xi$, dual to the R-symmetry. Remarkably, this cone always admits a symplectic structure, which descends to a contact structure on $Y$, with $xi$ as Reeb vector field. Moreover, the contact volumes of $Y$, which can be computed by localization, encode essential properties of the dual CFT, such as the central charge and the conformal dimensions of BPS operators corresponding to wrapped D3-branes. We then define a notion of ``generalized Sasakian geometry'', which can be characterized by a simple differential system of three symplectic forms on a four-dimensional transverse space. The correct Reeb vector field for an AdS$_5$ solution within a given family of generalized Sasakian manifolds can be determined---without the need of the explicit metric---by a variational procedure. The relevant functional to minimize is the type IIB supergravity action restricted to the space of generalized Sasakian manifolds, which turns out to be just the contact volume. We conjecture that this contact volume is equal to the inverse of the trial central charge whose maximization determines the R-symmetry of the dual superconformal field theory. The power of this volume minimization is illustrated by the calculation of the contact volumes for a new infinite family of solutions, in perfect agreement with the results of $a$-maximization in the dual mass-deformed generalized conifold theories.

Finite-energy pseudoholomorphic planes with multiple asymptotic limits

Abstract: It's known from from work of Hofer, Wysocki, and Zehnder [1996] and Bourgeois [2002] that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its non removable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $(M, \xi)$ with cooriented contact structure one can choose a contact form $\lambda$ with $\ker\lambda=\xi$ and a compatible complex structure $J$ on $\xi$ so that for the associated $\mathbb{R}$-invariant almost complex structure $\tilde J$ on $\mathbb{R}\times M$ there exist families of embedded finite-energy $\tilde J$-holomorphic cylinders and planes having embedded tori as limit sets.

Symplectic Asphericity, Category Weight, and Closed Characteristics of K-Contact Manifolds

Abstract: Let $M$ be a closed K-contact $(2n+1)$-manifold equipped with a quasi-regular K-contact structure. Rukimbira proved that the Reeb vector field $\xi$ of this structure has at least $n+1$ closed characteristics. We note that $\xi$ has at least $2n+1$ closed characteristics provided that the space of leaves of the foliation determined by $\xi$ is symplectically aspherical.

Basic Kirwan Surjectivity for K-Contact Manifolds

Abstract: We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free $S^1$-action, the $S^1$-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-Weitsman type description of the kernel of the basic Kirwan map for $S^1$-actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting.