arXiv research

New surgery operation preserves monotonicity of Lagrangians.

We use the Gromov-Witten invariants and a nonsqueezing theorem by the author to affirm a conjecture by P.Biran on the Lagrangian barriers.

Defines algebraic structures in Lagrangian Floer cohomology using differential forms.

Criterion for flat circle bundles using intrinsically harmonic forms.

We give short proofs of the following two facts: Iterated principal circle bundles are precisely the nilmanifolds. Every iterated circle bundle is almost flat, and hence diffeomorphic to an infranilmanifold.

Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all…

Eta invariant computed for circle bundles over Fano manifolds.

Let $X$ be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic forms on $X$ are isotopic. This implies that blow-ups of these manifolds are unique, thus extending work of Biran. We also e…

We describe a necessary and sufficient condition for a principal circle bundle over an even-dimensional manifold to carry an invariant contact structure. As a corollary it is shown that all circle bundles over a given base manifold carry an invariant contact structure, only provided the trivial bundle does. In particul…

Maps asymptotically embed conic transforms from circle bundles.

Uniformizes compact Sasakian manifolds into circle bundles.

We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classificat…

Knots in circle bundles are uniquely identified by their complements.

Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.

We proof that having boundary of standard 3-dimensional simplex $\partial Δ^3$ as a base of triangulation one can triangulate only trivial and Hopf circle bundles.

We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fiber…

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known…

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.

Stein fillability of circle bundles over symplectic manifolds is restricted.

We demonstrate an obstruction to finding certain splittings of four-manifolds along sufficiently twisted circle bundles over Riemann surfaces, arising from Seiberg-Witten theory. These obstructions are used to show a non-splitting result for algebraic surfaces of general type.

The paper derives a local formula for the Euler number of circle bundles.

In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.

The paper studies mapping class groups of 3-manifolds fibered over surfaces.

We extend topological T-duality to the case of general circle bundles. In this setting we prove existence and uniqueness of T-duals. We then show that T-dual spaces have isomorphic twisted cohomology, twisted $K$-theory and Courant algebroids. A novel feature is that we must consider two kinds of twists in de Rham coho…

Given a Poisson (or more generally Dirac) manifold $P$, there are two approaches to its geometric quantization: one involves a circle bundle $Q$ over $P$ endowed with a Jacobi (or Jacobi-Dirac) structure; the other one involves a circle bundle with a (pre-) contact groupoid structure over the (pre-) symplectic groupoid…

New proof of Milnor-Wood inequality for circle bundles.

An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the self-intersection of the class and the Thurston norm of the underlying 3-manifold.

The paper classifies manifolds with free torus actions and positive Ricci curvature.

We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual …

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $Σ$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the compleme…

The paper studies Kähler-Einstein metrics on circle bundles and their obstruction flatness.

We study special circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb R}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann …

We investigate a PL topology question: which circle bundles can be triangulated over a given triangulation of the base? The question got a simple answer emphasizing the role of minimal triangulations encoded by local systems of circular permutations of vertices of the base simplices. The answer is based on an experimen…

Study of flows on circle bundles over translation surfaces, showing decay of correlations.

We study surface groups $Γ$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $Γ$ so that its limit set $Λ_Γ$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - Λ_Γ) / Γ$ is a circle bundle over a surface. This circle bundl…

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain …

The analysis of holomorphic sections of high powers $L^N$ of holomorphic ample line bundles $L\to M$ over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. The Tian-Yau-Zelditch's asymptotic expansion of the Szegö kernel of a circle bundle plays an important role in Kähler-E…

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle act…

We consider Riemannian 4-manifolds $(X,g_X)$ with a Spin^c-structure and a suitable circle bundle $Y$ over $X$ such that the Spin^c-structure on $X$ lifts to a spin structure on $Y$. With respect to these structures a spinor $φ$ on $X$ lifts to an untwisted spinor $ψ$ on $Y$ and a U(1)-gauge field $A$ for the Spin^c-st…

We investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for the…

We classify conformally flat Riemannian $3-$manifolds which possesses a free isometric $S^1-$action.

We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction.

Maps between circle bundles are studied, proving fiber-preserving and finiteness results for mapping degrees.

New collapsing mechanism for G2-manifolds discovered.

The paper proves mass nonnegativity for certain asymptotically locally flat manifolds.

Study shows convergence of Lagrangian submanifolds under certain metrics.