The first two authors have recently defined Rabinowitz Floer homology groups $RF{H}_{*}(M,W)$ associated to a separating exact embedding of a contact manifold $(M,\xi )$ into a symplectic manifold $(W,\omega )$. These depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz Floer homology $RF{H}_{*}(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RF{H}_{*}(S{T}^{*}L,T^{*}L)$, where $S{T}^{*}L$ is the unit cosphere bundle of a closed manifold...

We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the...

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\mathbb{Z}$-graded chain complex of vector spaces over ${\mathbb{Z}}_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a...

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m\left(4ϵ\right)$. In the second, we give a complete classification of real hypersurfaces in $M_m\left(4ϵ\right)$ which satisfy the above geometric facts.

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type $\left(\mathrm{A}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for $M$ in $G_2\left({ℂ}^{m+2}\right)$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi {\phi }_1$ induced by two structure tensors $\phi$ and ${\phi }_1$. That is, this commuting shape operator is given by $\phi {\phi }_{1}A=A\phi {\phi }_1$. Using this condition, we prove that...

Real affine hypersurfaces of the complex space ${ℂ}^{n+1}$ are studied. Some properties of the structure determined by a J-tangent transversal vector field are proved. Moreover, some generalizations of the results obtained by V. Cruceanu are given.

We study real affine hypersurfaces $f:M\to {ℂ}^{n+1}$ with an almost contact structure (φ,ξ,η) induced by any J-tangent transversal vector field. The main purpose of this paper is to show that if (φ,ξ,η) is metric relative to the second fundamental form then it is Sasakian and moreover f(M) is a piece of a hyperquadric in ${ℝ}^{2n+2}$.

Real affine hypersurfaces of the complex space ${ℂ}^{n+1}$ with a J-tangent transversal vector field and an induced almost contact structure (φ,ξ,η) are studied. Some properties of hypersurfaces with φ or η parallel relative to an induced connection are proved. Also a local characterization of these hypersurfaces is given.

The aim of the present paper is to classify real hypersurfaces with pseudo-𝔻-parallel structure Jacobi operator, in non-flat complex space forms.

We define a procedure of reduction of locally conformal symplectic structures. We find a necessary and sufficient condition for this reduction to hold in terms of a special kind of de Rham cohomology class (tangent to the characteristic foliation) of the Lee form.