Contact flows and integrable systems

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.     

On a class of pseudodifferential operators on the product of compact Lie groups

In this paper, a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in his paper of 1975, is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen in their book of 2010 (see also V. Fischer's paper of 2015), in that it exploits the harmonic analysis of the groups involved.     

Generalized Trans-Sasakian manifolds

In this paper, we study the class of almost contact metric manifolds which are conformal to Trans-Sasakian manifolds, and we construct concrete examples from almost Hermitian manifolds using the product of manifolds. As a consequence, we obtain several properties for the three-dimensional case.     

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.     

On a relation between spectral theory of lens spaces and Ehrhart theory

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO($n$) is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.     

New fourfolds from F‐theory

In this paper, we apply Borcea–Voisin's construction and give new examples of fourfolds containing a del Pezzo surface of degree six, which admit an elliptic fibration on a smooth threefold. Some of these fourfolds are Calabi–Yau varieties, which are relevant for the compactification of Type IIB string theory known as F‐theory. As a by‐product, we provide a new example of a Calabi–Yau threefold with Hodge numbers .     

New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .