Ivan Minchev

Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem

A complete solution to the quaternionic contact Yamabe problem on the seven-dimen- sional sphere is given. Extremals for the Sobolev inequality on the seven-dimensional Heisenberg group are explicitly described and the best constant in theL 2 Folland-Stein embedding theorem is determined.

Quaternionic Kähler and hyperKähler manifolds with torsion and twistor spaces

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term.

Twistor and reflector spaces of almost para-quaternionic manifolds

The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds.We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.Under the action of the c-map, special K¨ahler manifolds are mapped into a class of quaternion-K¨ahler spaces. We explicitly construct the corresponding Swann bundle or hyperk¨ahler cone, and determine the hyperk¨ahler potential in terms of the prepotential of the special K¨ahler geometry.We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexion. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kaehler geometry, but from the perspective of non-integrable holonomy systems.An almost para-CR structure on a manifold

On the equivalence of quaternionic contact structures

M

Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem

is given by a distribution

The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group

HM \subset TM

Quaternionic contact Einstein manifolds

together with a field

Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group

K \in \Gamma({\rm End}(HM))

Quaternionic contact hypersurfaces in hyper-Kähler manifolds

of involutive endomorphisms of

THE TWISTOR SPACE OF A QUATERNIONIC CONTACT MANIFOLD

HM