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Quaternionic contact manifolds with a closed fundamental 4-form, with St. Ivanov, arXiv:0810.3888, 2008.
We show that the fundamental horizontal 4-form on a quaternionic contact manifold of dimension at least eleven is closed if and only if the torsion of the Biquard connection vanishes. This condition characterizes quaternionic contact structures which are locally qc homothetic to a 3-sasakian one.

 

Conformal paracontact curvature and the local flatness theorem, with St. Ivanov and S.  Zamkovoy, MPIM2007-90, 2007.

A curvature-type tensor invariant called para contact (pc) conformalcurvature is defined on a paracontact manifolds. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group iff the pc conformal curvature vanishes provided the dimension is bigger than three. A different proof of the Chern-Moser-Webster theorem is given, showing that the vanishing of the Chern-Moser invariant is necessary and sufficient condition a CR-structure to be CR equivalent to a quadric in $\mathbb{C}^{n+1}$ provided n>1. The proof establishes also Cartan's result in dimension three. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.

Conformal quaternionic contact curvature and the local sphere theorem, with St. Ivanov, MPIM2007-79 2007.

A curvature-type tensor invariant called quaternionic contact (qc) conformal curvature is defined on a qc manifolds in terms of the curvature and torsion of the Biquard connection. The discovered tensor is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser invariant in CR geometry. It is shown that a qc manifold is locally qc conformal to the standard flat qc structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure on the sphere if and only if the qc conformal curvature vanishes.

Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, with St. Ivanov and I. Minchev, MPIM2007-22, 2007.

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

Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem, with St. Ivanov and I. Minchev, International Centre for Theoretical Physics preprint IC/2006/117, 2006.

The paper is a study of the conformal geometry of quaternionic contact manifolds with the associated Biquard connection. We give a partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. In particular, the scalar curvature of the Biquard connection with vanishing torsion is a global constant. We consider interesting classes of functions on hypercomplex manifold and their restrictions to hypersurfaces. We show a '3-Hamiltonian form' of infinitesimal automorphisms of quaternionic contact structures and transformations preserving the trace-free part of the horizontal Ricci tensor of the Biquard connection. All conformal deformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing trace-free part of the horizontal Ricci tensor of the Biquard connection are explicitly described.

Lp estimates and asymptotic behavior for finite energy solutions of extremals toHardy-Sobolev inequalities, submitted.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp Lq regularity and asymptotic behavior for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of Rn. In addition, we find the best constant and extremals in the case of the considered L2 Hardy-Sobolev inequality. Finally we give some other possible applications and directions to explore concerning non-completeness of metrics with finite volume and bounded scalar curvature and stationary cylindrical states of the Vlasov-Poisson system. The results are related also to Hardy inequalities involving distance to the boundary.

Strong unique continuation for generalized Baouendi-Grushin operators, with N. Garofalo, 21 pages, to appear in Comm. PDE.

Our main result gives a quantitative control of the order of zero of a weak solution to perturbations of the Baouendi-Grushin operator.  The perturbations are tailored on the geometry of the Baouendi-Grushin operator and should be interpreted as a sort of Lipschitz continuity with respect to a suitable pseudo-distance associated to the system of vector fields. Our result generalizes the famous result due to Aronszaijn, Krzywicki and Szarski valid for elliptic operators in divergence form with Lipschitz continuous coefficients.

Regularity near the characteristic boundary for sub-laplacian operatorsPacific J Math.,  227 (2006), no. 2, 361--397.

We prove that the best constant in the Folland-Stein embedding theorem on Carnot groups is achieved. This implies the existence of a positive solution of the Yamabe type equation on Carnot groups. The second goal of the paper is to show  regularity of the Green's function and solutions of the Yamabe equation involving the sub-Laplacian near the characteristic boundary of a domain in the considered groups.

 A note on the stability of local zeta functions, Proc. Amer. Math. Soc, 134 (2006), 81-91.

We show the existence of an interval of stability under small perturbations of local zeta functions corresponding to non-trivial local solutions of an elliptic equation with Lipschitz coefficients.

Overdetermined BVP, quadrature domains, and applications, with D. Khavinson and A. Solynin,  Comput. Methods Funct. Theory, 5 (2005),  19-48.

We consider an overdetermined problem in planar multiply connected domains. This problem is solvable if and only if the domain is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution of the problem in some special cases and discuss some applications concerning the shape of electrified droplets and small air bubbles in a fluid flow.  Other cases, that are left open, include a variation of the  Serrin problem but with moving boundaries. This case is also of interest to computing the sharp constant in an inequality, finer than the isoperimetric inequality, concerning the analytic content of a set.

Symmetry properties of positive entire solutions of Yamabe type equations on groups of Heisenberg type , with N. Garofalo, Duke Math. J. 106 (2001), no. 3, 411--448.

The main result of this paper is the determination of the entire solutions with partial symmetry of the Yamabe equation on Iwasawa type groups. As a corollary we find the best constant in the L2 Folland-Stein Sobolev type inequality restricted to functions with partial symmetry.

Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groupsp , with N. Garofalo, Math. Ann. 318 (2000), no. 3, 453--516.

Part of this paper is devoted to the boundary regularity of weak solutions to the Yamabe equation on Carnot groups near characteristic points of the boundary. Another part concerns non-existence of solutions to the Yamabe equation results on certain domains in Heisenberg or Iwasawa type groups.

 

Proceedings papers


Strong unique continuation for generalized Baouendi-Grushin operators, with N. Garofalo, Proceedings of the 4th ISAAC Congress, Toronto, 2003.

The non-linear Dirichlet problem and the CR Yamabe problem, with N. Garofalo, Boundary value problems for elliptic and parabolic operators (Catania, 1998) Matematiche (Catania) 54 (1999), suppl., 75–93.

 


Theses

Yamabe type equations on Carnot groups, Ph. D. thesis Purdue University, 2000, advisor Professor Nicola Garofalo

Opérateur à puissances bornées et decroissance de l’énergie locale, DEA thesis Université Bordeaux I, 1994, advisor Professor Vesselin Petkov