Dimiter Vassilev

Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type

Here, G is a stratified, nilpotent Lie group, in short a Carnot group, of arbitrary step, and Ω ⊂ G is a domain which can be bounded or unbounded. The second order differential operator L represents a given sub-Laplacian on G. If g = r ⊕ j=1 Vj is a stratification of the Lie algebra g of G, with [V1, Vj ] ⊂ Vj+1 for 1 ≤ j < r, [V1, Vr] = {0}, we assume that a scalar product < ·, · > is given on g for which the V ′ j s are mutually orthogonal. The stratification allows to define a natural family of non-isotropic dilations ∆λ : g → g as follows ∆λ(X1 + ... + Xr) = λX1 + ... + λXr. The exponential map exp : g → G is an analytic diffeomorphism. It induces a group of dilations on G via the formula

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.

Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem

Analysis: Variational Problems Related to Sobolev Inequalities on Carnot Groups Groups of Heisenberg and Iwasawa Types Explicit Solutions to the Yamabe Equation Symmetries Solutions on Groups of Iwasawa Type Geometry: Quaternionic Contact Manifolds - Connection, Curvature and qc-Einstein Structures Quaternionic Contact Conformal Curvature Tensor The Quaternionic Contact Yamabe Pronlem and the Yamabe Constant of the qc Spheres CR Manifolds - Cartan and Chern-Moser Tensor and Theorem.

NON-KAEHLER HETEROTIC STRING SOLUTIONS WITH NON-ZERO FLUXES AND NON-CONSTANT DILATON

A bstractConformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (−)-connection on 6-dimensional non-Kähler nilmanifold are presented. In the conformally compact case the dilaton is determined by the real slices of the elliptic Weierstrass function. The dilaton of non-compact complete solutions is given by the fundamental solution of the Laplacian on R4. All solutions satisfy the heterotic equations of motion up to the first order of α′.

L p ESTIMATES AND ASYMPTOTIC BEHAVIOR FOR FINITE ENERGY SOLUTIONS OF EXTREMALS TO HARDY-SOBOLEV INEQUALITIES

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp L q regularity for finite energy solutions of p-Laplace equations involving critical exponents and possible singularity on a sub-space of ℝ n , which imply asymptotic behavior of the solutions at infinity. In addition, we find the best constant and extremals in the case of the considered L 2 Hardy-Sobolev inequality.

Quaternionic contact manifolds with a closed fundamental 4-form

We show that the fundamental 4-form on a quaternionic contact manifold of dimension at least eleven is closed if and only if the torsion endomorphism of the Biquard connection vanishes. This condition characterizes quaternionic contact structures which are locally qc homothetic to 3-Sasakian structures.

AN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD

We prove a CR version of the Obatas result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three dimensional unit sphere.

Overdetermined Boundary Value Problems, Quadrature Domains and Applications

We discuss an overdetermined problem in planar multiply connected domains Ω. This problem is solvable in Ω if and only if Ω 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.

Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D

We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four.

Strong Unique Continuation for Generalized Baouendi-Grushin Operators

must vanish identically in some neighborhood of zo. In other words non-trivial solutions can have at most finite order of vanishing. In this paper we study the strong unique continuation property for a class of “variable coefficient” operators whose “constant coefficient” model at one point is the so called Baouendi-Grushin operator [B], [Gr1], [Gr2]. We recall that the latter is the following operator on RN = Rn × Rm, N = n + m,