Related papers: A Paneitz-type operator for CR pluriharmonic funct…
We prove that the integral operators $R_r$ and $H_r$ constructed in \cite{P} and such that $$f = \bar\partial_{\bold M} R_r(f) + R_{r+1}(\bar\partial_{\bold M} f) + H_r(f),$$ for a differential form $f \in C_{(0,r)}^{\infty}({\bold M})$ on…
In this paper we prescribe a fourth order conformal invariant 9the Paneitz Curvature) on five and six spheres. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem,…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
Using functional analysis and a Friedrichs approximation lemma for first order differential operators, we derive a global homotopy formula in large degrees for the tangential Cauchy-Riemann operator from local homotopy formulas without loss…
In this paper, we establish the existence of conformal deformations that uniformize fourth order curvature on 4-dimensional Riemannian manifolds with positive conformal invariants. Specifically, we prove that any closed, compact Riemannian…
Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.
We study two kinds of curvature invariants of Riemannian manifold equip\-ped with a complex distribution $D$ (for example, a CR-submanifold of an almost Hermitian manifold) related to sets of pairwise orthogonal subspaces of the…
The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to…
In this paper we shall assume that the ambient manifold is a pseudo-Riemannian space form $N^{m+1}_t(c)$ of dimension $m+1$ and index $t$ ($m\geq2$ and $1 \leq t\leq m$). We shall study hypersurfaces $M^{m}_{t'}$ which are polyharmonic of…
In this paper, we genelize the Heintze-Karcher type inequalities for fractional Q-curvature $Q_{2\gamma}$ on conformally compact Einstein manifolds. Such inequality holds for all $\gamma\in (0,1]$. In particular, for $\gamma=\frac{1}{2}$…
A global secondary CR invariant is defined as the integral of a pseudo-hermitian invariant which is independent of a choice of pseudo-Einstein contact form. We prove that any global secondary CR invariant on CR five-manifolds is a linear…
Based on the analytic property of the symmetric $q$-exponent $e_q(x)$, a new symmetric $q$-deformed Kadomtsev-Petviashvili ($q$-KP) hierarchy associated with the symmetric $q$-derivative operator $\partial_q$ is constructed. Furthermore,…
We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier…
We derive the first and second variation formula for the Green's function pole's value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively…
In this paper we study bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that…
I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit…
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal…
In this short note, as a simple application of the strong result proved recently by B\"ohm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by the new invariant cone constructions of…
Let $\mathrm S^3$ be the unit sphere of $\mathbb C^2$ with its standard Cauchy-Riemann (CR) structure. This paper investigates the CR geometry of curves in $\mathrm S^3$ which are transversal to the contact distribution, using the local CR…
Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero…