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Ali Maalaoui

Publications and source records attributed to Ali Maalaoui.

At least 19 recordsLinked to original sources
arXiv2026

A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometry

Given a closed Riemannian Spin manifold $(M,g)$ of dimension greater or equal than four, we consider a generalized conformally invariant equation involving the Dirac operator with a non-linearity of convolution type. We show that the Aubib-type inequality corresponding to the problem is always strict, unless $(M,g)$ is conformal to the round sphere. In particular, this result provides an existence result for a ground state to the conformal Dirac-Einstein problem in dimension four. We point out that aside from some perturbative or special cases, this presents the first general existence result for the conformal Dirac-Einstein equations in dimension four.

math.DG↗
arXiv2025

Conformal Fractional Dirac Operator and Spinorial Q-curvature

In this paper we introduce the conformal fractional Dirac operator and its associated fractional spinorial Yamabe problem. We also present a Caffarelli-Silvestre type extension for this fractional operator, allowing us to express it as a Dirichlet-to-Neumann type operator. As a consequence, we exhibit energy inequalities associated to this operator along with a weighted type Sobolev inequality for spinors. In the second part of the paper, we focus on the critical operator (which can be local or non-local depending on the evenness of the dimension). We introduce a Q-curvature operator, acting on spinors generalizing the classical notion of the scalar Q-curvature.

math.DG↗
arXiv2025

Conformally Invariant Dirac Equation with Non-Local Nonlinearity

We study a conformally invariant equation involving the Dirac operator and a non-linearity of convolution type. This non-linearity is inspired from the conformal Einstein-Dirac problem in dimension 4. We first investigate the compactness, bubbling and energy quantization of the associated energy functional then we characterize the ground state solutions of the problem on the standard sphere. As a consequence, we prove an Aubin-type inequality that assures the existence of solutions to our problem and in particular the conformal Einstein-Dirac problem in dimension 4. Moreover, we investigate the effect of a linear perturbation to our problem, leading us to a Brezis-Nirenberg type result.

math.DG↗
arXiv2024

Singular Solutions for the Conformal Dirac-Einstein Problem on the Sphere

In this paper we investigate the existence of singular solutions to the conformal Dirac-Einstein system. Because of its conformal invariance, there are many similarities with the classical construction of singular solutions for the Yamabe problem. We construct here a family of singular solutions, on the three-dimensional sphere, having exactly two singularities.

math.DG↗
arXiv2023

Constructions of Delaunay-type solutions for the spinorial Yamabe equation on spheres

In this paper we construct singular solutions to the critical Dirac equation on spheres. More precisely, first we construct solutions admitting two points singularities that we call Delaunay-type solutions because of their similarities with the Delaunay solutions constructed for the singular Yamabe problem in \cite{MP1 , Schoen1989}. Then we construct another kind of singular solutions admitting a great circle as a singular set. These solutions are the building blocks for singular solutions on a general Spin manifold.

math.AP↗
arXiv2022

Conformal Dirac-Einstein equations on manifolds with boundary

In this paper we study Dirac-Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence result.

math.DG↗
arXiv2022

$\bar{Q}'$-curvature flow on Pseudo-Einstein CR manifolds

In this paper we consider the problem of prescribing the $\bar{Q}'$-curvature on three dimensional Pseudo-Einstein CR manifolds. We study the gradient flow generated by the related functional and we will prove its convergence to a limit function under suitable assumptions.

math.DG↗
arXiv2022

Compactness of Dirac-Einstein spin manifolds and horizontal deformations

In this paper we consider the Hilbert-Einstein-Dirac functional, whose critical points are pairs, metrics-spinors, that satisfy a system coupling the Riemannian and the spinorial part. Under some assumptions, on the sign of the scalar curvature and the diameter, we prove a compactness result for this class of pairs, in dimension three and four. This can be seen as the equivalent of the study of compactness of sequences of Einstein manifolds as in \cite{And0,N}. Indeed, we study the compactness of sequences of critical points of the Hilbert-Einstein-Dirac functional which is an extension of the Hilbert-Einstein functional having Einstein manifolds as critical points. Moreover we will study the second variation of the energy, characterizing the horizontal deformations for which the second variation vanishes. Finally we will exhibit some explicit examples.

math.DG↗
arXiv2021

Some properties of Dirac-Einstein bubbles

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $\mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-\frac{1}{2}$-Killing spinors on the round sphere $\mathbb{S}^3$.

math.AP↗
arXiv2020

Functional Determinant on Pseudo-Einstein 3-manifolds

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_θ$, related to the problem of prescribing the $Q'$-curvature, under the conformal change $θ\mapsto e^{w}θ$ with $w\in ¶$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in \cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in \cite{CY}.

math.DG↗
arXiv2020

Logarithmic Hardy-Littlewood-Sobolev Inequality on Pseudo-Einstein 3-manifolds and the Logarithmic Robin Mass

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we study the existence of a contact structure conformal to $θ$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green's function of the $P'$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. Then we exhibit an Aubin type result guaranteeing the existence of a minimizer for the total mass which yields the classical LHLS inequality.

math.DG↗
arXiv2019

Prescribing the $\bar Q^{\prime}$-Curvature on Pseudo-Einstein CR 3-Manifolds

In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove the existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality and non-normal ones that have a biharmonic leading term.

math.DG↗
arXiv2019

Singular CR structures of constant Webster curvature and applications

We consider the sphere $\Sph^{2n+1}$ equipped with its standard CR structure. In this paper we construct explicit contact forms on $\Sph^{2n+1}\setminus \Sph^{2k+1}$, which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular the curvature is positive if $2k< n-2$. As main applications, we provide two perturbative results. In the first one we prove the existence of infinitely many contact structures on $\Sph^{2n+1}\setminus τ(\Sph^{1})$ conformal to the standard one and having constant Webster curvature, where $τ(\Sph^{1})$ is a small perturbation of $\Sph^1$. In the second application, we show that there exist infinitely many bifurcating branches of periodic solutions to the CR Yamabe problem on $\Sph^{2n+1}\setminus \Sph^{1}$ having constant Webster curvature.

math.DG↗
arXiv2018

Palais-Smale sequences for the fractional CR Yamabe functional and multiplicity results

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely many solutions to the related equation.

math.AP↗