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Georges Habib

Publications and source records attributed to Georges Habib.

At least 19 recordsLinked to original sources
arXiv2026

Laplace and Dolbeault operators on Sasakian manifolds

Sasakian manifolds, the odd-dimensional analogues of K\"ahler manifolds, carry two natural pairs of first-order Dolbeault-type operators on the full complex of differential forms, extending the Kohn--Rossi differentials. On such Sasaki manifolds, we establish K\"ahler-type identities for these operators, and relate the resulting Dolbeault Laplacians to the Hodge Laplacian. Unlike the K\"ahler case, where $\Delta=2\Delta_{\overline\partial}$, the relation has extra terms coming from the Reeb flow. Using these formulas and a Lefschetz decomposition, we derive lower and upper eigenvalue estimates for $\Delta$ on forms depending on the eigenvalues of the Lie derivative of the Reeb vector field.

math.DG↗
arXiv2025

On the spectrum of the magnetic Dirac operator

The magnetic Dirac operator describes the relativistic motion of a charged particle in a magnetic field. Although this operator got a lot of attention in physics many of its fundamental mathematical properties remain unexplored and this article is a first step towards filling this gap. To this end we provide a number of eigenvalue estimates for the magnetic Dirac operator on closed Riemannian manifolds and explicitly compute its spectrum for specific choices of the magnetic field on the flat torus and on the three-dimensional round sphere.

math.DG↗
arXiv2025

Magnetic Steklov operator on differential forms

In this paper, we introduce the magnetic Steklov operator on differential forms and show that the underlying boundary value problem is well-posed. Moreover, we show that an analogue of the Diamagnetic Inequality does not always hold for this operator, and we present some spectral computations of magnetic Steklov operators for $2$-dimensional and $4$-dimensional balls in Euclidean space.

math.SP↗
arXiv2025

Harmonic unit vector fields on 3-manifolds

We investigate harmonic unit vector fields with totally geodesic integral curves on 3-manifolds. Under mild curvature assumptions, we classify both the vector fields and the manifolds that support them. Our results are inspired by Carriere's classification of Riemannian flows on compact three-manifolds, as well as by the works of Geiges and Belgun on Killing vector fields on Sasakian manifolds.

math.DG↗
arXiv2025

Eigenvalue estimates of the Kohn-Dirac operator on CR manifolds

In this paper, we establish a new eigenvalue estimate for the Kohn-Dirac operator on a compact CR manifold. The equality case of this estimate is characterized by the existence of a CR twistor spinor. We then classify CR manifolds carrying such spinors by showing that the Webster Ricci tensor has at most two eigenvalues. In this context, we construct several examples.

math.DG↗
arXiv2024

The Buckling and Clamped Plate Problems on Differential Forms

We extend the buckling and clamped-plate problems to the context of differential forms on compact Riemannian manifolds with smooth boundary. We characterize their smallest eigenvalues and prove that, in the case of bounded Euclidean domains, their spectra without multiplicities on forms coincide with the spectra of the corresponding problems on functions. We obtain various estimates involving the first eigenvalues of the mentioned problems and the ones of the Hodge Laplacian with respect to Dirichlet and absolute boundary conditions on forms. These estimates generalize previous ones in the case of functions.

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arXiv2024

A note on the magnetic Steklov operator on functions

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger-Jammes type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias and Savo. In addition, we compute the full spectrum in the case of the Euclidean $2$-ball and $4$-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the non-magnetic case due to Colbois, Girouard and Hassannezhad.

math.DG↗
arXiv2024

Geometric singularities and Hodge theory

We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic under uniform perturbations of the geometric data. We consider applications of this to the Hodge-Dirac operator on differential forms induced by so-called rough Riemannian metrics, which can be of only measurable coefficient in regularity, on both compact and non-compact settings. As a consequence, we show that the kernel of the associated non-smooth Hodge-Dirac operator with respect to a rough Riemannian metric remains isomorphic to smooth and singular cohomology when the underlying manifold is compact.

math.DG↗
arXiv2024

Transverse geometric formality

A Riemannian metric on a closed manifold is said to be geometrically formal if the wedge product of any two harmonic forms is harmonic; equivalently, the interior product of any two harmonic forms is harmonic. Given a Riemannian foliation on a closed manifold, we say that a bundle-like metric is transversely geometrically formal if the interior product of any two basic harmonic forms is basic harmonic. In this paper, we examine the geometric and topological consequences of this condition.

math.DG↗
arXiv2023

A Poincaré formula for differential forms and applications

We prove a new general Poincaré-type inequality for differential forms on compact Riemannian manifolds with nonempty boundary. When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and scalar curvatures of the boundary only and characterize its limiting case in codimension one. A new Ros-type inequality for differential forms is also derived assuming the existence of a nonzero parallel form on the manifold.

math.DG↗
arXiv2023

A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the $\mathbb Z_2$-graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square.

math.DG↗
arXiv2022

Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the Hodge Laplacian on differential forms, and discuss similarities and differences of this new ``magnetic-type'' operator.

math.DG↗
arXiv2022

Biharmonic Steklov operator on differential forms

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the first eigenvalue, some of which involve eigenvalues of other problems such as the Dirichlet, Neumann, Robin and Steklov ones. Independently, new inequalities relating the eigenvalues of the latter problems are proved.

math.DG↗
arXiv2022

Eigenvalue estimates on weighted manifolds

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for (f)-minimal hypersurfaces and the spectrum of the Hodge Laplacian.

math.DG↗
arXiv2020

Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $ψ$ is a spinor that satisfies the equation $\nabla$X$ψ$ = AX $\times$ $ψ$ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product R x N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to S 2 x R 2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.

math.DG↗
arXiv2020

An Obata-type characterization of doubly-warped product K\''ahler manifolds

We give a characterization {\sl à la Obata} for certain families of K\''ahler manifolds. These results are in the same line as other extensions of the well-known Obata's rigidity theorem from \cite{Obata62}, like for instance the generalizations in \cite{RanjSant97} and \cite{Santhanam07}. Moreover, we give a complete description of the so-called K\''ahler doubly-warped product structures whose underlying metric is Einstein.

math.DG↗