arXiv research

Classifies Spin(7) structures using spinorial equations.

Study on solutions to spinorial Yamabe equation on manifolds with boundary.

This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, …

The paper finds bounds for a spinorial equation and applies it to a Bär-Hijazi-Lott invariant.

Refines spinorial Sobolev inequality on sphere, proving stability and new properties of Killing spinors.

Study on spinor field equation on spheres, focusing on blow-up analysis.

We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hija…

New spinorial field equation reveals geometric properties of Sasaki manifolds.

Proves principles and estimates for initial data sets in Einstein equations.

Paper proves existence of minimum energy solutions in 5D contact spin manifolds.

The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the…

Let (M^n,g) be a Riemannian spin manifold. The basic equations in supergravity models of type IIa string theory with 4-form flux involve a 3-form T, a 4-form F, a spinorial covariant derivative \nabla depending on \nabla^g, T, F, and a \nabla-parallel spinor field Ψ. We classify and construct many explicit families of …

This paper studies non-compactness in spinorial Yamabe-type problems on manifolds.

We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for bu…

We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are c…

In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescal…

Study on prescribing mean curvature for $S^2$ to $\mathbb{R}^3$ using spinor fields.

The paper solves spinorial Yamabe-type problems on spheres, with applications in geometry.

This paper presents a spinorial representation for SpinC manifolds in Euclidean space.

Study of four-dimensional Lorentzian manifolds with real Killing spinors.

Spinorial methods prove inequalities for special hypersurfaces.

We use a modified Bochner technique to derive an inequality relating the nodal set of eigenspinors to eigenvalues of the Dirac operator on closed surfaces. In addition, we apply this technique to solutions of similar spinorial equations.

Study of solutions for a spinorial Dirac equation with critical exponent.

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in $\…

The Dirac field is studied in a Lyra space-time background by means of the classical Schwinger Variational Principle. We obtain the equations of motion, establish the conservation laws, and get a scale relation relating the energy-momentum and spin tensors. Such scale relation is an intrinsic property for matter fields…

Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.

Spinorial representation for submanifolds in complex Lie groups.

In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of minimal surfaces. We also obtain as particular cases the spinorial characterization…

We prove that an isometric immersion of a simply connected Riemannian surface M in four-dimensional Minkowski space, with given normal bundle E and given mean curvature vector H \in Γ(E), is equivalent to a normalized spinor field \varphi \in Γ(ΣE \otimes ΣM) solution of a Dirac equation D\varphi=H\cdot\varphi on the s…

We characterize certain CR structures of arbitrary codimension (different from 3, 4 and 5) on Riemannian Spin$^c$ manifolds by the existence of a Spin$^c$ structure carrying a strictly partially pure spinor field. Furthermore, we study the geometry of Riemannian Spin$^c$ manifolds carrying a strictly partially pure spi…

Researchers found a spinorial representation for surfaces in 3D Lorentzian spaces.

We develop a spinorial description of CR structures of arbitrary codimension. More precisely, we characterize almost CR structures of arbitrary codimension on (Riemannian) manifolds by the existence of a Spin$^{c, r}$ structure carrying a partially pure spinor field. We study various integrability conditions of the alm…

Spinorial approach characterizes submanifolds in product spaces of constant curvature.

The paper constructs solutions to a critical Dirac equation on spheres.

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special Kähler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also…

Solving polynomial equations finds circle packings on surfaces.

We define (higher rank) spinorially twisted spin structures and deduce various curvature identites as well as estimates for the eigenvalues of the corresponding twisted Dirac operators.

This is a companion paper to arXiv:1207.3529 where we introduced the spinorial energy functional and studied its main properties in dimensions equal or greater than three. In this article we focus on the surface case. A salient feature here is the scale invariance of the functional which leads to a plenitude of critica…

Study spinorial Yamabe problem on product manifolds, proving spike layer solutions.

Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists \cite{onish1} and we treat in detail the only new series of inclusions providing the spinorial structur…

We study the spinorial Killing equation of supergravity involving a torsion 3-form $\T$ as well as a flux 4-form $\F$. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel $\G_2$-geometries and on the homogeneous Aloff-Wallach space. The constraint $\F \c…

We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in $\mathbb{R}^{2,1}$ to other Lorentzian space forms. We also characterize immersions of Riemannia…

Develops complex spinorial forms for all dimensions and signatures, proving Brinkmann waves in supergravity.

New mass definition for negative cosmological constant spacetimes.

Paper proves a spinorial version of Aubin's estimate for the Yamabe problem.

Spinorially constructs Sasakian and 3-Sasakian structures in arbitrary dimensions.

Defines a spinorial quasilocal mass for compact manifolds.

The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.