arXiv research

Paper defines a new functional for spinors on Euclidean manifolds.

We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Weiß, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sp…

A new spinorial heat flow framework studies geometric degeneration on 3-manifolds.

New spinorial functional connects Perelman's W- and F-functionals.

Study torsion parallel spinors on Lorentzian 4-manifolds and their evolution flows.

We show stability of pairs of Ricci flat metrics and parallel spinor fields with respect to the spinor flow, i.e. we show that the spinor flow with initial conditions near such pairs converges to a critical point with exponential speed. Moreover, we show stability of certain volume constrained critical points of the sp…

On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, φ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor φ. We investigate the basic properties of this functional and study its negative gradient f…

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

We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze invariant $\mathrm{Spin(7)}$ structures on quasi abelian Lie algebras.

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 $\…

In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This g…

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…

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…

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, …

Spinorial approach characterizes submanifolds in product spaces of constant curvature.

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

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…

The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.

Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces.

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 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…

New mass definition for negative cosmological constant spacetimes.

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.

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

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

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

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G/H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G.$ This in particular includes $SL_n(\mathbb{C})/SU(n),$ and extends the previously known spinorial representation of a surfa…

Defines a spinorial quasilocal mass for compact manifolds.

Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the imme…

The paper studies parallel spinor flows on 3D Cauchy hypersurfaces and provides initial data characterizations.

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…

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We …

New spinorial field equation reveals geometric properties of Sasaki 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…

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

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…

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…

This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operat…

We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich for $\R^3$ and B. Morel for $\Ss^3$ and $\HH^3$…

Unified framework for Riemannian, Kahler, and hyper-Kahler geometries in 4D.

We derive necessary conditions for the spinorial Witten-Nester energy to be well-defined for asymptotically locally AdS spacetimes. We find that the conformal boundary should admit a spinor satisfying certain differential conditions and in odd dimensions the boundary metric should be conformally Einstein. We show that …

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

Derives generalizations of the long neck principle and spectral width inequality.

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