arXiv research

In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtl…

The paper extends Strichartz's conjecture to spinor bundles over real hyperbolic spaces.

Geometric correspondence between spinors and horospheres in hyperbolic space.

Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.

This paper connects spinors to horospheres in hyperbolic space.

Extends spinor-horosphere correspondence to higher dimensions and new spinor types.

Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.

We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in…

Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.

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

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

The paper proves geometric rigidity using harmonic twisted spinors and scalar curvature comparison.

New spinor fields reveal local or global geometric properties of manifolds.

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

In this paper we examine the structure of Riemannian manifolds with a special kind of Codazzi tensors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly irreducible holonomy representation with parallel spinors, i.e. with a holonomy group which is a semi…

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with thre…

The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.

We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfyin…

Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.

Simply connected 3-dimensional homogeneous manifolds $E(κ, τ)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $E(κ, τ)$. As…

Study of lambda lengths in figure eight knot complement using Eisenstein integers.

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on t…

We establish a type of positive energy theorem for asymptotically anti-de Sitter Einstein-Maxwell initial data sets by using Witten's spinoral techniques.

Solves Jang equation for hyperboloidal data, proving positive mass theorem.

The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asympto…

Suppose that $Σ=\partialΩ$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(Ω^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $Σ$ admits an isometric and isospin immersion $F$ into the hy…

New rigidity results for warped product domains.

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

Solves Jang's equation for hyperboloidal data in 4-7 dimensions, proving positive mass theorem.

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic form…

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

We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold …

The intention of this article is to give a flavour of some global problems in General Relativity. We cover a variety of topics, some of them related to the fundamental concept of 'Cauchy hypersurfaces': (1) structure of globally hyperbolic spacetimes, (2) the relativistic initial value problem, (3) constant mean curvat…

The article constructs Feynman propagators for normally hyperbolic operators on curved spacetimes.

New spinor types found on certain manifolds.

Spinor fields depending on tensor fields and other spinor fields are considered. The concept of extended spinor fields is introduced and the theory of differentiation for such fields is developed.

Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators …

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 study generalized Killing spinors on round spheres $\mathbb{S}^n$. We show that on the standard sphere $\mathbb{S}^8$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover we classify generalized Killing spinors on $\mathbb{S}^n$ whose associated symmetric endomorphism has at most two eigenva…

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X…

We provide explicit spinor representations for Clifford algebras.

This paper studies Sasakian quasi-Killing spinors on 3D Sasakian manifolds.

We consider gauged twistor spinors which are supersymmetry generators of supersymmetric and superconformal field theories in curved backgrounds. We show that the spinor bilinears of gauged twistor spinors satify the gauged conformal Killing-Yano equation. We prove that the symmetry operators of the gauged twistor spino…

Study on hypersurfaces in Einstein manifolds using Killing spinors.

In this review, basic definitions of spin geometry are given and some of its applications to supersymmetry, supergravity and condensed matter physics are summarized. Clifford algebras and spinors are defined and the first-order differential operators on spinors which lead to the definitions of twistor and Killing spino…