Related papers: Initial data rigidity via Dirac-Witten operators

In this article, we revisit the initial data rigidity theorem of Eichmair, Galloway and Mendes (arxiv:2009.09527). The goal is to strengthen their result by showing that the initial data sets concerned carry a vector field that is lightlike…

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is…

In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases. In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main…

We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs here is to establish conditions that ensure that a marginally outer trapped surface is "weakly outermost". A…

Using spinors, we show a dihedral type rigidity for polyhedral initial data sets. This rigidity connects spacetime positive mass theorem, dihedral rigidity and capillary marginally trapped surfaces. Our method is to extend the rigidity…

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we…

We establish a spacetime positive mass theorem and rigidity statement for asymptotically flat spin initial data sets with a codimension one singularity controlled by a matching Bartnik data condition involving spacetime rotations, and…

We pursue the idea of constructing higher spin fields as solutions to twisted Dirac operators. As general results we find that twisted prenormally hyperbolic first order operators (such as the Dirac operator) on globally hyperbolic…

Using Weitzenb\"ock techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields…

We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu-|J|$ has a positive lower bound). In a…

In this note, we consider some initial data rigidity results concerning marginally outer trapped surfaces (MOTS). As is well known, MOTS play an important role in the theory of black holes and, at the same time, are interesting spacetime…

Let $(\overline M,\overline g)$ be a time- and space-oriented Lorentzian spin manifold, and let $M$ be a compact spacelike hypersurface of $\overline M$ with induced Riemannian metric $g$ and second fundamental form $K$. If $(\overline…

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.

In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang…

In this work, we present several rigidity results for compact free boundary hypersurfaces in initial data sets with boundary. Specifically, in the first part of the paper, we extend the local splitting theorems from [G. J. Galloway and H.…

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to…

In this note we prove a theorem on non-vacuum initial data for general relativity. The result presents a ``rigidity phenomenon'' for the extrinsic curvature, caused by the non-positive scalar curvature. More precisely, we state that in the…

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…

Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with…

Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel…