The Dirac Operator on Untrapped Surfaces

Abstract

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 rigidity results for the constraint equations with spherical boundary as well as uniqueness results for constant mean curvature surfaces in Minkowski space.

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“…As a first consequence, we have the estimate proved by Raulot [2013] for the first eigenvalue of the Dirac operator on †.…”

“…A spinorial Reilly-type inequality for manifolds with boundary. Here, we prove a spinorial Reilly-type inequality (see [Liu and Yau 2003] and [Raulot 2013]).…”

“…It is by now standard (see [Hijazi et al 2001b;) to make use of (15) for a compact Riemannian spin manifold M with nonnegative scalar curvature R, together with the solution of an appropriate boundary value problem for the Dirac operator D of M , in order to establish a certain integral inequality for the induced Dirac operator = D of the boundary hypersurface @M D †. Raulot [2013] uses such arguments for compact manifolds whose scalar curvature satisfies (16). In this section, we generalize the holographic principle for the existence of parallel spinors proved in [Hijazi and Montiel 2014] in the context studied in [Raulot 2013].…”

“…Raulot [2013] uses such arguments for compact manifolds whose scalar curvature satisfies (16). In this section, we generalize the holographic principle for the existence of parallel spinors proved in [Hijazi and Montiel 2014] in the context studied in [Raulot 2013].…”