Eigenvalue Estimate of the Dirac operator and Rigidity of Poincare-Einstein Metrics

TL;DR

This paper refines eigenvalue estimates for the Dirac operator on spin manifolds with boundary, establishing rigidity results for Poincare-Einstein metrics, showing they must be standard models under certain eigenvalue conditions.

Contribution

It proves eigenvalue estimates under weaker assumptions and demonstrates that specific eigenvalue conditions imply the manifold is a standard model space.

Findings

Eigenvalue estimates hold with weaker boundary conditions.

Rigidity results show manifolds are standard models when eigenvalues reach extremal values.

Poincare-Einstein metrics are standard hyperbolic or Euclidean models under certain spectral conditions.

Abstract

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant.In two cases the Poincare-Einstein metrics are standard hyperbolic metric.