# Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions

## TL;DR

This paper proves that the Atiyah-Singer Dirac operator's spectral properties depend continuously on boundary condition perturbations, with bounds influenced by geometric and boundary regularity conditions.

## Contribution

It establishes Riesz continuity of the Dirac operator under boundary condition perturbations on manifolds with boundary, extending perturbation estimates for elliptic operators.

## Key findings

- Riesz continuity of Dirac operator under boundary perturbations
- Lipschitz bounds depend on geometric and boundary regularity
- Perturbation estimates for functional calculi of elliptic operators

## Abstract

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.