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

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.

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