A Weighted L^2-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds

TL;DR

The paper addresses how interior geometry affects the Witten spinor on complete, nonnegatively curved manifolds that are asymptotically Schwarzschild. By constructing a conformal compactification and relating the Witten spinor to the Dirac Green's function on the compactified manifold, the authors isolate the interior geometry's influence to the spectral gap $\inf\operatorname{spec}(\tilde{\mathcal{D}}^2)$. They develop a regularized Green's-function framework with explicit counter-terms derived from the sphere, establishing weighted $L^1$ and $L^2$ bounds for the spinor and its deviations. The results yield concrete, geometry-robust estimates that avoid dependence on interior isoperimetric constants and extend to general dimension, connecting spinorial analysis to spectral data of the conformal Dirac operator.

Abstract

We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold $(M^n,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.

Figures (1)

• Figure 1: The asymptotically Schwarzschild manifold $(M,g)$ and its conformal compactification $(\bar{M}, \tilde{g})$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.5
• Theorem 3.6
• Theorem 4.1