A note on the existence of nontrivial zero modes on Riemannian manifolds
Jonah Reuß
TL;DR
This work generalizes a sharp necessary condition for the existence of Dirac zero modes from Euclidean space to closed or bounded-geometry Riemannian manifolds by substituting the Sobolev constant with the Yamabe invariant. It derives a fundamental integral identity linking the Dirac operator, the twisting curvature $F^{\Sigma}$, and the Yamabe functional, yielding a universal bound for nontrivial solutions of $D\psi = i A\cdot_{\text{Cl}}\psi$; the bound is sharp for odd dimensions with equality realized on $\mathbb{S}^{n}$ and extended to manifolds of bounded geometry in the noncompact case. The equality analysis connects to Yamabe optimizers and Killing spinors, leading to a conformal-sphere classification of equality cases. Overall, the results provide intrinsic geometric criteria governing spinorial zero modes beyond the Euclidean setting and contribute to spinorial Yamabe-type problems in conformal geometry.
Abstract
We prove a necessary criterion for the (non-)existence of nontrivial solutions to the Dirac equation $Dψ=i A \cdot_{Cl} ψ$ on Riemannian manifolds that are either closed or of bounded geometry. This generalizes a result of Rupert Frank and Michael Loss on $\mathbb{R}^n$ where the criterion relates the $L^n$-norm of $A$ to the Sobolev constant on $\mathbb{R}^n$. On Riemannian manifolds the role of the Sobolev constant will be replaced by the Yamabe invariant. If $n$ is odd, we show that our criterion is sharp on $\mathbb{S}^n$.