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arXiv · 2503.01602

A note on the existence of nontrivial zero modes on Riemannian manifolds

Jonah Reuß
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Abstract

We prove a necessary criterion for the (non-)existence of nontrivial solutions to the Dirac equation $D\psi=i A \cdot_{Cl} \psi$ 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 $\mathbb{S}^n$.

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Jonah Reuß. 2025-03-03. A note on the existence of nontrivial zero modes on Riemannian manifolds. https://arxiv.org/abs/2503.01602

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