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

2025/03/03 by Reuß, Jonah · 3 citations
#53C18 (Secondary) #53C37 (Primary) 34L40 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.01602

Abstract

We prove a necessary criterion for the (non-)existence of nontrivial solutions to the Dirac equation Dψ=i A ⋅_Cl ψ on Riemannian manifolds that are either closed or of bounded geometry. This generalizes a result of Rupert Frank and Michael Loss on ℝⁿ where the criterion relates the Lⁿ-norm of A to the Sobolev constant on ℝⁿ. 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 \mathbbSⁿ.

Cited by

• Conformal invariants for the zero mode equation
• On the spectrum of the magnetic Dirac operator
• Stability of spinorial Sobolev inequalities on \mathbbSⁿ

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