On the Dirac spectrum on degenerating Riemannian surfaces

2024/09/09 by Anghel, Cipriana
Mathematics · #58J05 #58J40 #58J50 #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory (math.SP) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2409.05616

openalex publication_date 2024/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length t of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial. The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus, in the spirit of the celebrated b-algebra of Melrose, which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents. We obtain smoothness of the spectral projectors, and t² log t regularity for the cusp-surgery trace of the relative resolvent in the degeneracy process as t \searrow 0.

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