Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Alvarez--Wentworth boundary conditions

2022/11/08 by Mathieu Dutour Sikirić, Dutour, Mathieu
Mathematics · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2211.04040

openalex publication_date 2022/11/08 · openalex created_date 2022/11/14 · openalex updated_date 2026/07/28

Abstract

A cuspidal end is a type of metric singularity, described as a product S¹ × ] a, +∞ [ with the Poincaré metric. The underlying set can also be seen as ℝ × ] a, +∞ [ subject to the action of the translation T : ( x,y ) \longrightarrow ( x+1, y ). On it, one may consider a holomorphic line bundle L, coming from a unitary character of the group generated by T. The complex modulus induces a flat metric on L, and a pseudo-Laplacian Δ_L,0 acting on functions can be associated to the Chern connection. One needs to specify boundary conditions, and they are here chosen to be the Alvarez--Wentworth boundary conditions, which are a combination of Dirichlet and Neumann boundary conditions. The aim of this paper is to find the asymptotic behavior of the zeta-regularized determinant det ( Δ_L,0 + μ), as μ> 0 goes to infinity for any a, and also as a goes to infinity for μ= 0.

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