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

Abstract

A cuspidal end is a type of metric singularity, described as a product $S^1\times \left]a,+\infty \right[$ with the Poincaré metric. The underlying set can also be seen as $\mathbb{R}\times \left]a,+\infty \right[$ subject to the action of the translation $T:\left(x,y\right)⟶\left(x+1,y\right)$. 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 ${\Delta }_{L,0}$ can be associated to the Chern connection, with Dirichlet boundary conditions. The aim of this paper is to find the asymptotic behavior of the zeta-regularized determinant $\det \left({\Delta }_{L,0}+\mu \right)$, as $\mu >0$ goes to infinity for any $a$, and also as $a$ goes to infinity for $\mu =0$.

Minor changes made: - Typo corrected in the statement of theorem 3.76 and 3.77 (a term 1/2*log(a) had been forgotten) - Proposition 3.52 was added to remove a term which was previously dealt with in (what is now) Proposition 3.62 - Rearrangements in section 2.4 - Some material was removed from the appendices