Analysis and spectral theory of neck-stretching problems

TL;DR

This paper investigates the spectral properties of elliptic operators on manifolds with cylindrical ends as they are glued along a neck, providing new insights into eigenvalue decay, inverse construction, and applications to $G_2$-manifolds.

Contribution

It introduces a method to construct Fredholm inverses for elliptic operators in neck-stretching problems and offers geometric interpretations of obstructions, advancing spectral analysis in geometric gluing.

Findings

Constructed a Fredholm inverse with controlled growth for large neck length T.

Analyzed decay rates and density of low eigenvalues of the Laplacian on differential forms.

Provided improved estimates for compact $G_2$-manifolds built via twisted connected sum.

Abstract

We study the mapping properties of a large class of elliptic operators $P_T$ in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length $2T$. In the limit where $T \rightarrow \infty$, we reduce the question of constructing approximate solutions of $P_T u = f$ to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system. Under some assumptions on the real roots of the model operator $P_0$ on the cylinder, we construct a Fredholm inverse for $P_T$ with good control on the growth of its norm. As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact $G_2$-manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.