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Analysis and spectral theory of neck-stretching problems

Thibault Langlais

TL;DR

The paper develops a comprehensive analytic framework for neck-stretching gluing of two asymptotically cylindrical (EAC) manifolds along a neck of length $2T$, focusing on elliptic operators $P_T$ and the existence of a Fredholm inverse in the complement of a finite-dimensional substitute kernel/cokernel when the indicial operator has a single real root. It combines translation-invariant PDE analysis on cylinders with weighted-geometry techniques on EAC manifolds to construct approximate solutions and reduce obstructions to a finite-dimensional characteristic system, proving uniform $T$-dependent inverse estimates. The work yields precise descriptions of the low-lying spectrum of the Laplacian on differential forms in the neck limit, including density formulas tied to the cross-section topology, and applies these results to twisted connected sums that realize compact $G_2$-manifolds with improved quantitative control. It also connects to swampland physics by clarifying how towers of light Kaluza–Klein states emerge from neck geometry in $G_2$ compactifications, with the cross-section Betti numbers governing decay rates and densities of low eigenvalues.

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.

Analysis and spectral theory of neck-stretching problems

TL;DR

The paper develops a comprehensive analytic framework for neck-stretching gluing of two asymptotically cylindrical (EAC) manifolds along a neck of length , focusing on elliptic operators and the existence of a Fredholm inverse in the complement of a finite-dimensional substitute kernel/cokernel when the indicial operator has a single real root. It combines translation-invariant PDE analysis on cylinders with weighted-geometry techniques on EAC manifolds to construct approximate solutions and reduce obstructions to a finite-dimensional characteristic system, proving uniform -dependent inverse estimates. The work yields precise descriptions of the low-lying spectrum of the Laplacian on differential forms in the neck limit, including density formulas tied to the cross-section topology, and applies these results to twisted connected sums that realize compact -manifolds with improved quantitative control. It also connects to swampland physics by clarifying how towers of light Kaluza–Klein states emerge from neck geometry in compactifications, with the cross-section Betti numbers governing decay rates and densities of low eigenvalues.

Abstract

We study the mapping properties of a large class of elliptic operators in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length . In the limit where , we reduce the question of constructing approximate solutions of 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 on the cylinder, we construct a Fredholm inverse for 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 -manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.
Paper Structure (20 sections, 29 theorems, 264 equations)

This paper contains 20 sections, 29 theorems, 264 equations.

Table of Contents

  1. Introduction and motivation
  2. Setup and discussion of results
  3. Model gluing problem
  4. Substitute kernel and cokernel
  5. Results and strategy
  6. Translation-invariant elliptic PDEs on cylinders
  7. Analysis on cylinders by separation of variables
  8. Polyhomogeneous sections
  9. Existence of solutions
  10. Construction of solutions
  11. Analysis on EAC manifolds
  12. Characteristic system
  13. Main construction
  14. Spectral aspects
  15. Approximate harmonic forms
  16. ...and 5 more sections

Key Result

Proposition 2.4

With the above setup, let $p > 1$ and $l \in \mathbb{N}$. Then the map is uniformly bounded as $T \rightarrow \infty$. Moreover there exist constants $C, C' > 0$ such that for $T$ large enough and for any $u \in W^{k+l,p}(E_T)$ we have:

Theorems & Definitions (54)

  • Remark 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 44 more