Asymptotics of the self-dual deformation complex

TL;DR

This paper investigates the asymptotic behavior of the self-dual deformation complex on cylindrical manifolds, establishing decay rates, resolving a conjecture for hyperbolic 3-manifolds, and exploring implications for geometric gluing techniques.

Contribution

It provides a detailed analysis of indicial roots, confirms a conjecture in specific cases, and demonstrates the existence of both true and false instances, advancing understanding of self-dual structures.

Findings

Determined optimal decay rates for solutions on cylindrical self-dual manifolds.

Resolved the Kovalev-Singer conjecture for hyperbolic rational homology 3-spheres.

Showed existence of infinitely many cases where the conjecture holds and fails.

Abstract

We analyze the indicial roots of the self-dual deformation complex on a cylinder $(\mathbb{R} \times Y^3, dt^2 + g_Y)$, where $Y^3$ is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual manifold with cylindrical ends having cross-section $Y^3$. We also resolve a conjecture of Kovalev-Singer in the case where $Y^3$ is a hyperbolic rational homology 3-sphere, and show that there are infinitely many examples for which the conjecture is true, and infinitely many examples for which the conjecture is false. Applications to gluing theorems are also discussed.