An instability of hyperbolic space under the Yang-Mills flow

TL;DR

This paper investigates the stability of hyperbolic 3-space under the Yang-Mills flow, revealing that certain torsion-full perturbations grow exponentially, indicating an instability of the background geometry.

Contribution

It introduces a novel analysis of the Yang-Mills flow on hyperbolic space, identifying torsion-related unstable modes and confirming their growth through numerical simulations.

Findings

Exponential growth of certain perturbation modes indicating instability.

Unstable modes are characterized by non-zero torsion.

Numerical simulations support the theoretical instability results.

Abstract

We consider the Yang-Mills flow on hyperbolic 3-space. The gauge connection is constructed from the frame-field and (not necessarily compatible) spin connection components. The fixed points of this flow include zero Yang-Mills curvature configurations, for which the spin connection has zero torsion and the associated Riemannian geometry is one of constant curvature. Perturbations to the fixed point corresponding to hyperbolic 3-space can be expressed as a linear superposition of distinct modes, some of which are exponentially growing along the flow. The growing modes imply the divergence of the (gauge invariant) perturbative torsion for a wide class of initial data, indicating an instability of the background geometry that we confirm with numeric simulations in the partially compactified case. There are stable modes with zero torsion, but all the unstable modes are torsion-full. This leads us to speculate that the instability is induced by the torsion degrees of freedom present in the Yang-Mills flow.