Gap Eigenvalues and Asymptotic Dynamics of Geometric Wave Equations on Hyperbolic Space

TL;DR

This paper investigates the stability and spectral properties of equivariant wave maps and Yang-Mills fields on hyperbolic space, revealing conditions for stability and the existence of metastable states linked to eigenvalues in the spectral gap.

Contribution

It establishes stability criteria for stationary solutions based on their image in the target manifold and demonstrates the presence of positive eigenvalues indicating metastability in certain cases.

Findings

Stable when the image is in a geodesically convex subset

Existence of positive eigenvalues in the spectral gap for large image coverage

Evidence for metastable states with slow decay rates

Abstract

In this paper we study $k$-equivariant wave maps from the hyperbolic plane into the $2$-sphere as well as the energy critical equivariant $SU(2)$ Yang-Mills problem on $4$-dimensional hyperbolic space. The latter problem bears many similarities to a $2$-equivariant wave map into a surface of revolution. As in the case of $1$-equivariant wave maps considered in~\cite{LOS1}, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schr\"odinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.