Gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space (Q329683)

The authors consider the \(k\)-equivariant wave maps equation from \({\mathbb R}\times {\mathbb H}^{2}\to {\mathbb S}^{2}\) and the energy critical equivariant Yang-Mills problem on \({\mathbb R}\times {\mathbb H}^{4}\) with gauge group \(\mathrm{SU}(2)\). After the usual equivariant reductions, both equations take the form NEWLINE\[NEWLINE \psi_{tt}-\psi_{rr}-\coth r\psi_{r}+\frac{k^{2}g(\psi)g'(\psi)}{\sinh^{2}r}=0, \eqno{(1)} NEWLINE\]NEWLINE where \((\psi,\theta)\) are geodesic polar coordinates on a target surface of revolution \({\mathcal M}\), and \(g\) determines the metric. In the case of \(k\)-equivariant wave maps, the authors set \({\mathcal M} = S^{2}\) and \(g(\psi) = \sin\psi\). For the Yang-Mills problem, \(k = 2\) and \(g(\psi) = \psi-\frac{1}{2}\psi^{2}\). Both problems admit a family of stationary solutions to (1) depending on a parameter. The equation (1) is linearized around a stationary solution and the spectrum of the corresponding Schrödinger equation is studied. It is demonstrated that depending on the parameter the spectrum can be purely continuous and can contain a unique eigenvalue in the spectral gap. Some geometric consequences of these results are presented.