Memoirs of the American Mathematical Society 2014; 108 pp; softcover Volume: 228 ISBN10: 0821892150 ISBN13: 9780821892152 List Price: US$76 Individual Members: US$45.60 Institutional Members: US$60.80 Order Code: MEMO/228/1069
 The authors consider the Schrödinger Map equation in \(2+1\) dimensions, with values into \(\mathbb{S}^2\). This admits a lowest energy steady state \(Q\), namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that \(Q\) is unstable in the energy space \(\dot H^1\). However, in the process of proving this they also show that within the equivariant class \(Q\) is stable in a stronger topology \(X \subset \dot H^1\). Table of Contents  Introduction
 An outline of the paper
 The Coulomb gauge representation of the equation
 Spectral analysis for the operators \(H\), \(\tilde H\); the \(X,L X\) spaces
 The linear \(\tilde H\) Schrödinger equation
 The time dependent linear evolution
 Analysis of the gauge elements in \(X,LX\)
 The nonlinear equation for \(\psi\)
 The bootstrap estimate for the \(\lambda\) parameter
 The bootstrap argument
 The \(\dot H^1\) instability result
 Bibliography
