On the well-posedness of the wave map problem in high dimensions

Abstract

We construct a gauge theoretic change of variables for the wave map from $R\times R^n$ into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - $n\ge 4$ - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into {\it compact} Lie groups and symmetric spaces with small critical initial data and $n\ge 4$.