Following the work of Krieger, Schlag, and Tataru, we construct a family of blow-up solutions with finite energy norm to the equation

∂_t²u - Δg u = |u|⁴ u.

This family has a continuous rate of blow up, but In contrast to the case where g is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.

This is joint work with S. Shashahani.