  For compact, isometrically embedded Riemannian manifolds $ N \hookrightarrow
\mathbb{R}^L$, we introduce a fourth-order version of the wave map equation. By
energy estimates, we prove an priori estimate for smooth local solutions in the
energy subcritical dimension $ n = 1,2$. The estimate excludes blow-up of a
Sobolev norm in finite existence times. In particular, combining this with
recent work of local well-posedness of the Cauchy problem, it follows that for
smooth initial data with compact support, there exists a (smooth) unique global
solution in dimension $n = 1,2$. We also give a proof of the uniqueness of
solutions that are bounded in these Sobolev norms.
