  We study the gradient flow of the $L^2-$norm of the second fundamental form
of smooth immersions of two-dimensional surfaces into compact Riemannian
manifolds. By analogy with the results obtained for the Willmore flow in
Riemannian manifolds, we prove lifespan estimates in terms of the
$L^2-$concentration of the second fundamental form of the initial data and we
show existence of blowup limits. Under special condition both on the initial
data and on the target manifold, we prove a long time existence result for the
flow and subconvergence to a critical immersion.
