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

A Convergence Result for the Gradient Flow of $\int |A|^2$ in Riemannian Manifolds