  We identify the wave maps type nonlinearities of incompressible Hookean
elastodynamics euqations in Lagerangian coordinates, and iterate them in the
adapted $U^2$-type spaces to prove the small data global well-posedness in the
critical Besov space $\dot{B}^{\frac{n}{2}+1}_{2,1}(\mathbb{R}^n)\times
\dot{B}^{\frac{n}{2}}_{2,1}(\mathbb{R}^n)\ (n\ge 2)$.
