= Global regularity for one-dimensional wave maps
For smooth localized compatible <wave map Cauchy data> and compact Riemannian target, one-dimensional <wave maps> stay smooth globally. In null coordinates, the equation is $D_v\phi_u=D_u\phi_v=0$. Compatibility of the target metric with the <covariant derivative> makes $|\phi_u|^2$ independent of $v$ and $|\phi_v|^2$ independent of $u$. These transported derivative bounds and higher <energy estimates> prevent finite-time breakdown.
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