The two-dimensional wave maps global well-posedness conjecture

Let $(M,g)$ be a complete Riemannian manifold, and consider wave maps from $\mathbb R^2\times\mathbb R$ into $(M,g)$. The critical initial-data space is $H^1\times L^2$. Wave maps global well-posedness conjecture. (i) The two-dimensional wave maps equation is globally well-posed for small data in $H^1\times L^2$ for any complete target manifold. (ii) The two-dimensional wave maps equation is globally well-posed for large data in $H^1\times L^2$ for “good” target manifolds. Local and small-data global well-posedness at critical regularity had been established in several settings, while the stated two-dimensional critical global questions remained open in the source.