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I’ve just uploaded to the arXiv a new paper, “Global regularity of wave maps III. Large energy from ${\Bbb R}^{1+2}$ to hyperbolic spaces“, to be submitted when three other companion papers (“Global regularity of wave maps” IV, V, and VI) are finished. This project (which I had called “Heatwave”, due to the use of a heat flow to renormalise a wave equation) has a somewhat lengthy history to it, which I will now attempt to explain.

For the last nine years or so, I have been working on and off on the global regularity problem for wave maps $\phi: {\Bbb R}^{1+d} \to M$. The wave map equation $(\phi^* \nabla)^\alpha \partial_\alpha \phi=0$ is a nonlinear generalisation of the wave equation $\partial^\alpha \partial_\alpha \phi = 0$ in which the unknown field $\phi$ takes values in a Riemannian manifold $M = (M,h)$ rather than in a vector space (much as the concept of a harmonic map is a nonlinear generalisation of a harmonic function). This equation (also known as the nonlinear $\sigma$ model) is one of the simplest examples of a geometric nonlinear wave equation, and is also arises as a simplified model of the Einstein equations (after making a U(1) symmetry assumption). The global regularity problem seeks to determine when smooth initial data for a wave map (i.e. an initial position $\phi_0: {\Bbb R}^d \to M$ and an initial velocity $\phi_1: {\Bbb R}^d \to TM$ tangent to the position) necessarily leads to a smooth global solution.

The problem is particularly interesting in the energy-critical dimension d=2, in which the conserved energy $E(\phi) := \int_{{\Bbb R}^d} \frac{1}{2} |\partial_t \phi|_{h(\phi)}^2 + \frac{1}{2} |\nabla_x \phi|_{h(\phi)}^2\ dx$ becomes invariant under the scaling symmetry $\phi(t,x) \mapsto \phi(t/\lambda,x/\lambda)$. (In the subcritical dimension d=1, global regularity is fairly easy to establish, and was first done by Gu and by Ladyzhenskaya-Shubov; in supercritical dimensions $d \geq 3$, examples of singularity formation are known, starting with the self-similar examples of Shatah.)

It is generally believed that in two dimensions, singularities can form when M is positively curved but that global regularity should persist when M is negatively curved, in analogy with known results (in particular, the landmark paper of Eells and Sampson) for the harmonic map heat flow (a parabolic cousin of the wave map equation). In particular, one should always have global regularity when the target is a hyperbolic space. There has been a large number of results supporting this conjecture; for instance, when the target is the sphere, examples of singularity formation have recently been constructed by Rodnianski-Sterbenz and by Krieger-Schlag-Tataru, while for suitably negatively curved manifolds such as hyperbolic space, global regularity was established assuming equivariant symmetry by Shatah and Tahvildar-Zadeh, and assuming spherical symmetry by Christodoulou and Tahvildar-Zadeh. I will not attempt to mention all the other results on this problem here, but see for instance one of these survey articles or books for further discussion.