# Concentration Compactness for Critical Wave Maps

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*Joachim Krieger; Wilhelm Schlag*

A publication of the European Mathematical Society

Wave maps are the simplest wave equations taking their values in a
Riemannian manifold \((M,g)\). Their Lagrangian is the same as for the
scalar equation, the only difference being that lengths are measured with
respect to the metric \(g\). By Noether's theorem, symmetries of the
Lagrangian imply conservation laws for wave maps, such as conservation of
energy.

In coordinates, wave maps are given by a system of semilinear wave
equations. Over the past 20 years important methods have emerged which address
the problem of local and global wellposedness of this system. Due to weak
dispersive effects, wave maps defined on Minkowski spaces of low dimensions,
such as \(\mathbb R^{2+1}_{t,x}\), present particular technical
difficulties. This class of wave maps has the additional important feature of
being energy critical, which refers to the fact that the energy scales exactly
like the equation.

Around 2000 Daniel Tataru and Terence Tao, building on earlier work of
Klainerman-Machedon, proved that smooth data of small energy lead to global
smooth solutions for wave maps from 2+1 dimensions into target manifolds
satisfying some natural conditions. In contrast, for large data, singularities
may occur in finite time for \(M =\mathbb S^2\) as target. This
monograph establishes that for \(\mathbb H\) as target the wave map
evolution of any smooth data exists globally as a smooth function.

While the authors restrict themselves to the hyperbolic plane as target the
implementation of the concentration-compactness method, the most challenging
piece of this exposition, yields more detailed information on the solution.
This monograph will be of interest to experts in nonlinear dispersive
equations, in particular to those working on geometric evolution equations.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in discrete mathematics, geometry and topology.