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On the regularity of spherically symmetric wave maps. (English) Zbl 0744.58071
Wave maps are critical points $$U: M\to N$$ of the Lagrangian $${\mathcal L}[U]=\int_ M\| dU\|^ 2$$, where $$M$$ is an Einsteinian manifold and $$N$$ a Riemannian one. For the case $$M=\mathbb{R}^{2,1}$$ and $$U$$ a spherically symmetric map, it is shown that the solution to the Cauchy problem for $$U$$ with smooth initial data of arbitrary size is smooth for all time, provided the target manifold $$N$$ satisfies the two conditions that (1) it is either compact or there exists an orthonormal frame of smooth vectorfields on $$N$$ whose structure functions are bounded, and (2) there are two constants $$c$$ and $$C$$ such that the smallest eigenvalue $$\lambda$$ and the largest eigenvalue $$\Lambda$$ of the second fundamental form $$k_{AB}$$ of any geodesic sphere $$\Sigma(p,s)$$ of radius $$s$$ centered at $$p\in N$$ satisfy $$s\lambda \geq c$$ and $$s\Lambda \leq C(1+s)$$.
This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution.
Reviewer: V.Christodoulou

##### MSC:
 58J32 Boundary value problems on manifolds 58E20 Harmonic maps, etc. 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C20 Global Riemannian geometry, including pinching
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