## On the evolution of harmonic mappings of Riemannian surfaces.(English)Zbl 0595.58013

Let (M,$$\gamma)$$ be a Riemannian surface with metric tensor $$\gamma =(\gamma_{\alpha \beta})_{1\leq \alpha,\beta \leq 2}$$ and (N,g) an n- manifold with metric tensor $$g=(g_{ij})_{1\leq i,j\leq n}$$. For differentiable mappings $$u: M\to N$$ an energy is defined $E(u)=\int_{M}e(u)dM,\quad e(u)=(1/2)\gamma^{\alpha \beta}(x) g_{ij}(u) (\partial /\partial x^{\alpha})u^ i (\partial /\partial x^{\beta \quad})u^ j$ and the well-known equation for harmonic maps (1) $$-\Delta_ Mu=\Gamma (u)(\nabla u,\nabla u)_ M$$ where $$\Gamma^{\ell}_{ij}$$ are Christoffel symbols of the metric g and $$\Delta_ M$$ is the Laplace-Beltrami operator. The aim of the paper is twofold. First, for the evolution problem associated with (1), the existence of a unique global solution for finite initial energy $$E(u_ 0)<\infty$$ is established. Second, a local Palais-Smale type compactness result for the energy functional E which permits a direct proof of the Sacks-Uhlenbeck results is presented.
Reviewer: L.G.Vulkov

### MSC:

 5.8e+21 Harmonic maps, etc.
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