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A Liouville theorem for certain nonstationary maps. (English) Zbl 0659.49019

Let u: (M,g)\(\to (N,h)\) be a map between Riemannian manifolds and let \(E(u)=\int_{M}| du|^ p dv(g)\), \(p\geq 2\) be the \(L^ p\) energy of u, where (M,g) is the Euclidean space \({\mathbb{R}}^ n\) or the hyperbolic space \(H^ n\), \(n\geq p\). The author shows that if E(u) grows sufficiently slowly and if its first r-variation satisfies a certain integral inequality, then u is constant almost everywhere.
Reviewer: C.Udrişte

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53C20 Global Riemannian geometry, including pinching
58E99 Variational problems in infinite-dimensional spaces
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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