Abstract
We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the \(L^p\)-norm of the tension field is bounded and the n-energy of the map is sufficiently small, then every biharmonic map must be harmonic, where \(2<p<n\).
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Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The author gratefully acknowledges the support of the Austrian Science Fund (FWF) through the project P30749-N35 “Geometric variational problems from string theory”.
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Branding, V. A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies. Arch. Math. 111, 329–336 (2018). https://doi.org/10.1007/s00013-018-1189-6
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DOI: https://doi.org/10.1007/s00013-018-1189-6