A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

Abstract

In this note we prove a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds of dimension \(m=\dim M\geq 3\) with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifolds by assuming finiteness of $\Vert \tau (\varphi ){\Vert }_{L^p(M)},p>1$ and smallness of $\Vert d\varphi {\Vert }_{L^m(M)}$. This is an improvement of a recent result of the first named author, where he assumed $2<p<m$. As applications we also get several nonexistence results for proper biharmonic submersions from complete non-compact manifolds into general Riemannian manifolds.

12 pages. Assumption in Theorem 1.2 is changed. Add a Liouville type result, Theorem 1.3. arXiv admin note: text overlap with arXiv:1511.07231

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