Biharmonic submanifolds in manifolds with bounded curvature

Abstract

We consider a complete biharmonic submanifold $\varphi :(M,g)\to (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt{c}$. If (i) ${\int }_M(|\mathbf{H}{|}^2-c{)}^{p}d{v}_g<\infty$, for some $0<p<\infty$, or (ii) the Ricci curvature of $M$ is bounded from below, then the mean curvature is $\sqrt{c}$. Furthermore, if $M$ is compact, then we obtain the same result without the assumption (i) or (ii).

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