Biharmonic hypersurfaces with bounded mean curvature

TL;DR

This paper proves that complete biharmonic hypersurfaces in spheres with bounded second fundamental form and finite integral of inverse mean curvature must have constant mean curvature.

Contribution

It establishes a new rigidity result for biharmonic hypersurfaces under specific boundedness and integrability conditions.

Findings

Mean curvature is constant under given conditions.

Bounded second fundamental form and integrability imply rigidity.

Extends understanding of biharmonic hypersurfaces in spheres.

Abstract

We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $\phi:(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$ is bounded from above by m, and $\int_M H^{- p }dv_g<\infty$, for some $0<p<\infty$, then the mean curvature is constant.