Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riemannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the normal stability index of the known biharmonic hypersurfaces in a Euclidean sphere and to prove the nonexistence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.