Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

TL;DR

This paper derives a second variation formula for biharmonic hypersurfaces, computes their stability index in spheres, and supports Chen's conjecture by proving the non-existence of unstable proper biharmonic hypersurfaces in Euclidean and hyperbolic spaces.

Contribution

It provides an explicit second variation formula for biharmonic hypersurfaces and applies it to analyze stability, supporting Chen's conjecture.

Findings

Explicit second variation formula for biharmonic hypersurfaces

Computed stability index in Euclidean spheres

Proved non-existence of unstable proper biharmonic hypersurfaces in Euclidean and hyperbolic spaces

Abstract

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence 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.