Biharmonic hypersurfaces with three distinct principal curvatures in spheres

TL;DR

This paper classifies proper biharmonic hypersurfaces with up to three distinct principal curvatures in spheres, showing they are either a specific sphere or a Clifford hypersurface, and proves non-existence in hyperbolic spaces.

Contribution

It provides a complete classification of such hypersurfaces in spheres and establishes non-existence results in hyperbolic spaces.

Findings

Proper biharmonic hypersurfaces in spheres are either a sphere or a Clifford hypersurface.

No proper biharmonic hypersurfaces with up to three principal curvatures exist in hyperbolic spaces.

The classification extends known results and completes the understanding of biharmonic hypersurfaces with limited principal curvatures.

Abstract

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmu\c{s}-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces $\mathbb S^{n+1}$ are either the hypersphere $\mathbb S^n(1/\sqrt2)$ or the Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb S^{n_2}(1/\sqrt2)$ with $n_1+n_2=n$ and $n_1\neq n_2$. Moreover, we also show that there does not exist proper biharmonic hypersurface with at most three distinct principal curvatures in hyperbolic spaces $\mathbb H^{n+1}$.