Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space

TL;DR

This paper proves Chen's conjecture for biharmonic hypersurfaces with three distinct principal curvatures in any Euclidean space dimension, extending previous results and including certain symmetric hypersurfaces.

Contribution

It extends Chen's conjecture verification to hypersurfaces with three distinct principal curvatures in arbitrary dimensions, including symmetric cases.

Findings

Chen's conjecture holds for hypersurfaces with three distinct principal curvatures in all Euclidean dimensions.

The result applies to $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space.

The paper generalizes previous special case results to a broader class of hypersurfaces.

Abstract

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvatures ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.