Classification theorems for biharmonic real hypersurfaces in a complex projective space

TL;DR

This paper classifies various types of biharmonic real hypersurfaces in complex projective spaces, providing a comprehensive understanding of their geometric properties and conditions for biharmonicity.

Contribution

It offers new classification results for proper biharmonic Hopf, hypersurfaces with two principal curvatures, and ruled hypersurfaces in complex projective spaces.

Findings

Proper biharmonic Hopf real hypersurfaces in CP^2 are classified.

Proper biharmonic hypersurfaces with two principal curvatures in CP^n are classified.

Biharmonic ruled real hypersurfaces in CP^n are minimal.

Abstract

First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that biharmonic ruled real hypersurfaces in $\mathbb{C}P^n$ are minimal, where $n\geq 2$.