Biharmonic rotational surfaces in the four-dimensional Euclidean space are minimal

TL;DR

This paper proves that all biharmonic simple rotational surfaces in four-dimensional Euclidean space are minimal, providing partial support for Chen's conjecture through differential equation analysis.

Contribution

It demonstrates that biharmonic simple rotational surfaces in 4D Euclidean space are necessarily minimal, advancing understanding of biharmonic submanifolds.

Findings

All biharmonic simple rotational surfaces in 4D Euclidean space are minimal.

The proof reduces the biharmonic condition to a differential equation system.

Excluded all non-minimal solutions, confirming minimality.

Abstract

In this paper, we show that any biharmonic simple rotational surface in the four-dimensional Euclidean space is minimal. The proof is based on reducing the biharmonic equation to a system of ordinary differential equations for the profile curve and then excluding all possible non-minimal branches. This is a partial affirmative answer to Chen's conjecture.