Biminimal properly immersed submanifolds in the Euclidean spaces

TL;DR

This paper proves that complete, properly immersed, nonnegative biminimal submanifolds in Euclidean spaces are minimal, providing partial confirmation of Chen's conjecture, and constructs examples for the case when lambda<0.

Contribution

It establishes that such submanifolds are minimal under certain conditions and offers new examples for negative lambda cases, advancing understanding of biminimal submanifolds.

Findings

Complete nonnegative biminimal submanifolds are minimal if properly immersed.

Provides partial proof of Chen's conjecture.

Constructs examples for biminimal submanifolds with negative lambda.

Abstract

We consider a complete nonnegative biminimal submanifold M (that is, a complete biminimal submanifold with lambda>=0) in a Euclidean space E^N. Assume that the immersion is proper, that is, the preimage of every compact set in E^N is also compact in M. Then, we prove that M is minimal. From this result, we give an affirmative partial answer to Chen's conjecture. For the case of lambda<0, we construct examples of biminimal submanifolds and curves.