On p-biharmonic curves

TL;DR

This paper explores $p$-biharmonic curves, generalizing biharmonic curves, classifies special cases on surfaces, and investigates their existence, stability, and relation to elastic and magnetic geodesic curves.

Contribution

It introduces the concept of $p$-biharmonic curves, classifies $rac{1}{2}$-biharmonic curves on specific manifolds, and connects them to magnetic geodesics and elastic curves.

Findings

Classified $rac{1}{2}$-biharmonic curves on closed surfaces and space forms.

Proved existence of $rac{1}{2}$-biharmonic curves using magnetic geodesic connection.

Analyzed stability of $p$-biharmonic curves under normal variations.

Abstract

In this article we study $p$-biharmonic curves as a natural generalization of biharmonic curves. In contrast to biharmonic curves $p$-biharmonic curves do not need to have constant geodesic curvature if $p=\frac{1}{2}$ in which case their equation reduces to the one of $\frac{1}{2}$-elastic curves. We will classify $\frac{1}{2}$-biharmonic curves on closed surfaces and three-dimensional space forms making use of the results obtained for $\frac{1}{2}$-elastic curves from the literature. By making a connection to magnetic geodesic we are able to prove the existence of $\frac{1}{2}$-biharmonic curves on closed surfaces. In addition, we will discuss the stability of $p$-biharmonic curves with respect to normal variations. Our analysis highlights some interesting relations between $p$-biharmonic and $p$-elastic curves.