On p-biharmonic curves

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=12 in which case their equation reduces to the one of 12-elastic curves. We will classify 12-biharmonic curves on closed surfaces and three-dimensional space forms making use of the results obtained for 12-elastic curves from the literature. By making a connection to magnetic geodesic we are able to prove the existence of 12-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.