On $f$-biharmonic maps and $f$-biharmonic submanifolds

Ye-Lin Ou

Vol. 271 (2014), No. 2, 461–477

DOI: 10.2140/pjm.2014.271.461

Abstract

We consider $f‘$-biharmonic maps, the extrema of the $f‘$-bienergy functional. We prove that an $f‘$-biharmonic map from a compact Riemannian manifold into a nonpositively curved manifold with constant $f‘$-bienergy density is a harmonic map; that any $f‘$-biharmonic function on a compact manifold is constant; and that the inversion in the sphere $S^{m-1}$ is a proper $f‘$-biharmonic conformal diffeomorphism for $m\ge 3$. We derive equations for $f‘$-biharmonic submanifolds (that is, submanifolds whose defining isometric immersions are $f‘$-biharmonic maps) and prove that a surface in a manifold $\left(N^n,h\right)$ is an $f‘$-biharmonic surface if and only if it can be biharmonically conformally immersed into $\left(N^n,h\right)$. We also give a complete classification of $f‘$-biharmonic curves in three-dimensional Euclidean space. Examples are given of proper $f‘$-biharmonic maps and $f‘$-biharmonic surfaces and curves.

Keywords

$f`$-biharmonic maps, $f`$-biharmonic submanifolds, $f`$-biharmonic functions, $f`$-biharmonic hypersurfaces, $f`$-biharmonic curves

Mathematical Subject Classification 2010

Primary: 58E20

Secondary: 53C43

Milestones

Received: 22 June 2013

Accepted: 28 March 2014

Published: 20 September 2014