Biharmonic maps between Riemannian manifolds

TLDR: In this paper, the geometric properties of biharmonic curves and surfaces of some Thurston's geometries have been discussed, including the biharmonicity of maps between warped products.

Abstract: points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the mathematical scene since the early ’60, when some of its analytical aspects have been discussed, and regularity of its critical points is nowadays a well-developed fleld, a systematic study of the geometry of biharmonic maps has started only recently. In this lecture we focus on the geometric properties of biharmonic maps and describe some recent achievements on the subject: (a) We give the explicit classiflcations of biharmonic curves and surfaces of some Thurston’s geometries [2, 3, 4]. (b) We describe the biharmonicity of maps between warped products and using this setting we study three classes of axially symmetric biharmonic maps [1]. (c) Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy, show it derives from a variational problem on metrics, exhibit the peculiarity of dimension four, and use the stress-energy tensor to construct new examples of biharmonic maps [5].