Harmonic Maps and Biharmonic Maps

Abstract

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.

Biharmonic Submanifolds and Biharmonic Maps in Riemannian Geometry

“…For biharmonic Riemannian submersions from the product M 2 × R, where (M 2 , g) is a general 2-dimensional manifold, we have Theorem 3.7. [23] For a proper biharmonic Riemannian submersion π : M 2 × R → (N 2 , h) from the product space, we have either (i) (N 2 , h) is flat, and locally, up to an isometry of the domain and/or codomain, π is the projection of the special twisted product (18) π : (R 3 , e 2p(x,y) dx 2 + dy 2 + dz 2 ) → (R 2 , dy 2 + dz 2 ), π(x, y, z) = (y, z), with the twisting function p(x, y) such that p y = 0 and solves the PDE (19) ∆p y := p yyy + p yy p y + e −2p(x,y) (p xxy − p xy p x ) = 0, rmor (ii) (N 2 , h) is non-flat, and locally, up to an isometry of the domain and/or codomain, the map can be expressed as (20) π : (R 3 , e 2p(x,y) dx 2 + dy 2 + dz 2 ) → (R 2 , dy 2 + e 2λ(y,φ) dφ 2 ), π(x, y, z) = (y, F z − e ϕ(x) dx ), Note that the Riemannian submersion in Case (ii) of Theorem 3.7 is a map (20) depending on the function F , the following corollary shows that after changes of coordinates, a proper biharmonic Riemannian submersions from a product manifold onto a non-flat surface can be described explicitly as a simple map between two special warped product manifolds with the warping functions solving an ODE. Corollary 3.8.…”

An Intrinsic Version of the k-Harmonic Equation

“…The study of polyharmonic curves fits into the more general theory of polyharmonic maps between Riemannian manifolds, just as the theory of geodesics falls under that of harmonic applications. Polyharmonic maps have only recently became a subject of interest (see [1] and references therein), but biharmonic maps and, in particular, biharmonic submanifolds and curves have been extensively studied in the last decades (see, for instance, [2][3][4][5][6]).…”

Biharmonic Maps into S-Space forms

“…After that, Rawnsley [14] studied structure preserving harmonic maps between f-manifolds. Later on many authors studied harmonic maps (see [6] [10], [11], [12], [15] [16]). …”