Harmonic Maps and Biharmonic Maps

Harmonic Maps and Biharmonic Maps

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.

Stability of F-biharmonic maps

This paper studies some properties of F-biharmonic maps between Riemannian manifolds. By considering the first variation formula of the F-bienergy functional, F-biharmonicity of conformal maps are investigated. Moreover, the second variation formula for F-biharmonic maps is obtained. As an application, instability and nonexistence theorems for F-biharmonic maps are given.

Stable Exponentially Harmonic Maps Between Finsler Manifolds

Existence of Biharmonic Curves and Symmetric Biharmonic Maps

where n is the exterior normal direction of ∂Ω. In other words, we look for a “best” way to extend the boundary value φ with the prescribed normal derivative ψ. Typical examples of Ω and N are the unit ball and the unit sphere, respectively. In this case, ψ : ∂Ω → TφN means φ (x) · ψ (x) = 0 for all |x| = 1. With the given Dirichlet data φ, the most natural extension is perhaps the harmonic map...

On biharmonic maps and their generalizations

Abstract. We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove t...