Interior and boundary regularity of intrinsic biharmonic maps to spheres. (English) Zbl 1154.58306
Weakly intrinsic biharmonic maps \(u:B^4\to N\) (\(N\) a compact Riemannian manifold) are the stationary points of the bi-energy \(\int_{B^4}| \tau(u)| ^2\,dx\), where \(\tau(u)\) is the tension field, that is the tangential part of \(\Delta u\). The bi-energy is independent of the choice of an isometric embedding of \(N\), and harmonic maps are trivial examples of intrinsic biharmonic maps.
The paper finally makes accessible some work that has been around for some years and predates some of the published results about biharmonic maps. It proves interior and boundary regularity of weakly intrinsic biharmonic maps \(B^4\to S^n\). It employs a special form of the Euler-Lagrange equation which writes \(\Delta^2u\) as the sum of certain divergences. The interior regularity for biharmonic maps to general compact target manifolds has meanwhile been proven by C. Wang [Math. Z. 247, 65–87 (2004; Zbl 1064.58016)].
The paper finally makes accessible some work that has been around for some years and predates some of the published results about biharmonic maps. It proves interior and boundary regularity of weakly intrinsic biharmonic maps \(B^4\to S^n\). It employs a special form of the Euler-Lagrange equation which writes \(\Delta^2u\) as the sum of certain divergences. The interior regularity for biharmonic maps to general compact target manifolds has meanwhile been proven by C. Wang [Math. Z. 247, 65–87 (2004; Zbl 1064.58016)].
