Dimension reduction for the singular set of biharmonic maps

Abstract

We consider extrinsically biharmonic maps from an open domain Ω ⊂ ℝ^m into an arbitrary compact submanifold N ⊂ ℝ^L. We show that the singular set of a minimizing biharmonic map has Hausdorff dimension at most m – 5. Furthermore, we give conditions on the target manifold under which the dimension can be reduced further and conditions under which similar results hold for maps that are only stationary biharmonic or stable stationary biharmonic. The proof is based on the analysis of defect measures by tools of geometric measure theory. A byproduct of the proof is the theorem that weak limits of stationary biharmonic maps are weakly biharmonic.