Quantitative stratification and higher regularity for biharmonic maps

Abstract

Abstract. In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent L p bounds for ∇ k f that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in W 4,p for all 1 ≤ p < 5/4. Further, for minimizing biharmonic maps from Ω ⊂ R 5 , we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps into special targets.

Regularity theory for type I Ricci flows

“…In [7,8,5,6,3] the selfsimilar points line up close to a lower dimensional subspace. Taking the analogy to the Ricci flow naively, one might expect that selfsimilar points will tend to line up around a lower dimensional submanifold.…”

Boundary regularity for minimizing biharmonic maps

“…In a recent paper Breiner and Lamm [4] prove that each minimizing biharmonic map is locally in W 4,p for 1 ≤ p ≤ 5/4. Let us mention here two inconclusive results in the direction of boundary regularity.…”

“…First it was observed by Hong and C. Wang in [11] that for N = S −1 the singular set Σ has Hausdorff dimension at most m − 5. One can prove the optimality of this result considering a map x |x| : B 5 → S 4 (see [11,Proposition A1.]). Finally, Scheven in [21] reduced the dimension of singular set of minimizing mappings to an arbitrary target manifold N .…”

“…In a recent paper Breiner and Lamm [4] prove that each minimizing biharmonic map is locally in W 4,p for 1 ≤ p ≤ 5/4.…”

Heat Flow of Extrinsic Biharmonic Maps from a four Dimensional Manifold with Boundary

“…Wang's partial regularity result was reproved by Lamm and Riviére [12] and Struwe [21] extending the lower order gauge theory technique developed in [16,17]. See also Scheven [18] for partial regularity result for minimizing extrinsic biharmonic maps and Breiner and Lamm [2] for recent development and references therein.…”