Regularity and uniqueness of a class of biharmonic map heat flows

Abstract

In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere S L ⊂ R L+1 under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of biharmonic maps, we prove the properties of uniqueness, convexity of hessian energy, and unique limit at t = ∞. We also establish both regularity and uniqueness for the class of weak solutions u to the heat flow of biharmonic maps into any compact Riemannian manifold N without boundary such that ∇ 2 u ∈ L q t L p x for some p > n 2 and q > 2 satisfying (1.13). 7 2 ,2 (∂M, N ) (see [14]). Namely, there is a unique, global

$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

“…f = u t ), a parabolic version of smallness assumptions can also be verified in some cases, see e.g. Hineman-Huang-Wang [20]. In view of the embedding M 4,n−4+4α * (B 1 ) ⊂ M q,n−q+qα (B 1 ) for any 1 ≤ q < 4, the estimate (1.10) can be viewed as a slight improvement of Struwe [43,Estimate (37)], where it was proved that ∇u ∈ M q,n−q+qα (B 1/2 ) for some 1 < q < 2.…”

On uniqueness of heat flow of harmonic maps

“…It turns out that we can extend the ideas in this paper to study the uniqueness issue of heat flow of biharmonic maps, which will be discussed in a forthcoming paper [21].…”

Analyse des problèmes conformément invariants

“…Ce résultat était déjà partiellement démontré par Changyou Wang [25] dans le cas des applications biharmoniques extrinsèques. Toutefois dans notre travail [30] nous avons bonnes espoir d'étendre le résultat au applications biharmonique intrinsèque et par la suite de pouvoir la notion de width obtenu par Colding et Minicozzi via les surfaces minimales dans [13] aux variétés de dimension 4.…”