Sequential Weak Approximation for Maps of Finite Hessian Energy

TL;DR

This paper investigates the approximation of Sobolev maps into spheres, establishing conditions under which weak or strong $W^{2,2}$ approximations by smooth maps are possible, especially focusing on maps into $S^3$ in five dimensions.

Contribution

It introduces a new method to bound the length of connecting curves between singularities, enabling weak $W^{2,2}$ approximation of maps into $S^3$ in five dimensions.

Findings

Weak $W^{2,2}$ density of smooth maps into $S^3$ in five dimensions.

A bound relating the length of connecting curves to the Hessian energy.

Construction of approximating sequences using twisting of pull-back normal framings.

Abstract

Consider the space $W^{2,2}(\Omega;N)$ of second order Sobolev mappings $\ v\ $ from a smooth domain $\Omega\subset\R^m$ to a compact Riemannian manifold $N$ whose Hessian energy $\int_\Omega |\nabla^2 v|^2\, dx$ is finite. Here we are interested in relations between the topology of $N$ and the $W^{2,2}$ strong or weak approximability of a $W^{2,2}$ map by a sequence of smooth maps from $\Omega$ to $N$. We treat in detail $W^{2,2}(\B^5,S^3)$ where we establish the \underline{sequential weak} $W^{2,2}$ density of $W^{2,2}(\B^5,S^3)\cap{\mathcal C}^\infty$. The strong $W^{2,2}$ approximability of higher order Sobolev maps has been studied in the recent preprint \cite{BPV} of P. Bousquet, A. Ponce, and J. Van Schaftigen. For an individual map $v\in W^{2,2}(\B^5,S^3)$, we define a number $L(v)$ which is approximately the total length required to connect the isolated singularities of a strong approximation $u$ of $v$ either to each other or to $\p\B^5$. Then $L(v)=0$ if and only if $v$ admits $W^{2,2}$ strongly approximable by smooth maps. Our critical result, obtained by constructing specific curves connecting the singularities of $u$, is the bound $\ L(u)\leq c\int_{\B^5}|\nabla^2 u|^2\, dx\ $. This allows us to construct, for the given Sobolev map $v\in W^{2,2}(\B^5,S^3)$, the desired $W^{2,2}$ weakly approximating sequence of smooth maps. To find suitable connecting curves for $u$, one uses the twisting of a $u$ pull-back normal framing of a suitable level surface of $u$