Robert Hardt, Tristan Rivière

Acta Math. 200 (1), 15-83, (2008) DOI: 10.1007/s11511-008-0023-6

Let *N* be a compact simply connected smooth Riemannian manifold and, for *p* ∈ {2,3,...}, *W*^{1, p}(**R**^{p+1}, *N*) be the Sobolev space of measurable maps from **R**^{p+1} into *N* whose gradients are in *L*^{p}. The restriction of *u* to almost every *p*-dimensional sphere *S* in **R**^{p+1} is in *W*^{1, p}(*S*, *N*) and defines an homotopy class in π_{p}(*N*) (White 1988). Evaluating a fixed element *z* of Hom(π_{p}(*N*), **R**) on this homotopy class thus gives a real number Φ_{z, u}(*S*). The main result of the paper is that any *W*^{1, p}-weakly convergent limit *u* of a sequence of smooth maps in *C*^{∞}(**R**^{p+1}, *N*), Φ_{z, u} has a *rectifiable Poincaré dual*$ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $. Here Γ is a a countable union of *C*^{1} curves in **R**^{p+1} with Hausdorff $ {\user1{\mathcal{H}}}^{1} $-measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function *θ*: Γ→**R**. The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ and *S* evaluates Φ_{z, u}(*S*), for almost every *p*-sphere *S*. Moreover, we exhibit a non-negative integer *n*_{z}, depending only on homotopy operation *z*, such that $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of *N*, *p* and *z* for which this rational power *p*/(*p* + *n*_{z}) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to **R**^{m} for any *m* ⩾ *p* + 1, in which case the bubbling is described by an (*m*–*p*)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean *p*-sphere.