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.

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds$M$with the property that$M$×$R$^{2}has non-negative isotropic curvature.

In this paper we study surfaces in$R$^{3}that arise as limit shapes in random surface models related to planar dimers. These limit shapes are$surface tension minimizers$, that is, they minimize a functional of the form ∫$σ$(∇$h$)$dx$$dy$among all Lipschitz functions$h$taking given values on the boundary of the domain. The surface tension$σ$has singularities and is not strictly convex, which leads to formation of$facets$and$edges$in the limit shapes.

Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class (i.e. between$C$^{1}and$C$^{2}). In particular, we show some generalizations of Denjoy theorem and the classical Kopell lemma for abelian groups. These techniques are also applied to the study of codimension-1 foliations. For instance, we obtain several generalized versions of Sacksteder theorem in class$C$^{1}. We conclude with some remarks about the stationary measure.

Let$M$be a smooth manifold and$V$a Euclidean space. Let $ \overline{{{\text{Emb}}}} $ ($M$,$V$) be the homotopy fiber of the map Emb($M$,$V$) → Imm($M$,$V$). This paper is about the rational homology of $ \overline{{{\text{Emb}}}} $ ($M$,$V$). We study it by applying embedding calculus and orthogonal calculus to the bifunctor ($M$,$V$)↦$HQ$∧ $ \overline{{{\text{Emb}}}} $ ($M$,$V$)_{+}. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$ (where ED($M$) is the embedding dimension of$M$), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at$E$^{1}. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$ The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of$M$. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of $ \overline{{{\text{Emb}}}} $ (–,V).