The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [5], provides a way of representing immersed surfaces in ${ℝ}^3$, and equipping the set of these immersions with a “distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of ${ℝ}^3$. Thus, it induces a distance function on the shape space of immersions, i.e., the space...

Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the...

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

Cet article contient une démonstration géométrique simple de ${\pi }_2(B{\Gamma }_1^r)=0$ pour $r=0,\infty$.Ce résultat (démontré aussi par Mather comme corollaire d’un théorème beaucoup plus général) apparaît comme une conséquence du théorème de Michael Herman : $\frac{\mathrm{Diff}{S}^1}{[\mathrm{Diff}{S}^1,\mathrm{Diff}{S}^1]}=0$.L’appendice contient une étude des $\Gamma$ structures sur les surfaces et un résultat sur la cohomologie de $\mathrm{Diff}{S}^1$.

Soit $f$ un morphisme propre et de Nash d’un ouvert $\Omega$ de ${\mathbf{R}}^n$ dans un ouvert ${\Omega }^{\text{'}}$ de ${\mathbf{R}}^p$. Nous démontrons que l’image par $f^{*}$ de l’algèbre $C^{\infty }({\Omega }^{\text{'}})$ des fonctions réelles $C^{\infty }$ dans ${\Omega }^{\text{'}}$ est fermée dans $C\infty \left(\Omega \right)$ munie de sa topologie habituelle d’espace de Fréchet. Ce résultat généralise, dans le cas algébrique, un résultat de G. Glaeser sur les fonctions composées différentiables.