It is well known that a $k$-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of $k$-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behavior of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.

We obtain global extensions of the celebrated Nash–Kuiper theorem for $C^{1,\theta}$ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in $3$-space, we show that the Nash–Kuiper non-rigidity prevails upto exponent $\theta \lt 1/5$. This extends previous results on embedding $2$-discs as well as higher dimensional analogues.

The purpose of this paper is to study geometrically simply-connected homotopy $4$-spheres by analyzing $n$-component links in $S^3$ with a Dehn surgery realizing $\#^n (S^1 \times S^2)$. We call such links $n$R-links. Our main result is that a homotopy $4$-sphere that can be built without $1$-handles and with only two $2$-handles is diffeomorphic to the standard $4$-sphere in the special case that one of the $2$-handles is attached along a knot of the form $Q_{p,q} = T_{p,q} \# T_{-p,q}$, which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf.

Along the way, we use thin position techniques from Heegaard theory to give a characterization of $2$R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot $Q_{p,q}$. As a consequence, we produce large families, for all even $n$, of $n$R-links that are potential counter-examples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g \geqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne–Mumford compactification $\overline{\mathcal{M}}_g$ of the moduli $\mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $\mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V=M \times [-1,1]$ with scalar curvature bounded below by $\sigma \gt 0$, the distance between the boundary components of $V$ is at most $C_n / \sqrt{\sigma}$, where $C_n = \sqrt{(n-1)/n} \cdot C$ with $C \lt 8(1+\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \times \mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the “$\mathcal{KO}$-width” of a closed manifold and deduce that infinite $\mathcal{KO}$-width is an obstruction to positive scalar curvature.