We describe complemented copies of $\ell_2$ both in $C(K_1)\hat{\otimes}_{\pi} C(K_2)$ when at least one of the compact spaces $K_i$ is not scattered and in $L_1(\mu_1)\hat{\otimes}_{\epsilon} L_1(\mu_2)$ when at least one of the measures is not atomic. The corresponding local construction gives uniformly complemented copies of the $\ell_2^n$ in $c_0\hat{\otimes}_{\pi} c_0$. We continue the study of $c_0\hat{\otimes}_{\pi} c_0$ showing that it contains a complemented copy of Stegall's space $c_0(\ell_2^n)$ and proving that $(c_0\ppi c_0)''$ is isomorphic to $\ell_\infty(\ell_\infty^n\hat{\otimes}_{\pi} \ell_\infty^n)$, together with other results. In the last section we use Hardy spaces to find an isomorphic copy of $L_p$ in the space of compact operators from $L_q$ to $L_r$, where $1<p,q,r<\infty$ and $1/r=1/p+1/q$.

In this paper we study the metric geometry of the space $\Sigma$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $\tau$. The trace induces an incomplete Riemannian metric $\langle x,y\rangle_a=\tau (ya^{-1}xa^{-1})$, and, though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance, we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $\Sigma$; and under a suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained via a nonlinear orthogonal projection $\Pi_M:\Sigma\to M$, a map which turns out to be contractive for the geodesic distance.

This article introduces the notion of 2-ruled 4-folds: submanifolds of ${\mathbb{R}}^n$ fibred over a 2-fold $\Sigma$ by affine 2-planes. This is motivated by a paper by Joyce and previous work of the present author. A 2-ruled 4-fold $M$ is r-framed if an oriented basis is smoothly assigned to each fibre, and then we may write $M$ in terms of orthogonal smooth maps $\phi_1,\phi_2:\Sigma\rightarrow\mathcal{S}^{n-1}$ and a smooth map $\psi:\Sigma\rightarrow{\mathbb{R}}^n$. We focus on 2-ruled Cayley 4-folds in ${\mathbb{R}}^8$ as certain other calibrated 4-folds in ${\mathbb{R}}^7$ and ${\mathbb{R}}^8$ can be considered as special cases. The main result characterizes non-planar, r-framed, 2-ruled Cayley 4-folds, using a coupled system of nonlinear, first-order, partial differential equations that $\phi_1$ and $\phi_2$ satisfy, and another such equation on $\psi$ which is linear in $\psi$. We give a means of constructing 2-ruled Cayley 4-folds starting from particular 2-ruled Cayley cones using holomorphic vector fields. This is used to give explicit examples of ${\mathbin{\rm U}}(1)$-invariant 2-ruled Cayley 4-folds asymptotic to a ${\mathbin{\rm U}}(1)^3$-invariant 2-ruled Cayley cone. Examples are also given based on ruled calibrated 3-folds in ${\mathbb{C}}^3$ and ${\mathbb{R}}^7$ and complex cones in ${\mathbb{C}}^4$.

We present the classification of all real hypersurfaces in complex hyperbolic space $\mathbb{C}H^{n}$, $n \geq 3$, with three distinct constant principal curvatures.

We produce examples of Legendrian isotopy classes in $ST^*\mathbb{R}^n$ (the space of cooriented contact elements of $\mathbb{R}^n$, endowed with its standard contact structure) such that any element of those classes has a singular (non-immersed) wave front, that is, the projection to the base of the fibration $ST^*\mathbb{R}^n \rightarrow \mathbb{R}^n$, when restricted to such a Legendrian embedding, has local singularities. Furthermore, our examples are such that the underlying Legendrian homotopy classes contain some Legendrian embeddings whose wave fronts are immersed. These examples are motivated by a question of Arnold about the removability of singularities of wave fronts by means of Legendrian isotopy. The two key points, which are of independent interest, are as follows.

(1) A duality argument which allows the question to be translated into a problem about Legendrian submanifolds of the one-jets space of the sphere, and hence to use the technology of generating families.

(2) Several independent constructions of families of functions on compact manifolds such that the ‘critical sets’ of these families are connected. For example, we prove that given two compact, connected manifolds $M$ and $N$ of positive dimension, there exists

\[f~: N \times M \rightarrow \mathbb{R}, (x,y) \rightarrow f(x,y),\quad x\in N, \ y \in M\]

such that the equation $(\partial f/\partial y)(x,y)=0$ is regular and defines a smooth and connected submanifold of $M \times N$.

In this paper we consider an aperiodic integer-valued random walk $S$ and a process $S^{\ast }$ that is a harmonic transform of $S$ killed when it first enters the negative half; informally, $S^{\ast }$ is ‘$S$ conditioned to stay non-negative’. If $S$ is in the domain of attraction of the standard normal law, without centring, a suitably normed and linearly interpolated version of $S$ converges weakly to standard Brownian motion, and our main result is that under the same assumptions a corresponding statement holds for $S^{\ast }$, the limit of course being the three-dimensional Bessel process. As this process can be thought of as Brownian motion conditioned to stay non-negative, in essence our result shows that the interchange of the two limit operations is valid. We also establish some related results, including a local limit theorem for $S^{\ast }$, and a bivariate renewal theorem for the ladder time and height process, which may be of independent interest.

We introduce the concept of ‘geometrical spine’ for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of $L_{p,q}$ that is close enough to its geometrical spine contains at least $E(p,q)-3$ vertices, which is exactly the conjectured value for the complexity $c(L_{p,q})$. As a byproduct, we find the minimal rotation distance (in the Sleator–Tarjan–Thurston sense) between a triangulation of a regular $p$-gon and its image under rotation.

We prove that every limit group acts geometrically on a CAT(0) space with the isolated flats property.