We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.

As a consequence we prove that, for $3$-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.

We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.

AMS 2000 Mathematics subject classification: Primary 37D25; 37C70; 37C20; 37C29

The notion of gluing of abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied further by Polishchuk. We observe that this notion is a particular case of a general categorical construction.

We then apply this general notion to the study of the ring of global differential operators $\mathcal{D}$ on the basic affine space $G/U$ (here $G$ is a semi-simple simply connected algebraic group over $\mathbb{C}$ and $U\subset G$ is a maximal unipotent subgroup).

We show that the category of $\mathcal{D}$-modules is glued from $|W|$ copies of the category of $D$-modules on $G/U$ where $W$ is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that the algebra $\mathcal{D}$ is Noetherian, and get some information on its homological properties.

AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20

We consider the inverse connection problem consisting of determining a gauge field on $\mathbb{R}^d$ from its non-abelian Radon transform along oriented straight lines. The determination is considered modulo gauge transformations. Our results include: global uniqueness theorems for $d\geq3$, new local uniqueness theorems for $d=2$, constructive proofs (i.e. proofs containing reconstruction procedures), counterexamples to the global uniqueness for $d=2$, a reduction to the attenuated X-ray transform.

AMS 2000 Mathematics subject classification: Primary 53C65; 81U40

In this paper we will study solution pairs $(u,D)$ of the minimal surface equation defined over an unbounded domain $D$ in $R^2$, with $u=0$ on $\partial D$. It is well known that there are severe limitations on the geometry of $D$; for example $D$ cannot be contained in any proper wedge (angle less than $\pi$). Under the assumption of sublinear growth in a suitably strong sense, we show that if $u$ has order of growth $\alpha$ in the sense of complex variables, then the ‘asymptototic angle’ of $D$ must be at least $\pi/\alpha$. In particular, there are at most two such solution pairs defined over disjoint domains. If $\alpha<1$ then $u$ cannot change sign and there is no other disjoint solution pair. This result is sharp as can be seen by a suitable piece of Enneper’s surface which has order $\alpha=\tfrac{2}{3}$ and asymptotic angle $\tfrac{3}{2}\pi$.

AMS 2000 Mathematics subject classification: Primary 35J60; 53A10