We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.

We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.

We extend a recent result of Avelin, Hed, and Persson about approximation of functions f that are plurisubharmonic on a domain Ω and continuous on ˙Ω, with functions that are plurisubharmonic on (shrinking) neighborhoods of ˙Ω. We show that such approximation is possible if the boundary of Ω is C0 outside a countable exceptional set E⊂∂Ω. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous u, approximation is possible under less restrictive conditions on E. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

In this paper we prove new embedding results for compactly supported deformations of CR submanifolds of Cn+d: We show that if M is a 2-pseudoconcave CR submanifold of type (n,d) in Cn+d, then any compactly supported CR deformation stays in the space of globally CR embeddable in Cn+d manifolds. This improves an earlier result, where M was assumed to be a quadratic 2-pseudoconcave CR submanifold of Cn+d. We also give examples of weakly 2-pseudoconcave CR manifolds admitting compactly supported CR deformations that are not even locally CR embeddable.

We describe a natural decomposition of a normal complex surface singularity ($X$, 0) into its “thick” and “thin” parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts.