Let $\bar M$ be a smoothly bounded orientable pseudoconvex CR manifold of finite type with at most one degenerate eigenvalue. Then we extend the given CR structure on $M$ to an integrable almost complex structure on the concave side of $M$. Therefore we may regard $M$ as the boundary of a complex manifold.

An algorithm to find a basis of the space of complex secondary classes of transversely holomorphic foliations is given. The mapping which relates the real secondary classes to the complex secondary classes is completely described when the complex codimension of the foliation is either two or three. Finally, it is shown that the image of real secondary classes under this mapping is naturally reduced under a certain condition.

In this work, complete constant mean curvature $1$ (\cmcone{}) surfaces in hyperbolic $3$-space with total absolute curvature at most $4\pi$ are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given.

Given a morphism of schemes which is flat, proper, and "fiber-by-fiber semi-stable'', we study the problem of extending the morphism to a morphism of fine log schemes, which is log smooth, integral, and vertical. The problem is rephrased in terms of a functor on the category of fine log schemes over the base, and the main result of the paper is that this functor is representable by a fine log scheme whose underlying scheme maps naturally to the base by a monomorphism of finite type. In the course of the proof, we also generalize results of Kato on the existence of log structures of embedding and semi-stable type.

A Hardy inequality is established for censored stable processes on a large class of bounded domains including bounded Lipschitz domains in $\RR^n$ with $n\geq 2$.

The aim of this article is to obtain a lower bound for the variance of a normalised $L^2$ function on the Heisenberg group under the assumption that its Fourier transform is small along a sequence of well distributed rays in the Heisenberg fan. This is achieved by proving an uncertainty inequality for Laguerre series which is analogous to the one obtained by Strichartz for spherical harmonic expansions. Applications to Hermite and special Hermite expansions are also given.