Giuseppe Della Sala

NON COMPACT BOUNDARIES OF COMPLEX ANALYTIC VARIETIES

We treat the boundary problem for complex varieties with isolated singularities, of dimension greater than one, which are contained in a certain class of strongly pseudoconvex, not necessarily bounded open subsets of ℂn. We deal with the problem by cutting with a family of complex hyperplanes and applying the classical Harvey–Lawsons theorem for the bounded case [6].

Semiglobal extension of maximally complex submanifolds

Let A be a domain of the boundary of a (weakly) pseudoconvex domain O of C^n and M a smooth, closed, maximally complex submanifold of A We find a subdomain E of \C^n, depending only on O and A, and a complex variety W contained in E such that bW = M. Moreover, a generalization to analytic sets of depth at least 4 is given. doi:10.1017/S0004972711002498

ASYMPTOTIC APPROXIMATIONS AND A BOREL-TYPE RESULT FOR CR FUNCTIONS

We show that for any smooth CR manifold which has a peak function (in a weak sense) at some point p, formal power series at p can be approximated asymptotically by continuous CR functions. Furthermore, if the peak function satisfies a certain growth property, the asymptotic approximation is actually smooth. This in fact allows to invert, in a Borel-type theorem, the natural map taking a smooth CR function to its formal Taylor series.

On the vanishing rate of smooth CR functions

Let M be a lineally convex hypersurface of Cn of finite type, 0 ∈ M . Then there exist non-trivial smooth CR functions on M that are flat at 0, i.e. whose Taylor expansion about 0 vanishes identically. Our aim is to characterize the rate at which flat CR functions can decrease without vanishing identically. As it turns out, non-trivial CR functions cannot decay arbitrarily fast, and a possible way of expressing the critical rate is by comparison with a suitable exponential of the modulus of a local peak function.

The star function for meromorphic functions of several complex variables

ABSTRACT We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.

Local equivalence problem for Levi flat hypersurfaces

In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence: S 1 ∼ S 2 if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant for this relation, which allows us to prove some characterizations of triviality (i.e. equivalence to the hyperplane). Then, we employ the same invariant to construct infinitely many non-trivial classes, including an infinite family of non-equivalent hypersurfaces which are almost everywhere analytic.