arXiv subjects
Publications and source records attributed to Giuseppe Della Sala.
We prove that if every chain on a strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ coincides with the boundary of a stationary disc, then $M$ is locally spherical.
The aim of the paper is to study the level sets of the solutions of Dirichlet problems for the Levi operator on strongly pseudoconvex domains $Ω$ in $\mathbb C^2$. Such solutions are generically non smooth, and the geometric properties of their level sets are characterized by means of hulls of their intersections with $bΩ$, using as main tool the local maximum property introduced by Slodkowski (PJM, 1988). The same techniques are then employed to study the behavior of the complete Levi operator for graphs in $\mathbb C^2$.
We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.
This paper is devoted to Riemann-Hilbert problems with constraints. We obtain results characterizing the existence of solutions as well as the dimension of the solution space in terms of certain indices. As an application, we show how such results may be used to construct analytic discs attached to singular manifolds.
We prove finite jet determination for (finitely) smooth CR diffeomorphisms of (finitely) smooth Levi degenerate hypersurfaces in $\mathbb{C}^{n+1}$ by constructing generalized stationary discs glued to such hypersurfaces.
We generalize Lempert's and Poletsky's works on the description of extremal discs for the Kobayashi metric to a higher order setting.
A holomorphic mapping $H$ between two real-analytic CR manifolds $M$ and $M'$ is said to be locally rigid if any other holomorphic map $F\colon M \to M'$ which is close enough to $H$ is obtained by composing $H$ with suitable automorphisms of $M$ and $M'$. With the aim of reducing the local rigidity problem to a linear one, we provide sufficient infinitesimal conditions. Furthermore we study some topological properties of the action of the automorphism group on the space of nondegenerate mappings from $M$ to $M'$.
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.
We call a subset $K$ of $\mathbb C$ \emph{biholomorphically homogeneous} if for any two points $p,q\in K$ there exists a neighborhood $U$ of $p$ and a biholomorphism $ψ:U\to ψ(U)\subset \mathbb C$ such that $ψ(p)=q$ and $ψ(K\cap U)= K\cap ψ(U)$. We show that a biholomorphically homogeneous smooth curve $γ\subset \mathbb C$ is necessarily real-analytic. Furthermore we show that the same holds for the homogeneity with respect of a wider class of groups $G$ of real-analytic transformations of the plane. The result also extends to subsets $K\subset \mathbb R^2$ which are just locally closed.
We study local rigidity properties of holomorphic embeddings of real hypersurfaces in $\mathbb C^2$ into real hypersurfaces in $\mathbb C^3$ and show that infinitesimal conditions imply actual local rigidity in a number of (important) cases. We use this to show that generic embeddings into a hyperquadric in $\mathbb C^3$ are locally rigid.
We construct a finitely dimensional invariant manifold of holomorphic discs attached to a certain class of smooth pseudconvex hypersurfaces of finite type in $\C^2$, generalizing the notion of stationary discs. The discs we construct are determined by a finite jet at a given boundary point and their centers fill an open set. As a consequence, we obtain a finite jet determination result for this class of smooth hypersurfaces.
Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G--> M-->X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if there exists a point p in bM such that T_p(G) is contained in the complex tangent space of bM at p, then the Bergman space of M is large. Natural examples include the gauged G-complexifications of Heinzner, Huckleberry, and Kutzschebauch.
For an arbitrary unimodular Lie group $G$, we construct strongly continuous unitary representations in the Bergman space of a naturally constructed strongly pseudoconvex neighborhood of $G$ in the complexification of its underlying manifold.
On the basis of a result of Barrett, we show that members of certain classes of abstract Levi flat manifolds with boundary, whose Levi foliation contains a compact leaf with contracting, flat holonomy, admit no $CR$ embedding as a hypersurface of a complex manifold.
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 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 not equivalent hypersurfaces which are almost everywhere analytic.
We discuss various problems regarding the structure of the foliation of some foliated submanifolds S of C^n, in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the leaves of its foliation are complex planes.
Let A be a domain of the boundary of a strictly pseudoconvex domain Ωof C^n and M a smooth, closed, maximally complex submanifold of A. We find a subdomain \widetilde A of Ω, depending only on Ωand A, and a complex subvariety W of \widetilde A such that bW\cap A = M. Moreover, a generalization to analytic sets of depth at least 4 is given.
We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.