Alberto Saracco

Università di Parma | UNIPR

We construct automorphisms of C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}^2$$\end{document} of constant Jacobian with a cycle of escaping Fatou co...

A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our p-energy is defined through a relaxation process, where a suitable p-rotation of inscribed polygons is adopted. The discrete p-rotation we choose has a geometric flavour: a polygon is viewed as an approximation to a smooth curve,...

We construct automorphisms of $\mathbb{C}^2$ with a cycle of escaping Fatou components, on which there are exactly two limit functions, both of rank 1. On each such Fatou component, the limit sets for these limit functions are two disjoint hyperbolic subsets of the line at infinity.

A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our $p$-energy is defined through a relaxation process, where a suitable $p$-rotation of inscribed polygonals is adopted. The discrete $p$-rotation we choose has a geometric flavor: a polygonal is viewed as an approximation to a smoo...

Citations are getting more and more important in the career of a researcher. But how to use them in the best possible way? This is a satirical paper, showing a bad trend currently happening in citation trends, due to intensive use of citation metrics. I am putting this on the arXiv and on Researchgate. Appeared in The Mathematical Intelligencer (1...

Is math useful? might sound as a trick question. And it is. Of course math is useful, we live in a data-filled world and every aspect of life is totally entwined with math applications, both trivial and subtle applications, of both basic and advanced math. But we need to ask once again that question, in order to truly understand what is math useful...

We construct automorphisms of ${\mathbb C}^2$ , and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms...

We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram–Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated with the Gauss hyper-sph...

A Correction to this paper has been published 10.1007/s12215-020-00516-3.

The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of some new integral-geometric formulas concerning both the Euclidean and geodesic total curvature of spherical cur...

Is math useful? might sound as a trick question. And it is. Of course math is useful, we live in a data-filled world and every aspect of life is totally entwined with math applications, both trivial and subtle applications, of both basic and advanced math. But we need to ask once again that question, in order to truly understand what is math useful...

In this paper we introduce, via a Phragmén-Lindelöf type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the pluricomplex Poisson kernel because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero o...

We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental H\'enon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of...

We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram-Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated to the Gauss hyper-spher...

In this paper we introduce, via a Phragmen-Lindel\"of type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the {\sl pluricomplex Poisson kernel} because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it i...

It is well known that students have difficulties with the concept of continuity, specifically on points of discontinuity, and concepts like limits and infinity. In Italian textbooks, the continuity of functions is usually defined using limits, while an intuitive characterization of continuous functions is proposed without providing the students wit...

This paper is devoted to proposing some possible educational paths in geometry, on graph theory, based on the Disney comics Paperino e i ponti di Quackenberg (Donald Duck and Quackenberg's bridges) to the realization of which I collaborated. The subject of the story is due to a collaboration between myself and the Disney writer Artibani, who also w...

The outcome of elections is strongly dependent on the dis-tricting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e. politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently...

Comics and illustrated stories are a communicative means of great impact. They conjugate the immediacy of the image with the possibility to tell a story and explain things with words. It is without doubts a great mean for dissemination and popularization of mathematics, other than-in certain contexts-a possible strong teaching tool. Nevertheless ,...

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its total variation is equal to an...

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then make use of a density argument....

The outcome of elections is strongly dependent on the districting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e. politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently p...

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes into account area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean...

Carleson measures were introduced by Lennart Carleson in 1962 to solve an interpolation problem about bounded holomorphic function called the corona problem. Since then Carleson measures have been both a powerful tool and a mathematical object worth of study per se and research on the corona problem or Carleson measures has gone a long way in the l...

In questo articolo si usa la probabilità per analizzare il gioco d'azzardo e si usa il gioco d'azzardo come motivazione per parlare di probabilità. Vuole fornire uno spunto per presentare la probabilità in maniera laboratoriale interessante per gli studenti delle scuole superiori. Abstract. In this paper we use probability to analyze gambling and u...

Discrete sequences with respect to the Kobayashi distance in a strongly pseudoconvex bounded domain $D$ are related to Carleson measures by a formula that uses the Euclidean distance from the boundary of $D$. Thus the speed of escape at the boundary of such sequence has been studied in details for strongly pseudoconvex bounded domain $D$. In this n...

We study a characterization of slice Carleson measures and of Carleson measures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces $\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc...

We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H. We deal with the problem by cutting with a family of complex hyperplanes in the fashion of [2] and applying the...

We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D in n complex variables. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A^p(D) into A^r(D) with r>p, generalizing and making more pre...

Let A be a domain of the boundary of a (weakly) pseudoconvex domain Ω of ℂn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain E of ℂn, depending only on Ω and A, and a complex variety W⊂E such that bW=M in E. Moreover, a generalization to analytic sets of depth at least 4 is given.

Let X be a (connected and reduced) complex space. A q-collar of X is a bounded domain whose boundary is a union of a strongly q-pseudoconvex, a strongly q-pseudoconcave and two flat (i.e. locally zero sets of pluriharmonic functions) hypersurfaces. Finiteness and vanishing cohomology theorems obtained in [Saracco and Tomassini, Math. Z. 256: 737–74...

We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds. Comment: 11 pages

We study suitable deformations of the complex structure on a compact balanced manifold (M, J, g, ω), naturally associated with cohomology classes in H 2 (M, R). We construct curves of balanced structures on a compact holomorphically parallelizable 5-dimensional complex nilmanifold and of half-flat structures on a compact complex 3-dimensional solvm...

We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in several complex variables, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measur...

We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

We deal with the cohomology of semi 1-coronae. Semi 1-coronae are domains whose boundary is the union of a Levi flat part, a 1-pseudoconvex part and a 1-pseudoconcave part. Using the main result in [C. Laurent-Thiébaut, J. Leiterer, Uniform estimates for the Cauchy-Riemann equation on q-concave wedges, in: Colloque d'Analyse Complexe et Géométrie,...

We transfer several characterizations of hyperbolic convex domains given in a recent joint paper by Bracci and one of the authors to analogous one for $\Bbb C$-convex domains.

We prove some extension theorems for analytic objects, in particular sections of a coherent sheaf, defined in semi q-coronae of a complex space. Semi q-coronae are domains whose boundary is the union of a Levi flat part, a q-pseudoconvex part and a q-pseudoconcave part. Such results are obtained mainly using cohomological techniques.

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.