arXiv subjects

Samuele Mongodi

Publications and source records attributed to Samuele Mongodi.

30 records · Page 2Linked to original sources

arXiv2016

Some properties of Grauert type surfaces

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Remmert-Stein reduction of a holomorphically convex space. Moreover, in our classification Theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where it is necessary to do so.

math.CV↗

arXiv2015

Weakly complete complex surfaces

A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can be modifications of Stein spaces or proper over a non compact (possibly singular) complex curve or foliated with real analytic Levi-flat hypersurfaces which in turn are foliated by dense complex leaves (these we call surfaces of Grauert type). In the last case, we also show that such Levi-flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. An example of Brunella shows that not every weakly complete surface can be endowed with a real analytic plurisubharmonic exhaustion function. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the minimal singular set, i.e the set where every plurisubharmonic exhaustion function has a degenerate Levi form.

math.CV↗

arXiv2014

1-Complete semiholomorphic foliations

A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in C^N . In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CP^N .

math.CV↗

arXiv2014

The Cotton tensor and the Ricci flow

We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.

math.DG↗

arXiv2014

Rigidity of gradient Einstein shrinkers

In this paper we consider a perturbation of the Ricci solitons equation proposed in \cite{jpb1} and studied in \cite{CaMa} and we classify noncompact gradient shrinkers with bounded nonnegative sectional curvature.

math.DG↗

arXiv2013

Homology by metric currents

Metric currents are, in a certain sense, a metric analogous of flat currents, therefore are related to the geometry of the space and of their support. In this short note, we try to give some evidence for the previous statement, by showing that the homology which can be defined by means of metric normal currents coincides, on nice enough metric spaces, with the usual singular homology.

math.AT↗

arXiv2013

Non compact boundaries of complex analytic varieties in Hilbert spaces

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 first named author's result for the compact case [13].

math.CV↗

arXiv2013

A low rank property and nonexistence of higher dimensional horizontal Sobolev sets

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of the Heisenberg group. This provides a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. Our approach differs from the previous ones. Its technical aspect consists in performing an "exterior differentiation by blow-up", where the standard distributional exterior differentiation is not possible. This method extends to higher dimensional Sobolev mappings taking values in higher dimensional Heisenberg groups.

math.AP↗

arXiv2012

Application of metric currents to complex analysis

In 2000, Ambrosio and Kirchheim, with the paper "Currents in metric spaces", settled the foundations of a theory of currents on metric spaces and used it to pose and solve Plateau's problem in a wide class of Banach spaces. Following an idea of De Giorgi, they gave a new definition of current which was meaningful on any metric space. A metric current is a multilinear functional on (k+1)-tuples of Lipschitz functions (with the first one bounded) satisfying a continuity property, a locality property and a finite mass property. The second chapter of the thesis is devoted to the theory of local metric currents. We introduce the basic concepts and adapt the theory to the complex case, defining the bidimension, the Dolbeault decomposition and related notions. A characterization of holomorphic currents is given. The development of a Sobolev theory on singular space is our main concern in the third chapter. We give a characterization of Sobolev functions in terms of their behavior and growth on the regular part; this leads to a capacity theory for the singular set which allows us to obtain an approximation result, using functions with support in the regular part. The second part of the chapter deals with the L2 theory on singular spaces. In the fourth chapter, some applications of the theory are discussed. We solve Cauchy-Riemann equation on completely reducible singularities, by means of a structure theorem for metric currents; we also treat the equation in Lp on complex curves and outline a possible approach for the same problem on complex spaces which can be embedded as divisors in C^n. The final chapter tries to spread some light on the complex geometry in infinite dimensional spaces. After solving the Cauchy-Riemann equation in Banach spaces, in terms of the quasi-local metric currents, we turn to the study of positive currents, holomorphic chains and their boundaries in Hilbert spaces.

math.CV↗

arXiv2012

Positive metric currents and holomorphic chains in Hilbert spaces

We present some results concerning currents of integration on finite-dimensional analytic spaces in Hilbert spaces, using the setting of metric currents. In particular, we obtain the characterization of such currents as positive closed $(k,k)-$rectifiable currents and solve the boundary problem for holomorphic chains.

math.CV↗

arXiv2012

Some applications of metric currents to complex analysis

The aim of this paper is to show two applications of metric currents to complex analysis. After recalling the basic definitions, we give a detailed proof of the comparison theorem between metric currents and classical ones on a manifold. In Section 3 we introduce the concept of bidimension for a metric current on a finite dimensional space, showing that the usual properties of $(p,q)-$currents still hold, except for the existence of a Dolbeault decomposition. Section 4 is devoted to the analysis of a particular class of complex spaces, whose structure allows us to give a structure theorem for currents, solve the Cauchy-Riemann equation and characterize holomorphic currents. In Section 5, we introduce the concept of bidimension of (global) metric currents on a Banach space and relate it to the behaviour of the finite dimensional projections of the currents. In section 6 we define a new class of currents, the \emph{quasi-local} metric currents, which are usual metric currents when restricted to bounded sets, and we give a definition of $(p,q)-$current in this new class. The last Section shows how to employ these newly defined quasi-local currents in order to obtain a solution to the equation $\debar U=T$, when $T$ is of bidimension $(0,q)$ and its support is bounded; finally, we extend the result to a current $T$ with generic bidimension, $\debar-$closed, with bounded support.

math.CV↗

arXiv2011

On $L^r$ hypoellipticity of solutions with compact support of the Cauchy-Riemann equation

In one complex variable, the existence of a compactly supported solution to the Cauchy-Riemann equation is related to the vanishing of certain integrals of the data; trying to generalize this approach, we find an explicit construction, via convolution, for a compactly supported solution in $\mathbb{C}^n$, which allows to estimate the $L^p$ norm of the solution. We also investigate the possible generalizations of this method to domains of the form $P\setminus Z$, where $P$ is a polydisc and $Z$ is the zero locus of some holomorphic function.

math.CV↗