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Michela Zedda

Publications and source records attributed to Michela Zedda.

At least 19 recordsLinked to original sources
arXiv2026

Radial Projectively Induced Canonical K\"ahler Metrics: Rigidity and Classification

We study radial K\"ahler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a K\"ahler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini-Study metric, the flat metric, or, in complex dimension two, generalized Burns-Simanca metrics. We also prove that every radial projectively induced K\"ahler-Einstein metric has constant holomorphic sectional curvature and is therefore a Fubini-Study, flat, or complex hyperbolic metric. Finally, we show that a radial infinitely projectively induced extremal K\"ahler metric has unbounded maximal radial domain if and only if it is scalar-flat.

math.DG↗
arXiv2024

Projectively induced Kähler cones over regular Sasakian manifolds

Motivated by a conjecture in [9] we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to $\mathcal D_a$-homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.

math.DG↗
arXiv2023

Kähler geometry of scalar flat metrics on Line bundles over polarized Kähler-Einstein manifolds

In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed in [10] by A. D. Hwang and M. A. Singer on $\mathbb C^{n+1}$ and on $\mathcal O(-k)$. For the metrics in both the families, we prove the existence of an asymptotic expansion for their $ε$-functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on $\mathbb C^{n+1}$ are not projectively induced, and that the Burns-Simanca metric is characterized among the scalar flat metrics on $\mathcal O(-k)$ to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the $ε$-function vanishes.

math.DG↗
arXiv2023

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi's rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

math.DG↗
arXiv2022

CR relatives Kaehler manifolds

In this paper we show that two Kähler manifolds which do not share a Kähler submanifold, do not share either a Levi degenerate CR-submanifold with constant dimension Levi kernel. In particular, they do not share a CR-product. Further, we obtain that a Levi degenerate CR-submanifold of $\mathbb C^n$ cannot be isometrically immersed into a flag manifold.

math.DG↗
arXiv2021

A Cartan-Hartogs version of the Polydisk Theorem

We extend the Polydisk Theorem for symmetric bounded domains to Cartan-Hartogs domains, and apply it to prove that a Cartan-Hartogs domain inherits totally geodesic submanifolds from the bounded symmetric domain which is based on, and to give a characterization of Cartan-Hartogs's geodesics with linear support.

math.DG↗
arXiv2020

Invariant escaping Fatou components with two rank 1 limit functions for automorphisms of $\mathbb{C}^2$

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 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):\mathbb{C}^2\rightarrow\mathbb{C}$ holomorphic.

math.DS↗
arXiv2020

Symplectic geometry of Cartan-Hartogs domains

This paper studies the geometry of Cartan-Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan-Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan-Hartogs and their dual. Further, we compute their symplectic capacity and show that a Cartan-Hartogs admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

math.DG↗
arXiv2019

Ricci flat Calabi's metric is not projectively induced

We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.

math.DG↗
arXiv2017

Kähler immersions of Kähler manifolds into complex space forms

The study of Kähler immersions of a given real analytic Kähler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally Kähler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a Kähler immersion into another. Although, a complete classification of Kähler manifolds admitting a Kähler immersion into complex space forms is not known, not even when the Kähler manifolds involved are of great interest, e.g. when they are Kähler-Einstein or homogeneous spaces. In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application. Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained. The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.

math.DG↗
arXiv2016

Strongly not relatives Kaehler manifolds

In this paper we study Kaehler manifolds that are strongly not relative to any projective Kaehler manifold, i.e. those Kaehler manifolds that do not share a Kaehler submanifold with any projective Kaehler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kaehler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relative to projective Kaehler manifolds.

math.DG↗
arXiv2016

Stability with respect to actions of real reductive Lie groups

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real non-compact reductive Lie group G on a real compact submanifold M of a Kaehler manifold Z and to the action of G on measures of M.

math.DG↗
arXiv2016

On Calabi's diastasis function of the cigar metric

We show that the Cigar metric on $\mathbb{C}$ is an example of real analytic Kähler manifold with globally defined and positive Calabi's diastasis function which cannot be Kähler immersed into any (finite or infinite dimensional) complex space form.

math.DG↗
arXiv2015

On the convergence of the Sasaki J-flow

This paper investigates the $C^\infty$-convergence of the Sasaki $J$-flow. The result is applied to prove a lower bound for the $K$-energy map in the Sasakian context.

math.DG↗
arXiv2014

On the J-flow in Sasakian manifolds

We study the space of Sasaki metrics on a compact manifold $M$ by introducing an odd-dimensional analogue of the $J$-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold there exists at most one normalised critical metric. The flow is a tool for texting the existence of such a metric. We show that some results proved by Chen in [7] can be generalised to the Sasakian case. In particular, the Sasaki $J$-flow is a gradient flow which has always a long-time solution minimising the distance on the space of Sasakian potentials of a polarized Sasakian manifold. The flow minimises an energy functional whose definition depends on the choice of a background transverse Kähler form $χ$. When $χ$ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.

math.DG↗