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Leonardo Biliotti

Publications and source records attributed to Leonardo Biliotti.

At least 19 recordsLinked to original sources
arXiv2025

Reduction principles for proper actions

We study the core of a proper action by a Lie group $G$ on a smooth manifold $M$, extending the construction for $G$ compact by Skjelbred and Straume. Moreover, we show that many properties of a proper $G$-action on $M$ are determined by the action of a group $G'$ on the corresponding core $_cM$. We say that such properties admit a reduction principle. In particular, we prove that a proper isometric $G$-action on $M$ is polar (resp. hyperpolar) if and only if the $G'$-action on $_cM$ is polar (resp. hyperpolar). In the case of a proper action by syplectomorphisms on a symplectic manifold, we show that a reduction principle holds for coisotropic and infinitesimally almost homogeneous actions. We further study the coisotropic condition for the case of a proper Hamiltonian action and its relation with the symplectic stratification described by Lermann and Bates. In particular, we obtain several characterizations for coisotropic actions, some of which extend known results for the action of a compact group of holomorphic automorphisms on a compact Kähler manifold obtained by Huckleberry and Wurzbacher. Finally, we study some applications of the core construction for the action of a compact Lie group on a Kähler manifold by holomorphic isometries.

math.DG↗
arXiv2025

Reduction principles for proper actions

Let $G$ be a Lie group acting properly on a smooth manifold $M$. If $M/G$ is connected, then we exhibit some simple and basic constructions for proper actions. In particular, we prove that the reduction principle in compact transformation groups holds for proper actions. As an application, we prove that a reduction principle holds for polar actions and for the integral invariant for isometric actions of Lie groups, called copolarity, which measures how far from being polar the action is. We also investigate symplectic actions. Hence we assume that $(M,ω)$ is a symplectic manifold and the $G$ action on $M$ preserves $ω$. %If $G$ is Abelian, we generalize results proved in \cite{Ben,DP1,DP2} The main result is the Equivalence Theorem for coisotropic actions, generalizing \cite[Theorem 3 p.267]{HW}. Finally, we completely characterize asystatic actions generalizing results proved in \cite{pg}.

math.DG↗
arXiv2024

Complex Structures on Product Manifolds

Let $M_i$, for $i=1,2$, be a Kähler manifold, and let $G$ be a Lie group acting on $M_i$ by Kähler isometries. Suppose that the action admits a momentum map $μ_i$ and let $N_i:=μ_i^{-1}(0)$ be a regular level set. When the action of $G$ on $N_i$ is proper and free, the Meyer--Marsden--Weinstein quotient $P_i:=N_i/G$ is a Kähler manifold and $π_i:N_i\to P_i$ is a principal fiber bundle with base $P_i$ and characteristic fiber $G$. In this paper, we define an almost complex structure for the manifold $N_1\times N_2$ and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for $N_1\times N_2$. As applications, we consider a non integrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds $\mathbb S^{2n+1}\times S(\mathcal{H})$, for $n\geq 1$, where $S(\mathcal{H})$ denotes the unit sphere of an infinite dimensional Hilbert space $\mathcal{H}$

math.DG↗
arXiv2023

A Hilbert-Mumford Criterion for polystability for actions of real reductive Lie groups

We presented a Hilbert-Mumford criterion for polystablility associated with an action of a real reductive Lie group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of the complexified group $U^\mathbb{C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $μ_\mathfrak{p}: X\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Under some mild restrictions on the $G$-action on $X,$ we characterize which $G$-orbits in $X$ intersect $μ_\mathfrak{p}^{-1}(0)$ in terms of the maximal weight function, which we viewed as a collection of maps defined on the boundary at infinity ($\partial_\infty G/K$) of the symmetric space $G/K$.

math.DG↗
arXiv2023

Properties of Gradient maps associated with Action of Real reductive Group

Let $(Z,ω)$ be a \Keler manifold and let $U$ be a compact connected Lie group with Lie algebra $\mathfrak{u}$ acting on $Z$ and preserving $ω$. We assume that the $U$-action extends holomorphically to an action of the complexified group $U^{\mathbb C}$ and the $U$-action on $Z$ is Hamiltonian. Then there exists a $U$-equivariant momentum map $μ: Z\to \mathfrak{u}$. If $G\subset U^{\mathbb C}$ is a closed subgroup such that the Cartan decomposition $U^{\mathbb C} = U\text{exp}(i\mathfrak{u})$ induces a Cartan decomposition $G = K\text{exp}(\mathfrak{p}),$ where $K = U\cap G$, $\mathfrak{p} = \mathfrak{g}\cap i\mathfrak{u}$ and $\mathfrak{g}=\mathfrak k \oplus \mathfrak p$ is the Lie algebra of $G$, there is a corresponding gradient map $μ_\mathfrak{p} : Z\to \mathfrak{p}$. If $X$ is a $G$-invariant compact and connected real submanifold of $Z,$ we may consider $μ_{\mathfrak p}$ as a mapping $μ_\mathfrak{p} : X\to \mathfrak{p}.$ Given an $\mathrm{Ad}(K)$-invariant scalar product on $\mathfrak p$, we obtain a Morse like function $f=\frac{1}{2}\parallel μ_{\mathfrak p} \parallel^2$ on $X$. We point out that, without the assumption that $X$ is real analytic manifold, the Lojasiewicz gradient inequality holds for $f$. Therefore the limit of the negative gradient flow of $f$ exists and it is unique. Moreover, we prove that any $G$-orbit collapses to a single $K$-orbit and two critical points of $f$ which are in the same $G$-orbit belong to the same $K$-orbit. We also investigate convexity properties of the gradient map $μ_\mathfrak{p}$ in the Abelian cases. In particular, we study two orbits variety $X$ and we investigate topological and cohomological properties of $X$.

math.DG↗
arXiv2022

Stability, analytic stability for real reductive Lie groups

We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group $G$ on a real submanifold $X$ of a Kähler manifold $Z$. More precisely, we suppose the action of a compact connected Lie group $U$ with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of the complexified group $U^{\mathbb C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^{\mathbb C}$ is closed and compatible, there is a corresponding gradient map $μ_\mathfrak{p} : X\longrightarrow \mathfrak{p}$, where $\mathfrak g = \mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of the Lie algebra of $G$. The concept of energy complete action of $G$ on $X$ is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a $G$-equivariant function associated with a gradient map, called maximal weight function. We also prove the classical Hilbert-Mumford criteria for semistabilty and polystability conditions. We thank the anonymous referee for carefully reading our paper and for giving such constructive comments which substantially helped improving the quality of the paper.

math.DG↗
arXiv2022

Satake-Furstenberg compactifications and gradient map

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $τ$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $τ(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $μ_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of $τ(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of $G$ induced by $τ$ are completely determined by the facial structure of the polar orbitope $\mathcal E=\mathrm{conv}(μ_{\mathfrak p} (\mathcal O))$. Moreover, any parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to $\mathcal O$ and $μ_{\mathfrak p}$ respectively. These results are new also in the complex reductive case. The connection between $\mathcal E$ and $τ$ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure $γ$ on $\mathcal O$, we construct a map $Ψ_γ$ from the Satake compactification of $G/K$ associated to $τ$ and $\mathcal E$. If $γ$ is a $K$-invariant measure then $Ψ_γ$ is an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we prove that for a large class of measures the map $Ψ_γ$ is surjective.

math.DG↗
arXiv2022

Projective representations of real reductive Lie groups and the gradient map

Let $G$ be a connected semisimple noncompact real Lie group and let $ρ: G \longrightarrow \mathrm{SL}(V)$ be a representation on a finite dimensional vector space $V$ over $\mathbb R$, with $ρ(G)$ closed in $\mathrm{SL}(V)$. Identifying $G$ with $ρ(G)$, we assume there exists a $K$-invariant scalar product $\mathtt g$ such that $G=K\exp(\mathfrak p)$, where $K=\mathrm{SO}(V,\mathtt g)\cap G$, $\mathfrak p=\mathrm{Sym}_o (V,\mathtt g)\cap \mathfrak g$ and $\mathfrak g$ denotes the Lie algebra of $G$. Here $\mathrm{Sym}_o (V,\mathtt g)$ denotes the set of symmetric endomorphisms with trace zero. Using the $G$-gradient map techniques we analyze the natural projective representation of $G$ on $\mathbb P(V)$.

math.RT↗
arXiv2021

Compact orbits of Parabolic subgroups

We study the action of a real reductive group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. We suppose that the action of a compact connected Lie group $U$ with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of the complexified group $U^{\mathbb C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^{\mathbb C}$ is compatible there exists a gradient map $μ_{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$. In this paper we describe compact orbits of parabolic subgroups of $G$ in term of the gradient map $μ_\mathfrak p$.

math.DG↗
arXiv2021

A splitting result for real submanifolds of a Kahler manifold

Let $(Z,ω)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariant connected submanifold of $Z$. Let $x\in M$. If $G$ is a real form of $U^{\mathbb C}$, we investigate conditions such that $G\cdot x$ compact implies $U^{\mathbb C} \cdot x$ is compact as well. The vice-versa is also investigated. We also characterize $G$-invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $(Z,ω)$ generalizing a result proved in \cite{pg}, see also \cite{bg,bs}.

math.DG↗
arXiv2020

A group theoretic proof of a compactness lemma and existence of nonradial solutions for semilinear elliptic equations

Symmetry plays a basic role in variational problems (settled e.g. in $\mathbb R^{n}$ or in a more general manifold), for example to deal with the lack of compactness which naturally appear when the problem is invariant under the action of a noncompact group. In $\mathbb R^n$, a compactness result for invariant functions with respect to a subgroup $G$ of $\mathrm{O}(n)$ has been proved under the condition that the $G$ action on $\mathbb R^n$ is compatible, see \cite{willem}. As a first result we generalize this and show here that the compactness is recovered for particular subgroups of the isometry group of a Riemannian manifold. We investigate also isometric action on Hadamard manifold $(M,g)$ proving that a large class of subgroups of $\mathrm{Iso}(M,g)$ is compatible. As an application we get a compactness result for ``invariant'' functions which allows us to prove the existence of nonradial solutions for a classical scalar equation and for a nonlocal fractional equation on $\mathbb R^n$ for $n=3$ and $n=5$, improving some results known in the literature. Finally, we prove the existence of nonradial invariant functions such that a compactness result holds for some symmetric spaces of non compact type.

math.AP↗
arXiv2019

Convexity properties of gradient maps associated to real reductive representations

Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson. A similar result is proved for the gradient map associated to the natural $G$ action on P(V). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result

math.RT↗
arXiv2019

Meromorphic limits of automorphisms

Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probability measures on $X$. In particular we obtain an extension of Furstenberg lemma to manifolds in the class $\mathscr{C}$.

math.CV↗
arXiv2018

Remarks on the abelian convexity theorem

This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.

math.DG↗
arXiv2018

Convexity theorems for the gradient map on probability measures

Given a Kähler manifold $(Z,J,ω)$ and a compact real submanifold $M\subset Z$, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group ${\rm G}$ on the space of probability measures on $M.$ In particular, we prove convexity results for such map when ${\rm G}$ is Abelian and we investigate how to extend them to the non-Abelian case.

math.DG↗
arXiv2016

Stability of measures on Kähler manifolds

Let $(M,ω)$ be a Kähler manifold and let $K$ be a compact group that acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on probability measures on $M$. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of $K^\mathbb{C}$ on measures. We get various stability criteria for measures on Kähler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied by Hersch and Bourguignon-Li-Yau.

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↗