Convexity theorems for the gradient map on probability measures
Leonardo BiliottiComplex Manifolds
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Given a Kähler manifold (Z, J, ω) and a compact real submanifold M ⊂ Z, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group G on the space of probability measures on M. In particular, we prove convexity results for such map when G is Abelian and we investigate how to extend them to the non-Abelian case.
Related papers
Convexity properties of gradient maps
Advances in Mathematics, 2010
We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of the complexified group G C . We assume that the induced action of a compatible maximal compact subgroup U of G C on Z is Hamiltonian. We have an associated gradient map µp : Z → p where g = k ⊕ p is the Cartan decomposition of g. For a G-stable subset Y of Z we consider convexity properties of the intersection of µp (Y ) with a closed Weyl chamber in a maximal abelian subspace a of p. Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z = P(V ) where V is a unitary representation of U .
Properties of Gradient maps associated with Action of Real reductive Group
2021
Let (Z, ω) be a Kähler manifold and let U be a compact connected Lie group with Lie algebra u acting on Z and preserving ω. We assume that the U -action extends holomorphically to an action of the complexified group U and the U -action on Z is Hamiltonian. Then there exists a U -equivariant momentum map μ : Z → u. If G ⊂ U is a closed subgroup such that the Cartan decomposition U = Uexp(iu) induces a Cartan decomposition G = Kexp(p), where K = U ∩ G, p = g ∩ iu and g = k ⊕ p is the Lie algebra of G, there is a corresponding gradient map μp : Z → p. If X is a G-invariant compact and connected real submanifold of Z, we may consider μp as a mapping μp : X → p. Given an Ad(K)-invariant scalar product on p, we obtain a Morse like function f = 1 2 ‖ μp ‖ 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 ...
Convexity properties of gradient maps associated to real reductive representations
2019
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
Stability of measures on Kähler manifolds
Advances in Mathematics, 2017
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 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 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. Contents 1. Introduction 1 2. Kempf-Ness functions 4 3. Stability 10 4. Polystability and semi-stability 13 5. Measures 17 6. The construction of Hersch and Bourguignon-Li-Yau 25 7. Applications 31 References 39
Convolution of Probability Measures on Lie Groups and Homogenous Spaces
Potential Analysis, 2015
We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.
Exponential Families on sharp convex cones & homogeneous manifolds and Koszul-Souriau- Fisher Metric: From Information Geometry to Statistical Mechanics of Dynamical Systems
▌ 1955 « Sur la forme hermitienne canonique des espaces homogènes complexes » Koszul considers the Hermitian structure of a homogeneous G/B manifold (G related Lie group and B a closed subgroup of G, associated, up to a constant factor, to the single invariant G, and to the invariant complex structure by the operations of G).
Factors, Roots and Embeddings of Measures on Lie Groups
Perspectives in Mathematical Sciences I, 2009
Openness and convexity for momentum maps
Transactions of the American Mathematical Society, 2008
The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the "Lokal-global-Prinzip" that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to non-compact manifolds.
Satake-Furstenberg compactifications, the moment map and \lambda_1
arXiv: Differential Geometry, 2010
Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of K. For the K-invariant measure, this map is a homeomorphism of the Satake compactification onto the convex envelope of O. For a large class of measures the image of the map is the convex envelope. As an application we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for an arbitrary Kaehler metric on a Hermitian symmetric space.
Convergence of Semigroups of Complex Measures on a Lie Group
2020
A theorem of Siebert in its essential part asserts that if μ n (t) are semigroups of probability measures on a Lie group G, and P n are the corresponding generating functionals, then for every unitary representation π of G on a Hilbert space E, where C ∞ (E, π) denotes the space of smooth vectors for π. The aim of this note is to give a simple proof of the theorem and propose some improvements, the most important being the extension of the theorem to semigroups of complex measures. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis for bounded twice differentiable functions f . As a corollary, the above thesis of Siebert is extended to bounded strongly continuous representations of G on Banach spaces.
Related topics
Related papers
Remarks on the abelian convexity theorem
Proceedings of the American Mathematical Society
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.
A remark on the gradient map
For a Hamiltonian action of a compact group U of isometries on a compact Kähler manifold Z and a compatible subgroup G of U C , we prove that for any closed G-invariant subset Y ⊂ Z the image of the gradient map µ p (Y ) is independent of the choice of the invariant Kähler form ω in its cohomology class [ω].
A Splitting Result for Real Submanifolds of a Kähler Manifold
2021
Let (Z, ω) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group U C , where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of U C and let M be a G-invariant connected submanifold of Z. Let x ∈ M. If G is a real form of U C , we investigate conditions such that G • x compact implies U C • 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 [7], see also [1, 3].
Projective representations of real reductive Lie groups and the gradient map
2022
Let G be a connected semisimple noncompact real Lie group and let ρ : G −→ SL(V) be a representation on a finite dimensional vector space V over R, with ρ(G) closed in SL(V). Identifying G with ρ(G), we assume there exists a K-invariant scalar product g such that G = K exp(p), where K = SO(V, g) ∩ G, p = Sym o (V, g) ∩ g and g denotes the Lie algebra of G. Here Sym o (V, 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 P(V).
Kählerian potentials and convexity properties of the moment map
Inventiones Mathematicae, 1996
Since Kostant proved his convexity theorems for torus actions on ag varieties ([Ko]), there have been numerous contributions to this subject in the more general symplectic and K ahlerian setting. For example, let K be a compact Lie group acting in a Hamiltonian fashion on a connected compact symplectic manifold X . Then the intersection (X ) + of the image of the moment map : X → (Lie K) * with a positive Weyl chamber t * + is convex. Thus (X ) = K · (X ) + , where (X ) + is a natural convex section. This was proved by Atiyah and Guillemin-Sternberg ([A], [G-S]) in the abelian case, i.e., where K = T is a compact torus, by Mumford [M]) for X projective algebraic with an integral K ahlerian structure and K not necessarily abelian and in its ÿnal form in the compact symplectic case by Kirwan ([K]).
On the embeddability of certain infinitely divisible probability measures on Lie groups
2011
We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain Lie groups, including the so called Walnut group (Corollary 1.5). The embeddability is concluded also under certain other conditions (Corollary 1.4 and Theorem 1.6).
A convexity theorem for Poisson actions of compact Lie groups
Annales scientifiques de l'École normale supérieure
<http://www.numdam.org/item?id=ASENS_1996_4_29_6_787_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1996, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens), implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Differentiation of measures on complete Riemannian manifolds
arXiv: Functional Analysis, 2020
In this note we give a new proof of a version of the Besicovitch covering theorem, given in \cite{EG1992}, \cite{Bogachev2007} and extended in \cite{Federer1969}, for locally finite Borel measures on finite dimensional complete Riemannian manifolds $(M,g)$. As a consequence, we prove a differentiation theorem for Borel measures on $(M,g)$, which gives a formula for the Radon-Nikodym density of two nonnegative locally finite Borel measures $\nu_1, \nu_2$ on $(M, g)$ such that $\nu_1 \ll \nu_2$, extending the known case when $(M, g)$ is a standard Euclidean space.
Metric convexity in the symplectic category
2006
We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. This extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino in [8] and allows us to obtain the most general convexity statement available in the literature for momentum maps associated to a symplectic Lie group action.
Stability with respect to actions of real reductive Lie groups
Annali di Matematica Pura ed Applicata (1923 -)
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 noncompact 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 noncompact reductive Lie group G on a real compact submanifold M of a Kähler manifold Z and to the action of G on measures of M.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.