On the extension problem for weak moment maps

Convexity properties of generalized moment maps

In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, M…

Circumcenter extension of Moebius maps to CAT(-1) spaces

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to coincide with the $(1, \log 2)$-quasi-isometric extension constructed in [biswas3], and is locally $1/2$-Holder continuous. When $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$ for some $b \geq 1$ then the extension $…

Lipschitz extensions of maps between Heisenberg groups

Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.

Geometric quantization for proper moment maps: the Vergne conjecture

We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.

Duistermaat-Heckman distributions for group valued moment maps

We introduce equivariant Liouville forms and Duistermaat-Heckman distributions for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat connections on 2-manifolds.

广义矩映射、约化与复商

Generalized moment maps, reduction and complex quotients

引入动量闭形式概念，将辛形式和矩映射推广到近Hermitian框架，证明Darboux-Weinstein定理变体，建立广义矩映射凸性、构造约化空间并研究Kirwan-Ness分层。

重访Manin对与动量图

Manin pairs and moment maps revisited

本文主要证明了带有D/G值动量图的拟泊松G-空间与带有D值动量图的拟泊松G×G-空间范畴间的双射对应提升定理，并据此给出了融合与共轭的简单构造及模空间新例子。

The Ding functional, Berndtsson convexity and moment maps

We explain how the formal aspects of the theory of Kahler-Einstein metrics can be developed in the framework of moment maps. The central result we use is the Berndtsson convexity theorem, which is interpreted as defining a metric on the space of complex structures. We discuss some applications of these ideas to the Kahler-Ricci flow.

Moment maps and Isoparametric hypersurfaces in spheres --- Hermitian cases

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. We prove that the Cartan-Münzner polynomials of these hypersurfaces can be written as squared-norms of the moment maps for some Hamiltonian actions. The proof is based on the structure theory of symmetric spaces.

Lectures on pure spinors and moment maps

The goal of this article is to give an elementary introduction to Dirac geometry and group-valued moment maps, via pure spinors. The material is based on my lectures at the summer school on 'Poisson geometry in Mathematics and Physics' at Keio University, June 2006.

Circumcenter extension maps for non-positively curved spaces

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a continuous map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve the quasi-isometry constants provided by Biswas in the case of CAT(-1)} spaces. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.

Dirac structures, moment maps and quasi-Poisson manifolds

We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and hamiltonian actions, to the realm of Dirac geometry. As an example, we show how hamiltonian quasi-Poisson manifolds fit into this framework by constructing an ``inversion'' procedure relating quasi-Poisson bivectors to twisted Dirac structures.

A cohomological framework for homotopy moment maps

Given a Lie group acting on a manifold $M$ preserving a closed $n+1$-form $ω$, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of $L_{\infty}$-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of …

Mean Curvature Flow, Orbits, Moment Maps

Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: eg, finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for example, proves that the minimal Lagrangian orbits are isolated.

Differentiability of quantum moment maps

We study quantum moment maps of $G$-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a $G$-invariant star product is differentiable. This property gives us a new method for the classification of $G$-invariant star products on regular coadjoint orbits of compact semisimple Lie groups.

Conformally Natural extensions revisited

In this note we revisit the notion of conformal barycenter of a measure on $\SS^n$ as defined by Douady and Earle in Acta Math. Vol 157, 1986. The aim is to extend rational maps from the Riemann sphere $\Cbar\isom\SS^2$ to the (hyperbolic) three ball $\BB^3$ and thus to $\SS^3$ by reflection. The construction which was pioneered by Douady and Earle in the case of homeomorphisms actually gives extensions for more general maps such as entire transcendental maps on $\C\subset\Cbar$. And it works for maps in any dimension.

Hypertoric manifolds and hyperKähler moment maps

We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension $4n$ with a tri-Hamiltonian action of a torus of dimension $n$, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperKähler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connecte…

Homogeneous spaces, multi-moment maps and (2,3)-trivial algebras

For geometries with a closed three-form we briefly overview the notion of multi-moment maps. We then give concrete examples of multi-moment maps for homogeneous hypercomplex and nearly Kaehler manifolds. A special role in the theory is played by Lie algebras with second and third Betti numbers equal to zero. These we call (2,3)-trivial. We provide a number of examples of such algebras including a complete list in dimensions up to and including five.

A Residue Formula for SU(2)-valued Moment Maps

Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently introduced by Alekseev, Malkin and Meinrenken.

Multi-moment maps

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by…

Scalar curvature, moment maps and the Deligne Pairing

In this paper, we the improve the bound for the moment map derivative proved by Donaldson in his recent proof of the Hilbert-Mumford stability of complex manifolds with constant scalar curvature. The proof depends on the identification of Donaldson's symplectic form with the curvature of a certain Deligne pairing.

Manin pairs and moment maps

A Lie group G in a group pair (D,G), integrating a Lie algebra g in a Manin pair (d,g) has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-hamiltonian. These moment maps take values in the homogeneous space D/G. We prove an analogue of the hamiltonian reduction theorem for quasi-Poisson group actions, and we study the symplectic leaves of the orbit spaces of quasi-hamiltonian spaces.

Moment maps and non-compact cobordisms

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of the normal bundles of the connected components of the fixed points set. We use this to give simple new proofs of two formulas: Guillemin's topological version of the abelian Jeffrey-Kirwan localization, and the Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measur…

Trivializations of moment maps

We study various trivializations of moment maps. First in the general framework of a reductive group $G$ acting on a smooth affine variety. We prove that the moment map is a locally trivial fibration over a regular locus of the center of the Lie algebra of $H$ a maximal compact subgroup of $G$. The construction relies on Kempf-Ness theory and Morse theory of the square norm of the moment map studied by Kirwan, Ness-Mumford and Sjamaar. Then we apply it together with ideas from Nakajima and Kronheimer to trivialize the hyperkaehler moment map for Nakajima quiver varieties. Notice this trivializ…

Closed forms and multi-moment maps

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.