We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. They are given by representations of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum polytope. There is a tight relation with parabolic subgroups: the set of extreme points of a face is the closed orbit of a parabolic subgroup of G and for any parabolic subgroup the closed orbit is of this form. Contents

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

Abstract. We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ⊂ p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted momentum map.

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 Kähler manifold Z and to the action of G on measures of M .2010 Mathematics Subject Classification. 53D20; 14L24.

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [27]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds. CONTENTS 2 1 0 g(γ,γ) dt ∈ R, defined on the Hilbert manifold Ω p,q of all curves of Sobolev class H 1 in M joining p and q, is a Morse function. Standard Morse theory does not apply to the semi-Riemannian geodesic action functional, due to the fact that in the non positive definite case all its critical points have infinite Morse index. Recent developments of Morse theory, mostly due to the work of Abbondandolo and Majer (see [1,2]) have shown that, under suitable assumptions, one can construct a doubly infinite chain complex (Morse-Witten complex) out of the critical points of a strongly indefinite Morse functional, using the dynamics of the gradient flow. The Morse relations for the critical points are obtained by computing the homology of this complex, which in the standard Morse theory is isomorphic to the singular homology of the base manifolds. Such computation is one of the central and highly non trivial issues of the theory. Remarkably, Abbondandolo and Majer have also shown that this homology is stable by "small" perturbations, so that in several concrete examples one can reduce its computation to a simpler case. This occurs for instance in the case of the geodesic action functional in a globally hyperbolic Lorentzian manifold, in which case the homology of the Morse-Witten complex is stable by small C 0 perturbations of the metric. Thus, it becomes a relevant issue to discuss under which circumstances a given metric tensor can be perturbed in a given class in such a way that the nondegenericity property for its geodesics between two prescribed points is preserved. This problem is the original motivation for the results developed in this paper; we basically give an affirmative answer to the genericity questions posed above, with three remarkable exceptions that will be discussed below.The idea for proving the genericity of the nondegeneracy property for the critical points of a family of functionals, which follows a standard transversality approach (see the classical reference [5, Chapter 4], or the more recent [3, Section 2.11]), is the following. Assume that one is given a Hilbert manifold Y , and a family of functionals f x : Y → R parameterized by points x in an open subset A of a Banach space X. In the geodesic case, Y is the Hilbert manifold Ω p,q (M ) of curves between two fixed poin...