Let M be a compact Hamiltonian T−space, with finite fixed point set MT . An equivariant class is determined by its restriction to MT , and to each fixed point p ∈ MT and generic component of the moment map, there corresponds a canonical class τp. For a special class of Hamiltonian T−spaces, the value τp,q of τp at a fixed point q can be determined through an iterated interpolation procedure, and we obtained a formula for τp,q as a sum over ascending chains from p to q. In general the number of such chains is huge, and the main result of this paper is a procedure to reduce the number of relevant chains, through a systematic degeneration of the interpolation direction. The resulting formula for τp,q resembles, via the localization formula, an integral over a space of chains, and we prove that, for complex Grassmannians, τp,q can indeed be expressed as the integral of an equivariant form over a smooth Schubert variety.

We prove that if a connected Lie group action on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses K¨ahler identities, the Rumin-de Rham complex and spectral geometry.

We consider a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space,through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.

We extend recent existence results for compact constant mean curvature normal geodesic graphs with boundary in space forms in a unified way to a large class of ambient spaces. That extension is achieved by solving a constant mean curvature existence problem in a general setting presented in two isometrically equivalent versions.