We study a natural counterpart of the Nirenberg problem, namely to prescribe the Q-curvature of a conformal metric on the standard S⁴ as a given function f. Our approach uses a geometric flow within the conformal class, which either leads to a solution of our problem as, in particular, in the case when f ≡ const, or otherwise induces a blow-up of the metric near some point of S⁴. Under suitable assumptions on f, also in the latter case the asymptotic behavior of the flow gives rise to existence results via Morse theory.

Let M be a smooth manifold and D_m, m ≥ 2, be the set of rank m distributions on M endowed with the Whitney C^∞ topology. We show the existence of an open set O_m dense in D_m, so that every nontrivial singular curve of a distribution D of O_m is of minimal order and of corank one. In particular, for m > 3, every distribution of O_m does not admit nontrivial rigid curves. As a consequence, for generic sub-Riemannian structures of rank greater than or equal to three, there do not exist nontrivial minimizing singular curves.

Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N, α) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set α of simple closed curves. We prove that the map which associates to each structure in P(N, α) the lengths of the curves in the bending locus α is one-to-one. If α is maximal, the traces of the curves in α are local parameters for the representation space R(N).

We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded as a standard hyperbolic subspace. Given a complex number t, we obtain a corresponding object in hyperbolic 3-space by earthquaking along the lamination, parametrized by the real part of t, and then bending along the image lamination, parametrized by the complex part of t. In the literature, it is usually assumed that there is a quasifuchsian group that preserves the structure, but this paper is more general and makes no such assumption. Our deformation is holomorphic, as in the λ-lemma, which is a result that underlies the results in this paper. Our deformation is used to produce a new, more natural proof of Sullivan's theorem: that, under standard topological hypotheses, the boundary of the convex hull in hyperbolic 3-space of the complement of an open subset U of the 2-sphere is quasi-conformally equivalent to U, and that, furthermore, the constant of quasiconformality is a universal constant. Our paper presents a precise statement of Sullivan's Theorem. We also generalize much of McMullen's Disk Theorem, describing certain aspects of the parameter space for certain parametrized spaces of 2-dimensional hyperbolic structures.

Let H^p_l(M) be the space of polynomial growth harmonic forms. We proved that the dimension of such spaces must be finite and can be estimated if the metric is uniformly equivalent to one with a nonnegative curvature operator. In particular, this implies that the space of harmonic forms of fixed growth order on the Euclidean space with any periodic metric must be finite dimensional.