We consider the Yamabe flow $\frac{\partial g}{\partial t} = -(R_g - r_g) \, g$ where $g$is a Riemannian metric on a compact manifold $M, R_g$ denotes its scalar curvature, and $r_g$ denotes the mean value of the scalar curvature. We prove convergence of the Yamabe flow if the dimension $n$ satisfies $3 \leq n \leq 5$ or the initial metric is locally conformally flat.

We study discrete group actions on coarse Poincaré duality spaces, e.g., acyclic simplicial complexes which admit free cocompact group actions by Poincaré duality groups. When G is an (n−1) dimensional duality group and X is a coarse Poincaré duality space of formal dimension n, then a free simplicial action G ↷ X determines a collection of “peripheral” subgroups H₁, … H_k ⊂ G so that the group pair (G, {H₁,…H_k }) is an n-dimensional Poincaré duality pair. In particular, if G is a 2-dimensional 1-ended group of type FP₂, and G ↷ X is a free simplicial action on a coarse PD(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse PD(3) spaces. In the process, we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasi-isometries and geometric group theory.

Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an area-length duality between a class of metric spaces and a class of symplectic manifolds. Using this duality, it is shown that there is a twistor-like correspondence between Finsler metrics on ℝPⁿ whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2-planes in ℝⁿ⁺¹.

We study local holomorphic mappings sending a piece of a real hyperquadric in a complex space into a hyperquadric in another complex space of possibly larger dimension. We show that these mappings possess strong super-rigidity properties when the hyperquadrics have positive signatures. These results are applied in the context of holomorphic mappings between classical domains in complex projective spaces of different dimensions.