We determine the isometric congruence classes of homogeneous Riemannian foliations of codimension one on connected irreducible Riemannian symmetric spaces of noncompact type. As an application we show that on each connected irreducible Riemannian symmetric space of noncompact type and rank greater than two there exist noncongruent homogeneous isoparametric systems with the same principal curvatures, counted with multiplicities.

We construct examples of closed negatively curved manifolds M which are homeomorphic but not diffeomorphic to Cayley locally symmetric spaces. Given ∊ > 0, we can construct such an M with sectional curvatures all in [−4 − ∊,−1].

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

In this paper, we study the holomorphic de Rham cohomology of a compact strongly pseudoconvex CR manifold X in ℂ^N with a transversal holomorphic S¹-action. The holomorphic de Rham cohomology is derived from the Kohn-Rossi cohomology and is particularly interesting when X is of real dimension three and the Kohn-Rossi cohomology is infinite dimensional. In Theorem A, we relate the holomorphic de Rham cohomology H^k_h(X) to the punctured local holomorphic de Rham cohomology at the singularity in the variety V which X bounds. In case X is of real codimension three in ℂⁿ⁺¹, we prove that Hⁿ⁻¹_h(X) and Hⁿ_h(X) have the same dimension while all other H^k_h(X), k > 0, vanish (Theorem B). If X is three-dimensional and V has at most rational singularities, we prove that H¹_h(X) and H²_h(X) vanish (Theorem C). In case X is three-dimensional and N = 3, we obtain in Theorem D a complete characterization of the vanishing of the holomorphic de Rham cohomology of X.