Abstract. In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f 1 for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if S is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain irregular toric Sasaki-Einstein metrics on the unit circle bundles of the powers of the canonical bundle of the two-point blow-up of the complex projective plane.

Link to this article: http://journals.cambridge.org/abstract_S0956792501004363How to cite this article: DIRK HORSTMANN and GUOFANG WANG (2001). Blow-up in a chemotaxis model without symmetry assumptions.In this paper we prove the existence of solutions of the Keller-Segel model in chemotaxis, which blow up in finite or infinite time. This is done without assuming any symmetry properties of the solution.

We introduce a functional that couples the nonlinear sigma model with a spinor field: L = M [|dφ| 2 + ψ, D //ψ ]. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.

Abstract. We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σ k -Yamabe problem for locally conformal flat manifolds when k = n/2.