It is proved that the diffeomorphism type of the minimal symplectic fillings of the link of a simple singularity is unique. In the proof, the uniqueness of the diffeomorphism type of CP² ∖ D, where D is a pseudo holomorphic rational curve with one (2,3)-cusp, is discussed.

The first part of this paper is devoted to proving a comparison theorem for Kähler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are ℙℂ^m, ℙ^m, and ℙℍ^m. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m² for a complete, m-dimensional, Kähler manifold with holomorphic bisectional curvature bounded from below by −1. The second part of the paper is to show that if this upper bound is achieved and when m=2, then it must have at most four ends.

Let $\Sigma \in \mathbf{R}^3$ be a smooth compact connected surface without boundary and denote by $A$ its second fundamental form. We prove the existence of a universal constant $C$ such that \begin{equation} \inf_{\lambda\in {\bf R}}\Vert A - \lambda \rm{Id} \Vert_{L^2(\Sigma)} \leq C \Vert A - \frac{\rm{tr}A}{2} \rm {Id} \Vert_{L^2(\Sigma)^\cdot} \end{equation} Building on this, we also show that, if the right-hand side of (1) is smaller than a geometric constant, Σ is W^2,2–close to a round sphere.

We show that C² conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.

In this paper, we show that the notion of moment map for the Hamiltonian action of a Lie group on a symplectic manifold is a special case of a much more general notion. In particular, we show that one can associate a moment map to a family of Hamiltonian symplectomorphisms, and we prove that its image is characterized, as in the classical case, by a generalized “energy-period” relation.

In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the Kähler–Einstein metric, we show that the logarithmic cotangent bundle over the Deligne–Mumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the Kähler–Einstein metric by using the Kähler–Ricci flow and a priori estimates of the complex Monge-Ampere equation.