Donaldson conjectured [16] that the space of Kähler metrics is geodesically convex by smooth geodesics and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in each Kähler class if the first Chern class is either strictly negative or 0. Furthermore, if C₁ ≤ 0, the constant scalar curvature metrics: realizes the global minimum of the Mabuchi K energy functional; thus it provides a new obstruction for the existence of constant curvature metrics: if the infimum of the K energy (taken over all metrics in a fixed Kähler class) is not bounded from below, then there does not exist a constant curvature metric. This extends the work of Mabuchi and Bando [3]: they showed that K energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class.

We prove the existence of contact submanifolds realizing the Poincaré dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact category to Donaldson's construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfy a transversality with estimates property directly in the contact category. The description of the obtained contact submanifolds allows us to prove an extension of the Lefschetz hyperplane theorem which completes their topological characterization.

In this paper we give a new method to convert results dealing with graph theoretic (or Markov chain) Laplacians into results concerning Laplacians in analysis, such as on Riemannian manifolds. We illustrate this method by using the results of [6] to prove

$$ \λ ₁\leq {1/over dist²(X,Y)} /Big(\cosh ^-1\sqrt {\mu X^c\mu Y^c\over \mu X\mu Y}\Big)². $$

for λ₁ the first positive Neumann eigenvalue on a connected compact Riemannian manifold, and X, Y any two disjoint sets (and where X^c is the complement of X). This inequality has a version for the k-th positive eigenvalue (involving k + 1 disjoint sets), and holds more generally for all "analytic" Laplacians described in [6]. We show that this inequality is optimal "to first order," in that it is impossible to obtain an inequality of this form with the right-hand-side divided by 1 + ∊ for any fixed constant ∊ > 0.

This paper continues the study of holomorphic semistable principal G-bundles over an elliptic curve. In this paper, the moduli space of all such bundles is constructed by considering deformations of a minimally unstable G-bundle. The set of all such deformations can be described as the ℂ^*-quotient of the cohomology group of a sheaf of unipotent groups, and we show that this quotient has the structure of a weighted projective space. We identify this weighted projective space with the moduli space of semistable G-bundles, giving a new proof of a theorem of Looijenga.