This paper, the second of a series, deals with the function space \mathcal{H} of all smooth Kähler metrics in any given n-dimensional, closed complex manifold V, these metrics being restricted to a given, fixed, real cohomology class, called a polarization of V. This function space is equipped with a pre-Hilbert metric structure introduced by T. Mabuchi [10], who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series [4], the second author showed that the same space is a path length space. He also proved that \mathcal{H} is geodesically convex in the sense that, for any two points of \mathcal{H}, there is a unique geodesic path joining them, which is always length minimizing and of class C^1,1. This partially verifies two conjectures of Donaldson [8] on the subject. In the present paper, we show first of all, that the space is, as expected, a path length space of nonpositive curvature in the sense of A. D. Aleksandrov. A second result is related to the theory of extremal Kähler metrics, namely that the gradient flow in \mathcal{H} of the "K energy" of V has the property that it strictly decreases the length of all paths in \mathcal{H}, except those induced by one parameter families of holomorphic automorphisms of M.

Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by −1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, closed hyperbolic manifolds minimize volume over all hyperbolic cone-manifolds in the same bilipschitz class with cone angles ≤ 2π. The proof uses techniques developed by Besson-Courtois-Gallot. In 3 dimensions, this result provides a partial solution to a conjecture in Kleinian groups concerning acylindrical manifolds.

The main purpose of this paper is to show that Bridgeland's moduli space of perverse point sheaves for certain flopping contractions gives the flops, and the Fourier-Mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories.

In this article, we prove that for any compact Kähler manifold Mⁿ with real analytic metric and nonpositive bisectional curvature, there exists a finite cover M^′ of M such that M^′ is a holomorphic and metric fiber bundle over a compact Kähler manifold N with nonpositive bisectional curvature and c₁(N) < 0, and the fiber is a flat complex torus. This partially confirms a conjecture of Yau.

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number (in the appropriate cases) and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra $\mathbb{A}$. For example, special Riemannian $\mathbb{A}$-manifolds are oriented Riemannian, Calabi-Yau, hyperkähler and $G_2$-manifolds respectively.

For vector bundles over such manifolds, we introduce (special) $\mathbb{A}$-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) $\frac{1}{2}\mathbb{A}$-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds.

We also discuss geometric dualities from this viewpoint: Fourier transformations on $\mathbb{A}$-geometry for flat tori and a conjectural SYZ mirror transformation from (special) $\mathbb{A}$-geometry to (special) $\frac{1}{2} \mathbb{A}$-Lagrangian geometry on mirror special $\mathbb{A}$-manifolds.

We prove Hersch's type isoperimetric inequality for the second positive eigenvalue on a two dimensional sphere.