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We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori’s Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifolds. As a by-product, we confirm the conjecture of Guedj [20] for automorphisms on 3-dimensional projective manifolds, and also determine π1(X).

In this paper, we prove a scalar curvature rigidity result for geodesic balls in Sn. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in [7].

We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n  10, is highly nonconnected.

In this article we show that any hyperbolic Inoue surface (also called Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result holds for any of its small deformations as far as its anti-canonical system is non-empty. Similar results are obtained for parabolic Inoue surfaces. Our method also yields a family of anti-self-dual hermitian metrics on any half Inoue surface. We use the twistor method of Donaldson-Friedman [13] for the proof.