Under open and dense conditions we show that Arnold diffusion orbits exist in a priori unstable and time-periodic Hamiltonian systems with two degrees of freedom.

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete non-compact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C ². This gives a partial affirmative answer to the well-known conjecture of Yau [41] on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in [42], which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.