We show that for a projective toric manifold with the ample second Chern character, if there exists a Fano contraction, then it is isomorphic to the projective space. For the case that the second Chern character is nef, the Fano contraction gives either a projective line bundle structure or a direct product structure. We also show that, for a toric weakly 2-Fano manifold, there does not exist a divisorial contraction to a point.

We extend the Hardy inequalities to the classical Hardy spaces and the rearrangement-invariant Hardy spaces.

We prove for $n\geq 3$ that every nonatomic ergodic measure of an $n$-dimensional flow whose Lyapunov exponents off the flow direction are all negative is supported on an attracting periodic orbit.

Let $\Omega$ be a smoothly bounded domain in $\mathbf{C}^{n}$, for $n\ge 2$. For a given continuous function $\phi$ on $b\Omega$, and a non-negative continuous function $\Psi$ on $\mathbf{R}\times \overline{\Omega}$, the main purpose of this note is to seek a plurisubharmonic function $u$ on $\Omega$, continuous on $\overline{\Omega}$, which solves the following Dirichlet problem of the complex Monge-Ampère equation \begin{equation*} \begin{cases} \det\left[\dfrac{\partial^{2}(u)}{\partial z_{i}\partial\bar{z}_{j}}\right](z)=\Psi(u(z),z)\geqslant 0 & \text{in}\quad\Omega,\\ u=\phi & \text{on}\quad b\Omega. \end{cases} \end{equation*} In particular, the boundary regularity for the solution of this complex, fully nonlinear equation is studied when $\Omega$ belongs to a large class of weakly pseudoconvex domains of finite and infinite type in $\mathbf{C}^{n}$.