We define complex constant mean curvature immersions in complex three space using a natural extension of constant mean curvature immersions in Euclidean three space via loop group techniques. We then discuss the fundamental properties of these complex constant mean curvature immersions. In particular, we prove that these immersions are doubly ruled by holomorphic null curves. We present a construction of minimal immersions from constant mean curvature immersions in Euclidean three space via the associated complex constant mean curvature immersions.

We show that if a connected closed orientable manifold $M$ admits a codimension one locally free smooth action $\phi$ of a connected nilpotent Lie group such that any orbit of $\phi$ is non-compact, then $M$ is homeomorphic to a nilmanifold. And as an example of such an action, we study also a homogeneous action.

We consider the existence and multiplicity of positive solutions to the quasilinear system $$ \begin{cases} -\Delta_{p_{i}}u_{i} = \mu_{i}a_{i}(x)f_{i}(u_{1},\dots,u_{n})~\text{in}~\Omega,\;i=1,\dots,n, \\[1pt] u_{i} = 0~\text{on}~\partial \Omega , \end{cases} $$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with a smooth boundary $\partial \Omega$, $\Delta_{p_{i}}u_{i}={\rm div}(|\nabla u_{i}|^{p_{i}-2}\nabla u_{i})$, $p_{i}>1$, $\mu_{i}$ are positive parameters, and $f_{i}$ are allowed to change sign.

The $\alpha$-parabolic Bergman space ${\boldsymbol b}_{\alpha}^{p}(\lambda)$ is the Banach space of solutions of the parabolic equation $L^{(\alpha)} = \partial/\partial t+(-\Delta_{x})^{\alpha}$ on the upper half space $H$ which have finite $L^{p}(H,t^{\lambda}dV)$ norms, where $t^{\lambda}dV$ is the weighted Lebesgue volume measure on $H$. It is known that ${\boldsymbol b}^{p}_{1/2}(\lambda)$ coincide with the harmonic Bergman spaces. In this paper, we introduce the extension of notion of conjugate functions of ${\boldsymbol b}_{\alpha}^{p}(\lambda)$-functions and study their properties. As an application, we give estimates of tangential derivative norms on ${\boldsymbol b}_{\alpha}^{p}(\lambda)$.

We introduce a ``weak $CMO$'' space $WCMO$ and consider the boundedness of the singular integral operators from $CMO$ to $WCMO$. Our result is the best possible.

This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation $$ f''+A_{1} (z) e^{P (z)}f'+A_{0} (z) e^{Q (z)}f = F, $$ where $P (z)$, $Q (z)$ are nonconstant polynomials such that $\deg P=\deg Q=n$ and $A_{j} (z)$ $( \not\equiv 0 )$ $(j=0,1)$, $F\not\equiv 0$ are entire functions with $\rho ( A_{j} ) < n$ $( j=0,1 )$. We also investigate the relationship between small functions and differential polynomials $g_{f} (z)=d_{2}f''+d_{1}f'+d_{0}f$, where $d_{0} (z)$, $d_{1} (z)$, $d_{2} (z)$ are entire functions that are not all equal to zero with $\rho ( d_{j} ) < n$ $( j=0,1,2 )$ generated by solutions of the above equation.