In this paper, we first consider relations between signatures of pseudo- Kähler metrics on a flag manifold and complex structures on a nilpotent Lie algebra corresponding to the flag manifold. On the nilpotent Lie algebra, we also consider complex structures which do not correspond to invariant complex structures on the flag manifold.

We consider function spaces which consist of two parabolic Bloch spaces, and present reproducing formulas. As an application, we introduce Bloch type spaces which consist of solutions of a partial differential equation $(L^{(\alpha)})^2u=0$, and investigate several properties.

In this paper, we show that every locally solvable subnormal subgroup or locally solvable quasinormal subgroup of the multiplicative group of a division ring is central.

In this article we study a three dimensional contact metric manifold $M^3$ with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with $Q\xi = \rho\xi$, where $\rho$ is a smooth function on $M$ constant along Reeb vector field $\xi$ and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with $\xi$ or is a gradient vector field. Moreover, if $\rho$ is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: $SU(2)$ or $SO(3)$ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field $\xi$. Secondly, it is proved that a $(\kappa,\mu,\nu)$-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that $M$ is Sasakian, flat, a contact metric $(0,−4)$-space or a contact metric $(\kappa,0)$-space with $\kappa \lt 1$ and $\kappa\neq0$. For the potential vector field being orthogonal to $\xi$, if $\nu$ is constant we prove that $M$ is either Sasakian, or a $(\kappa,\mu)$-contact metric space.

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathscr E$ of rank $r \leq n - 2$ and an ample line bundle $H$. A numerical character $m_2 = m_2(X,\mathscr E, H)$ of the triplet $(X,\mathscr E, H)$ is defined, extending the well-known second class of a polarized manifold $(X, H)$, when either $n = 2$ or $H$ is very ample. Under some additional assumptions on $\mathscr F:= \mathscr E \oplus H^{\oplus(n-r-2)}$, triplets $(X,\mathscr E, H)$ as above whose $m_2$ is small with respect to the invariants $d := c_{n-2}(\mathscr F)H^2$ and $g := 1 + \frac{1}{2}(K_X + c_1(\mathscr F)+H)\cdot c_{n-2}(\mathscr F)\cdot H$ are studied and classified.

In this paper, we study two eigenvalue problems of the weighted Laplacian and get the Reilly-type bounds and isoperimetric type bounds for the first nonzero $n$ eigenvalues on hypersurfaces of the Euclidean space. Besides, we give lower bounds for the first eigenvalue of weighted $p$-Laplacian on submanifolds with locally bounded weighted mean curvature. Meanwhile, several applications of these estimates have also been given.