In this paper, we generalize the concept of Kähler-Einstein metrics for Fano manifolds with nonvanishing Futaki character. Similar to Kähler-Einstein metrics, these new metrics have various nice properties. In addition, the equations for the metrics are in general neither those of extremal Kähler metrics nor those of Kähler-Ricci solitons.

We consider the asymptotic behaviour of the transition density for processes of jump type as the time parameter $t$ tends to 0. We use Picard's duality method, which allows us to obtain the lower and upper bounds of the density even for the case where the support of Lévy measure is singular. The main result is that, under certain restrictions, the density behaves in polynomial order or may decrease in exponential order as $t\to0$ according to geometrical conditions of the objective points.

We establish a formula for homomorphisms and extensions of group schemes and formal groups, related to deformations of the multiplicative group to the additive group. As an application, we give an explicit description of the theory unifying the Kummer and Artin-Schreier-Witt theories of degree $p^2$.

Let $G$ be a non-unimodular solvable Lie group which is a semidirect product of $R^m$ and $R^n$. We consider a codimension one locally free volume preserving action of $G$ on a closed manifold. It is shown that, under some conditions on the group $G$, such an action is homogeneous. It is also shown that such a group $G$ has a homogeneous action if and only if the structure constants of $G$ satisfy certain algebraic conditions.

We consider Marcinkiewicz integrals arising from rough kernels satisfying the $L\log L$ condition on the unit $(n-1)$-sphere and prove the weak type (1, 1) estimates. We also prove the weighted weak type (1, 1) estimates with certain $A_1$-weights. In this case the $L\log L$-condition is replaced by the $L^q$-condition with $q>1$.

We give an improved proof for the result established recently by the present author that the scattering operators are well-defined in the whole energy space for a class of nonlinear Klein-Gordon and Schrödinger equations in any spatial dimension. Using some Sobolev-type inequatilies, we can simplify and somewhat enhance the Morawetz-type estimates and thereby weaken the required repulsivity conditions.

We investigate surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends, and prove that their irregular ends must self-intersect, which answers affirmatively a conjecture of Umehara and Yamada. Moreover we also obtain an explicit representation of a constant mean curvature one surface and a new minimal surface in the Euclidean three-space.

For each Painlevé system, we have a manifold, called the defining manifold, on which the system defines a uniform foliation. In this paper, we describe confluence processes in these manifolds as deformations of manifolds compatible to those in Painlevé systems.