For an odd prime number $p$, we classify the isomorphism classes of finitely generated torsion $\Lambda=\mathbb{Z}_{p}[[T]]$-modules with $\lambda=3$ and $\mu=0$, which are free over $\mathbb{Z}_{p}$. We apply this classification to the Iwasawa module associated to the cyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic field.

Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Möbius automorphisms, the coverings $\mathbb{P}^{1}\to\mathbb{P}^{1}$ that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases with Liouvillian solutions). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 are compositions of smaller degree transformations between hypergeometric and Heun functions. The 61 transformations are realized by 48 different Belyi coverings (though 2 coverings should be counted twice as their moduli field is quadratic). 38 of these coverings appear in Herfurtner's list of elliptic surfaces over $\mathbb{P}^{1}$ with four singular fibers, as their $j$-invariants. In passing, we show in an elegant way that there are no coverings with some branching patterns.

We prove that amenability of a discrete group is equivalent to dimension flatness of certain ring inclusions naturally associated with measure preserving actions of the group. This provides a group-measure space theoretic solution to a conjecture of Lück stating that amenability of a group is characterized by dimension flatness of the inclusion of its complex group algebra into the associated von Neumann algebra.

We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of compact Lie groups. We show that up to conjugation the faces are completely determined by the geometry of the faces of the convex hull of Weyl group orbits. We also consider the geometry of the faces and show that they are themselves coadjoint orbitopes. From the complex geometric point of view the sets of extreme points of a face are realized as compact orbits of parabolic subgroups of the complexified group.

Let $G$ be an open set of $\mathbb{R}^{d}$ ($d\ge 2$) and $dx$ denotes the Lebesgue measure on it. We construct a diffusion process with jumps associated with diffusion data (diffusion coefficients $\{a_{ij}(x)\}$, a drift coefficient $\{b_{i}(x)\}$ and a killing function $c(x)$) and a Lévy kernel $k(x,y)$ in terms of a lower bounded semi-Dirichlet form on $L^{2}(G;dx)$. When $G$ is the whole space, we allow that the diffusion coefficients may degenerate. We also show some Sobolev inequalities for the Dirichlet form and then show the absolute continuity of its resolvent.

In many problems of PDE involving the Laplace--Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which includes asymptotically conical and hyperbolic manifolds. We study the decay rate to the metric at infinity associated to radial coordinates and also show that the latter metric is always conformally equivalent to the metric at infinity associated to the original coordinate system. We finally give several examples illustrating the sharpness of our results.

We consider relations between two families of flat manifolds with holonomy group $\mathbb{Z}_{2}^{k}$ of diagonal type: the family $\mathcal{RBM}$ of real Bott manifolds and the family $\mathcal{GHW}$ of generalized Hantzsche--Wendt manifolds. In particular, we prove that the intersection $\mathcal{GHW}\cap \mathcal{RBM}$ is not empty. Moreover, we consider some class of real Bott manifolds without $\operatorname{Spin}$ and $\operatorname{Spin}^{\mathbb{C}}$ structure. There are given conditions (Theorem 1) for the existence of such structures. As an application a list of all 5-dimensional oriented real Bott manifolds without $\operatorname{Spin}$ structure is given.

We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, equipped with either one or two conformal vector fields. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally geodesic hypersurfaces in terms of their $r$-th mean curvatures. For instance, for a timelike geodesically complete conformally stationary spacetime endowed with a closed conformal timelike vector field $V$, under appropriate restrictions on the flow and the norm of the tangential component of $V$, we are able to prove that totally geodesic spacelike hypersurfaces must be, in fact, leaves of the distribution determined by $V$. Applications to the so-called generalized Robertson--Walker spacetimes are also given. Furthermore, we extend our approach in order to obtain a lower estimate of the relative nullity index.

The self-similar solutions to the mean curvature flow have been defined and studied on the Euclidean space. In this paper we propose a general treatment of the self-similar solutions to the mean curvature flow on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend results on special Lagrangian submanifolds on $\mathbb{C}^{n}$ to the toric Calabi--Yau cones over Sasaki--Einstein manifolds.

When the cohomology ring of a generalized Bott manifold with $\mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $\mathbb{C}P^{n_{i}}$, the generalized Bott manifold is said to be $\mathbb{Q}$-trivial. We find a necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$-trivial. In particular, every $\mathbb{Q}$-trivial generalized Bott manifold is diffeomorphic to a $\prod_{n_{i}>1}\mathbb{C}P^{n_{i}}$-bundle over a $\mathbb{Q}$-trivial Bott manifold.

In this paper we address the following questions for smooth Fano threefolds of Picard number 1: \begin{itemize} \item \textit{When does such a threefold $X$ possess an open cylinder $U \simeq Z\times\mathbb{A}^{1}$, where $Z$ is a surface?} \item \textit{When does an affine cone over $X$ admit an effective action of the additive group of the base field?} \end{itemize} A geometric criterion from [26] (see also [27]) says that the two questions above are equivalent. In [26] we found some interesting families of Fano threefolds carrying a cylinder. Here we provide new such examples.

Let $X$ be a complete toric variety of dimension $n$ and $\Delta$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\Delta$ there corresponds an orbit closure $V(\sigma)$ of the action of complex torus on $X$. The homology classes $\{[V(\sigma)] \mid \dim \sigma = k\}$ form a set of specified generators of $H_{n-k}(X, \mathbb{Q})$. Then any $x \in H_{n-k}(X, \mathbb{Q})$ can be written in the form \begin{equation*} x = \sum_{\sigma \in \Delta_{X}{,} \dim\sigma = k}\mu(x, \sigma)[V(\sigma)]. \end{equation*} A question occurs whether there is some canonical way to express $\mu(x, \sigma)$. Morelli [12] gave an answer when $X$ is non-singular and at least for $x = \mathcal{T}_{n-k}(X)$ the Todd class of $X$. However his answer takes coefficients in the field of rational functions of degree $0$ on the Grassmann manifold $G_{n-k+1}(N_{\mathbb{Q}})$ of $(n-k+1)$-planes in $N_{\mathbb{Q}}$. His proof uses Baum--Bott's residue formula for holomorphic foliations applied to the action of complex torus on $X$ On the other hand there appeared several attempts for generalizing non-singular toric varieties in topological contexts [4, 10, 7, 11, 9, 2]. Such generalized manifolds of dimension $2n$ acted on by a compact $n$ dimensional torus $T$ are called by the names quasi-toric manifolds, torus manifolds, toric manifolds, toric origami manifolds, topological toric manifolds and so on. Similarly torus orbifold can be considered. To a torus orbifold $X$ a simplicial set $\Delta_{X}$ called multi-fan of $X$ is associated. A question occurs whether a similar expression to Morelli's formula holds for torus orbifolds. It will be shown the answer is yes in this case too at least when the rational cohomology ring $H^{*}(X)_{\mathbb{Q}}$ is generated by $H^{2}(X)_{\mathbb{Q}}$. Under this assumption the equivariant cohomology ring with rational coefficients $H^{*}_{T}(X, \mathbb{Q})$ is isomorphic to $H^{*}_{T}(\Delta_{X}, \mathbb{Q})$, the face ring of the multi-fan $\Delta_{X}$, and the proof is carried out on $H^{*}_{T}(\Delta_{X}, \mathbb{Q})$ by using completely combinatorial terms.