We obtain a partial classification result for generic $3$-dimensional conformally flat hypersurfaces in the conformal $4$-sphere: explicit analytic data are obtained for conformally flat hypersurfaces with Bianchi-type canonical Guichard net. This is the first classification result for conformally flat hypersurfaces without additional symmetry. We discuss the curved flat associated family for conformally flat hypersurfaces and show that it descends to an associated family of conformally flat hypersurfaces. The associated family of conformally flat hypersurfaces with Bianchi-type Guichard net is investigated.

In this paper, we study a connected non-parabolic, or transient, network compactified with the Kuramochi boundary, and show that the random walk converges almost surely to a random variable valued in the harmonic boundary, and a function of finite Dirichlet energy converges along the random walk to a random variable almost surely and in $L^{2}$. We also give integral representations of solutions of Poisson equations on the Kuramochi compactification.

The quintic-mirror family is a well-known one-parameter family of Calabi--Yau threefolds. A complete description of the global monodromy group of this family is not yet known. In this paper, we give a presentation of the global monodromy group in the general linear group of degree 4 over the ring of integers modulo 5.

We show that if a normal $\mathbb{Q}$-acyclic complex surface has negative Kodaira dimension then its smooth locus is not of general type. This generalizes an earlier result of Koras--Russell for contractible surfaces.

We determine the moduli space of the surfaces of general type studied by Catanese, Ciliberto and Hirotaka Ishida by using the family of Hesse cubic curves.

In this paper we discuss a classification problem of homogeneous 2-spheres in the complex Grassmann manifold $G(k + 1, n + 1)$ by theory of unitary representations of the 3-dimensional special unitary group $\mathit{SU}(2)$. First we observe that if an immersion $x\colon S^{2} \to G(k + 1, n + 1)$ is homogeneous, then its image $x(S^{2})$ is a 2-dimensional $\rho(\mathit{SU}(2))$-orbit in $G(k + 1, n + 1)$, where $\rho\colon \mathit{SU}(2) \to U(n + 1)$ is a unitary representation of $\mathit{SU}(2)$. Then we give a classification theorem of homogeneous 2-spheres in $G(k + 1, n + 1)$. As an application we describe explicitly all homogeneous 2-spheres in $G(2, 4)$. Also we mention about an example of non-homogeneous holomorphic 2-sphere with constant curvature in $G(2, 4)$.

We present some classification results for quasitoric manifolds $M$ with $p_{1}(M) = -\sum a_{i}^{2}$ for some $a_{i}\in H^{2}(M)$ which admit an action of a compact connected Lie-group $G$ such that $\dim M/G \leq 1$. In contrast to Kuroki's work [7, 6] we do not require that the action of $G$ extends the torus action on $M$.

We show that antipodal sets of symmetric $R$-spaces have the following properties. Any antipodal set is included in a great antipodal set and any two great antipodal sets are congruent.

We give a formula for the Donaldson--Futaki invariants of certain type of semi test configurations, which essentially generalizes the Ross--Thomas slope theory [28]. The positivity (resp. non-negativity) of those ``a priori special'' Donaldson--Futaki invariants implies K-stability (resp. K-semistability). As an application, we prove the K-(semi)stability of certain polarized varieties with semi-log-canonical singularities, which generalizes some results of [28].

The canonical circle action on complex Stiefel manifolds is considered. The corresponding mod $p$ index for all primes $p$ is computed and some Borsuk--Ulam type theorems are proved.

We determine the group of conformal automorphisms of the self-dual metrics on $n \# \mathbb{CP}^{2}$ due to LeBrun for $n \geq 3$, and Poon for $n = 2$. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space $\mathcal{H}^{3}$ minus a finite number of points, called monopole points. We show that for $n \geq 3$, any conformal automorphism is a lift of an isometry of $\mathcal{H}^{3}$ which preserves the set of monopole points. Furthermore, we prove that for $n = 2$, such lifts form a subgroup of index $2$ in the full automorphism group, which we show to be a semi-direct product $(\mathrm{U}(1) \times \mathrm{U}(1)) \rtimes \mathrm{D}_{4}$, where $\mathrm{D}_{4}$ is the dihedral group of order $8$.

It is well known that a group of automorphisms $G$ of an unbordered Klein surface $X$ of topological genus $g\geq 2$ in the orientable case and $g\geq 3$ otherwise has at most $84(g-\varepsilon)$ elements, where $\varepsilon =1$ or $2$ respectively. In the middle of the fifties, Oikawa used the cardinality $k$ of a $G$-invariant subset to introduce the bound $\lvert G\rvert \leq 12(g-1)+6k$ in the orientable case. Much later, T. Arakawa has generalized this result, involving $s=2$ or $3$ such subsets and showing in addition that the bound for $s=3$ is sharp for infinitely many configurations. Here we improve the bound of Arakawa for $s=2$, showing in particular that the last is never attained. In both orientable and non-orientable case, we also find bounds for arbitrary $s$ and show their sharpness for infinitely many topological configurations. Using another well known theorem of Oikawa and the canonical Riemann double cover, we get similar results for bordered Klein surfaces.

An ICON surface is an incompressible compact orientable nonseparating surface properly embedded in a knot exterior. We show that for any odd positive number $n$, there exist plenty of knots whose exteriors $E$ contain an ICON surface $F$ with $\lvert \partial F\rvert =n$. We also show that our examples satisfy the $\mathbb{Z}$-conjecture, that is, $\pi_{1}(E/F)\cong \mathbb{Z}$.