We discuss a complete noncompact hypersurface $\Sigma^n$ in a product manifold $\mathbf{S}^{n} \times \mathbf{R} (n \geq 3)$. Suppose that the inner product of the unit normal to $\Sigma$ and $\frac{\partial}{\partial t}$ has a positive lower bound $\delta_0$, where $t$ denotes the coordinate of the factor $\mathbf{R}$ of $\mathbf{S}^{n} \times \mathbf{R}$. We prove that there is no nontrivial $L^2$ harmonic 1-form if the total curvature or the length of the traceless $\Phi$ of the second fundamental form is bounded from above by a constant depending only on $n$ and $\delta_0$. These results are extensions of results on hypersurfaces in Hadamard manifolds and spheres. These results are also generalization of results on hypersurfaces in $\mathbf{S}^{n} \times \mathbf{R}$ without minimality.

We fix an integer $n \geq 1$, a prime number $\ell$ with $\ell \not\mid 2n$ and an integer $s \geq 0$. We deal with a prime number $p$ of the form $p=2n\ell^f+1$. For $0 \leq t \leq f$, let $K_t$ be the real cyclic field of degree $\ell^t$ contained in the $p$th cyclotomic field, and let $h_t$ be the class number of $K_t$. We show that when $p$ (or $f$) is large enough with respect to $n$, $\ell$ and $s$, a prime number $r$ does not divide the ratio $h_f/h_{f-(s+1)}$ whenever $r$ is a primitive root modulo $\ell^2$.

For a generalized Cantor set $E(\omega)$ with respect to a sequence $\omega=\{q_n\}_{n=1}^{\infty} \subset (0,1)$, we consider Riemann surface $X_{E(\omega)}:=\hat{\mathbf{C}} \setminus E(\omega)$ and metrics on Teichmüller space $T(X_{E(\omega)})$ of $X_{E(\omega)}$. If $E(\omega) = \mathcal{C}$ (the middle one-third Cantor set), we find that on $T(X_{\mathcal{C}})$, Teichmüller metric $d_T$ defines the same topology as that of the length spectrum metric $d_L$. Also, we can easily check that $d_T$ does not define the same topology as that of $d_L$ on $T(X_{E(\omega)})$ if $\sup q_n =1$. On the other hand, it is not easy to judge whether the metrics define the same topology or not if $\inf q_n =0$. In this paper, we show that the two metrics define different topologies on $T(X_{E(\omega)})$ for some $\omega=\{q_n\}_{n=1}^{\infty}$ such that $\inf q_n =0$.

In this paper, we consider linear combinations of harmonic K-quasiregular mappings $f_j=h_j+\overline{g}_j$ $(j=1, 2)$ of the class ${\mathrm{Har}} (k; \phi_j)$, where $k\in [0,1)$, $\|\omega_{f_j}\|_{\infty}=\|g'_j/h'_j\|_{\infty}\leq k<1$, $k=(1-K)/(1+K)$, and $\phi_j=h_j+e^{i\theta}g_j$ is a univalent analytic function. We provide sufficient conditions for the linear combinations of mappings in these classes to be univalent and for the image domains to be linearly connected. Furthermore, we consider under which conditions the linear combination $f$ is bi-Lipschitz.

We determine the connectivity of the set of germs of generic cuspidal edges (resp. cuspidal edges with positive or negative extrinsic Gaussian curvatures) in a 3-dimensional space form.

Motivated by a recent work of Zhou-Zheng [20], we study compact balanced threefolds with constant holomorphic sectional curvature with respect to the Chern connection. We prove that a compact normal balanced threefold with constant holomorphic sectional curvature is Kähler if the constant is negative and Chern flat when the constant is zero.

In the present paper, we first prove that, for an arbitrary reducible Hodge-Tate $p$-adic representation of dimension two of the absolute Galois group of a $p$-adic local field and an arbitrary continuous automorphism of the absolute Galois group, the $p$-adic Galois representation obtained by pulling back the given $p$-adic Galois representation by the given continuous automorphism is Hodge-Tate. Next, we also prove the existence of an irreducible Hodge-Tate $p$-adic representation of dimension two of the absolute Galois group of a $p$-adic local field and a continuous automorphism of the absolute Galois group such that the $p$-adic Galois representation obtained by pulling back the given $p$-adic Galois representation by the given continuous automorphism is not Hodge-Tate.

We show finiteness of families of meromorphic functions sharing two points CM and one point IM if the families satisfy some conditions.