We discuss a complete noncompact hypersurface Σⁿ in a product manifold Sⁿ × R (n ≥ 3). Suppose that the inner product of the unit normal to Σ and ∂/∂t has a positive lower bound δ₀, where t denotes the coordinate of the factor R of Sⁿ × R. We prove that there is no nontrivial L² harmonic 1-form if the total curvature or the length of the traceless Φ of the second fundamental form is bounded from above by a constant depending only on n and δ₀. These results are extensions of results on hypersurfaces in Hadamard manifolds and spheres. These results are also generalization of results on hypersurfaces in Sⁿ × R without minimality.

We fix an integer n ≥ 1, a prime number ℓ with ℓ 2n and an integer s ≥ 0. We deal with a prime number p of the form p = 2nℓ^f + 1. For 0 ≤ t ≤ f, let K_t be the real cyclic field of degree ℓ^t contained in the pth 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, ℓ and s, a prime number r does not divide the ratio h_f/h_{f - (s + 1)} whenever r is a primitive root modulo ℓ².

For a generalized Cantor set E(ω) with respect to a sequence , we consider Riemann surface and metrics on Teichmüller space T(X_E(ω)) of X_E(ω). If E(ω) = (the middle one-third Cantor set), we find that on , 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(ω)) 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(ω)) for some such that inf q_n = 0.

In this paper, we consider linear combinations of harmonic K-quasiregular mappings f_j = h_j + g_i (j = 1, 2) of the class Har(k; φ_j), where k ∈ [0,1), ||ω_fj||_∞ = ||g'_j/h'_j||_∞ ≤ k < 1, k = (1 - K)/(1 + K), and φ_j = h_j + e^iθg_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.