By using the potential theory and Pólya peaks, we will investigate the indirect singular points of meromorphic functions. This is a continuous work of M. Tsuji.

We prove that the sequence of MacLane key polynomials constructed in [7] and [3] for a valuation extension $(K, ν) \subset (K(x), μ)$ is finite, provided that both ν and μ are divisorial and μ is centered over an analytically irreducible local domain $(R, \mathfrak{m}) \subset K[x]$. As a corollary, we prove Izumi's theorem on comparison of divisorial valuations. We give explicit bounds for the Izumi constant in terms of the key polynomials of the valuations. We show that this bound can be attained in some cases.

Most of the classical calibrations possess a property which does not seem to be recognized. Making this property explicit, we define what we call the strong calibrations and prove that a strong 2-calibration field on R²ⁿ is a constant field.

In this paper, by modifying Cheng-Yau's technique to complete spacelike submanifolds in Q_p^n+p(c), we prove a rigidity theorem for complete spacelike submanifolds in the de Sitter space with parallel normalized mean curvature vector. As a corollary, we have the Corollary 1.1 of [7].

Let X be a non-singular projective (n + 1)-fold defined over an algebraically closed field k of characteristic p ≥ 0, and B be a non-singular complete curve defined over k. A surjective morphism f : X → B is said to be an n-abelian fiber space if almost all fibers are n-dimensional abelian varieties. We examine the canonical bundle formula for n-abelian fiber spaces.

Let G be a complex affine algebraic group, and let E_G be a holomorphic principal G-bundle on the complement M$\backslash$S, where S is a closed complex analytic subset, of complex codimension at least two, of a connected complex manifold M. We give a criterion for E_G to extend to M as a holomorphic principal G-bundle. Two applications of this criterion are given.

We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of the complex Reeb vector field on a complex contact Lie algebra is diagonalizable, then it is trivial.

Let M be an n-dimensional closed submanifold with parallel mean curvature in S^n+p, $\tilde{h}$ the trace free part of the second fundamental form, and $\tilde{\sigma}$(u) = ||$\tilde{h}$(u, u)||² for any unit vector u $\in$ TM. We prove that there exists a positive constant C(n, p, H) (≥ 1/3) such that if $\tilde{\sigma}$(u) ≤ C(n, p, H), then either $\tilde{\sigma}$(u) ≡ 0 and M is a totally umbilical sphere, or $\tilde{\sigma}$(u) ≡ C(n, p, H). A geometrical classification of closed submanifolds with parallel mean curvature satisfying $\tilde{\sigma}$(u) ≡ C(n, p, H) is also given. Our main result is an extension of the Gauchman theorem [4].

We discuss an equivalence problem of pseudo-Hermitian structures on 3-dimensional manifolds, and develop a method of constructing equivalence maps by using systems of linear partial differential equations. It is proved that a pseudo-Hermitian structure is transformed to a standard model of pseudo-Hermitian structure constructed on the Heisenberg group if and only if it has the vanishing pseudo-Hermitian torsion and the pseudo-Hermitian curvature. A system of linear partial differential equations whose coefficients are associated with a given pseudo-Hermitian structure is introduced, and plays a central role in this paper. The system is integrable if and only if the pseudo-Hermitian structure has vanishing torsion and curvature. The equivalence map is constructed by using a normal basis of the solution space of the system.

Let $B$ be a Blaschke product with zeros {$a_n$}. If $B′ \in A^p_α$ for certain $p$ and $α$, it is shown that $Σ_n (1 - |a_n|)^β < ∞$ for appropriate values of β. Also, if {$a_n$} is uniformly discrete and if $B′ \in H^p$ or $B′ \in A^{1+p}$ for any $p \in (0,1)$, it is shown that $Σ_n (1 - |a_n|)^1-p < ∞$.

An infinite family of pseudo-Anosov diffeomorphisms over Heegaard surface of the 3-sphere is constructed, when genus is at least 3.

In this paper, we prove that there is no expanding gradient Ricci soliton with (positively) pinched Ricci curvature in dimension three. The same result is true for higher dimensions with the extra decay condition about the full curvature.

In this paper, we characterize biharmonic Legendre curves in 3-dimensional (κ, μ, ν)-contact metric manifolds. Moreover, we give examples of Legendre geodesics in these spaces. We also give a geometric interpretation of 3-dimensional generalized (κ, μ)-contact metric manifolds in terms of its Legendre curves. Furthermore, we study biharmonic anti-invariant surfaces of 3-dimensional generalized (κ, μ)-contact metric manifolds with constant norm of the mean curvature vector field. Finally, we give examples of anti-invariant surfaces with constant norm of the mean curvature vector field immersed in these spaces.