We calculate cuspidal values of the modular transform of the principal Eisenstein series of weight 1. We obtain this by looking at derivatives of the double sine function. Our result is considered to be an explicit calculation of values of a Stirling modular function.

An irreducible non-torus plane sextic with simple singularities is said to be special if its fundamental group factors to a dihedral group. There exist (exactly) ten configurations of simple singularities that are realizable by such curves. Among them, six are realizable by non-special sextics as well. We conjecture that for each of these six configurations there always exists a non-special curve whose fundamental group is abelian, and we prove this conjecture for three configurations (another one has already been treated in one of our previous papers). As a corollary, we obtain new explicit examples of Alexander-equivalent Zariski pairs of irreducible sextics.

In the present paper we prove that the Hermitian curvature tensor $\tilde{R}$ associated to a nearly Kähler metric g always satisfies the second Bianchi identity $\mathfrak{S}(\tilde{\nabla}_X\tilde{R})$ (Y, Z, ·, ·)=0 and that it satisfies the first Bianchi identity $\mathfrak{S}\tilde{R}$(X, Y, Z, ·)=0 if and only if g is a Kähler metric. Furthermore we characterize condition for $\tilde{R}$ to be parallel with respect to the canonical Hermitian connection $\tilde{\nabla}$ in terms of the Riemann curvature tensor and in the last part of the paper we study the curvature of some generalizations of the nearly Kähler structure.

Let Mⁿ (n ≥ 3) be an n-dimensional complete noncompact oriented submanifold in an (n+p)-dimensional Euclidean space R^n+p with finite total mean curvature, i.e, ∫_M|H|ⁿ < ∞, where H is the mean curvature vector of M. Then we prove that each end of M must be non-parabolic. Denote by φ the traceless second fundamental form of M. We also prove that if ∫_M|φ|ⁿ < C (n), where C (n) is an an explicit positive constant, then there are no nontrivial L² harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, such a submanifold has only one end. This implies that such a minimal submanifold is plane.

The aim of this work is to deal with index of closed orientable non-totally geodesic minimal hypersurface Σⁿ of the Euclidean unit sphere Sⁿ⁺¹ whose second fundamental form has squared norm bounded from below by n. In this case we shall show that the index of stability, denoted by Ind_Σ, is greater than or equal to n + 3, with equality occurring at only Clifford tori $\mathbf{S}^k(\frac{k}{n})\times\mathbf{S}^{n-k}(\sqrt{\frac{(n-k)}{n}})$. Moreover, we shall prove also that, besides Clifford tori, we have the following gap: Ind_Σ ≥ 2n + 5.

In this paper we will consider a holomorphic family of closed Riemann surfaces of genus two which is constructed by Riera. The goal of this paper is to estimate the number of holomorphic sections of this family.

Inspired by the recursive lemma due to Bressoud (1983), we present an iteration process for constructing transformations from unilateral multiple basic hypergeometric series to bilateral univariate one. Applications are illustrated to multiple series transformation formulae and multiple Rogers-Ramanujan identities.

In the setting of α-parabolic Bergman spaces, we consider weighted versions of averaging functions and Berezin transformations. Related norm equivalence relations are shown. They are very useful to study our Bergman spaces. As an application, we characterize the Schatten classes of compact Toeplitz operators.

In [1], B. Y. Chen provided a new method to construct Lagrangian surfaces in C² by using Legendre curves in S³(1) $\subset$ C². In this paper, we investigate the similar methods to construct some Lagrangian submanifolds in complex Euclidean spaces Cⁿ (n ≥ 3).

We compute the homotopy groups of the spaces of self-maps of SU(3) and Sp(2).

We study liftings of holomorphic maps into some Teichmüller spaces. We also study the relationship between universal holomorphic motions and holomorphic lifts into Teichmüller spaces of closed sets in the Riemann sphere.

Habiro arranged Kirby moves into a pair so that it preserves linking matrices. The author showed that two framed links of diagonal linking matrices yield homeomorphic 3-manifolds of linking form (±1/p) for an odd prime p if and only if they are related by a sequence of Habiro moves. We generalize this result to 3-manifolds of linking forms (±1/c) for any odd prime power c.