In this paper, we study gradient Ricci expanding solitons (X,g) satisfying

Rc = cg + D²f,

where Rc is the Ricci curvature, c < 0 is a constant, and D²f is the Hessian of the potential function f on X. We show that for a gradient expanding soliton (X,g) with non-negative Ricci curvature, the scalar curvature R has at most one maximum point on X, which is the only minimum point of the potential function f. Furthermore, R > 0 on X unless (X,g) is Ricci flat. We also show that there is exponentially decay for scalar curvature on a complete non-compact expanding soliton with its Ricci curvature being ε-pinched.

In 2008, we obtained an intrinsic characterization of the unit polydisc Δⁿ in Cⁿ from the viewpoint of the holomorphic automorphism group. In connection with this, A. V. Isaev investigated the structure of a complex manifold M with the property that every isotropy subgroup of the holomorphic automorhism group of M is compact, and obtained the same characterization of Δⁿ as ours among the class of all such manifolds. In this paper, we establish some extensions of these results. In particular, Isaev's characterization of the unit polydisc Δⁿ is extended to that of any bounded symmetric domain in Cⁿ.

A submanifold of Rⁿ whose tangent space makes constant angle with a fixed direction d is called a helix. Helix submanifolds are related with the eikonal PDE equation. We give a method to find every solution to the eikonal PDE on a Riemannian manifold locally. As a consequence we give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension. Also we characterize the ruled helix submanifolds and in particular we describe those which are minimal.

Let Mⁿ (n ≤ 7) be an n-dimensional complete immersed super stable minimal submanifold in an (n + p)-dimensional Euclidean space R^n+p with flat normal bundle. We prove that if the second fundamental form A of M satisfies ∫_M |A|³ < ∞, then M is an affine n-dimensional plane.

In the paper, we have two purposes. Firstly, we prove two theorems and two corollaries of normal families which improve and generalize some results of Pang and Zalcman [9], Zhang, Sun and Pang [13], Chang and Fang [2]. Secondly, we use the theory of normal families and differential equations to obtain a uniqueness theorem of entire function which is an improvement of Chang and Fang [1].

Baikoussis, Lyu and Suh [1] showed that a Hopf hypersurface M in a non-flat complex space form M_n(c) with constant mean curvature and with η-recurrent Ricci tensor is locally congruent to one of real hypersurfaces of type A and B. They also conjectured that the same result can be obtained even without the constancy assumption on the mean curvature (cf. [1, Remark 5.1.]). The purpose of this paper is to answer this question in the affirmative.

In this paper, we investigate the exponent of convergence of the zero-sequence of solutions of the second order linear differential equation

$$f''+\left(\sum_{j=1}^{l}Q_j(z)e^{P_j(z)}\right)f=0,$$

where P_j (z) (j= 1,2, ..., l ≥ 3) are polynomials of degree n ≥ 1, Q_j (z) are entire functions of order less than n, and obtain some results which improve and generalize the previous results in [8, 9, 13].

First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S²ⁿ⁺¹ admits a non-Killing conformal vector field, and (ii) a conformal vector field on the unit tangent bundle of a space-form of dimension > 2 is necessarily Killing.

The main purpose of the present paper is to study almost Norden structures on 4-dimensional Walker manifolds. We discuss the integrability and Kähler (holomorphic) conditions for these structures. The curvature properties for Norden-Walker metrics is also investigated. Then examples Norden-Walker metrics are constructed from an arbitrary harmonic function of two variables.

We are interested in a global version of Lê-Ramanujam μ-constant theorem for polynomials. We consider an analytic family {f_s}, s $\in$ [0, 1], of complex polynomials in two variables, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber of f_s is constant, then we show that the global monodromy fibrations of f_s are all isomorphic, and that the degree of f_s is constant (up to an algebraic automorphism of C²).

By making use of an integral operator due to Noor, a new subclass of analytic functions, denoted by k–$\mathscr{UCV}_n$ (n $\in$ N₀ := {0,1,2,...};0 ≤ k < ∞); is introduced. For this class the Fekete-Szegö problem is completely settled. The results obtained here also give the Fekete-Szegö inequalities for the classes of k-uniformly convex functions and k-parabolic starlike functions.