International Journal of Mathematics

In this article, unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables sharing small identical sets for moving targets are given.

This paper presents a unified approach to a class of identities which was first proved by Pohožaev. These identities have generalizations to Schrödinger's equation, the wave equation, the p-Laplace equation and the biharmonic equation. We also show that the systems can be proved under our unified approach.

In this paper, we study the approximation of negative plurisubharmonic functions with given boundary values. We want to approximate a plurisubharmonic function by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

We prove a conjecture of Bonavero et al. on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹ⁿ, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

Let (M, g) be a Riemannian manifold and T₁ M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T₁M the Sasaki metric . This metric, and other well-known Riemannian metrics on T₁ M, are particular examples of Riemannian natural metrics. In this paper we equip T₁ M with a Riemannian natural metric and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kähler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric . Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T₁S²ⁿ⁺¹ depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S²ⁿ⁺¹ to T₁S²ⁿ⁺¹.

The purpose of this paper is to describe a method for computing homotopy groups of the space of α-stable representations of a quiver with fixed dimension vector and stability parameter α. The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vector, and the choice of parameter. As a corollary we also compute low dimensional homotopy groups of the moduli space of α-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.