International Journal of Mathematics

Let denote the set of all closed curves of class C^r on the sphere S² whose geodesic curvatures are constrained to lie in (κ₁, κ₂), furnished with the C^r topology (for some r ≥ 2 and possibly infinite κ₁ < κ₂). In 1970, J. Li ttle proved that the space of closed curves having positive geodesic curvature has three connected components. Let ρ_i = arccot κ_i(i = 1, 2). We show that has n connected components where

We present new open manifolds that are not homeomorphic to leaves of any C⁰ codimension one foliation of a compact manifold. Among them are simply connected manifolds of dimension d ≥ 5 that are non-periodic in homotopy, namely in their two-dimensional homotopy groups.

Let μ be any weight function defined on the unit disk 𝔻 and let ϕ be an analytic self-map of 𝔻. In this paper, we show that the essential norm of composition operator C_ϕ mapping from the α-Bloch space, with α > 0, to μ-Bloch space is comparable to

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil₃. Also, we construct helicoidal surfaces in Nil₃ with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over which has prescribed core, it fixes the exceptional set E of D, and is dominating on . Each section in this spray is of class and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class and holomorphic on D can be approximated in the topology by π-sections that are holomorphic in open neighborhoods of . Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

In this paper, we establish the notion of (r, s)-stability concerning spacelike hypersurfaces with higher-order mean curvatures linearly related in conformally stationary spacetimes of constant sectional curvature. In this setting, we characterize (r, s)-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of an operator naturally attached to the higher-order mean curvatures. Moreover, we obtain sufficient conditions which assure the (r, s)-stability of complete spacelike hypersurfaces immersed in the de Sitter space.