When harmonic maps from the Riemann sphere into the complex projective space are energy bounded, it contains a subsequence converging to a bubble tree map f^I: T^I → CPⁿ. We show that their ∂-transforms and $\overline{\partial}$-transforms are also energy bounded. Hence their subsequences converge to harmonic bubble tree maps $f_1^{I_1}:T^{I_1}$ → CPⁿ and $f_{-1}^{I_{-1}}:T^{I_{-1}}$ → CPⁿ respectively. In this paper, we show relations between f^I, $f_1^{I_1}$ and $f_{-1}^{I_{-1}}$.

We show that M^2n-1 is a real hypersurface all of whose geodesics orthogonal to the characteristic vector ξ are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form $\widetilde{M}_n$(c) (= CPⁿ(c) or CHⁿ(c)) if and only if M is a Sasakian manifold with respect to the almost contact metric structure from the ambient space $\widetilde{M}_n$(c). Moreover, this Sasakian manifold M is a Sasakian space form of constant φ-sectional curvature c + 1 for each c (≠0).

In this paper, we will consider a family $\mathscr{Y}$ of complete CAT(0) spaces such that the tangent cone TC_p Y at each point p $\in$ Y of each Y $\in$ $\mathscr{Y}$ is isometric to a (finite or infinite) product of the Euclidean cones Cone(X_α) over elements X_α of some Gromov-Hausdorff precompact family {X_α} of CAT(1) spaces. Each element of such $\mathscr{Y}$ is a space presented by Gromov [4] as an example of a "CAT(0) space with "bounded" singularities". We will show that the Izeki-Nayatani invariants of spaces in such a family are uniformly bounded from above by a constant strictly less than 1.

Let X be a smooth complex projective variety of dimension n and let L₁, ..., L_n-i be ample line bundles on X, where i is an integer with 0 ≤ i ≤ n-1. In the previous paper, we defined the i-th sectional geometric genus g_i(X, L₁, ..., L_n-i) of (X, L₁, ..., L_n-i). In this part II, we will investigate a lower bound for g_i(X, L₁, ..., L_n-i). Moreover we will study the first sectional geometric genus of (X, L₁, ..., L_n-1).

Ruscheweyh extended the work of Becker and Ahlfors on sufficient conditions for a normalized analytic function on the unit disk to be univalent there. In this paper we refine the result to a quasiconformal extension criterion with the help of Becker's method. As an application, a positive answer is given to an open problem proposed by Ruscheweyh.

A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.

We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold (M, η), then under some natural assumptions of integrability M carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then M admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of a para-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the (κ, μ)-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric (κ, μ)-spaces.