In this paper, several versions of gluing theorems for quasiconformal mappings in the plane are obtained. The best possibility of gluing quasiconformal mappings is investigated. As an application, we provide a new short proof of the gluing theorem obtained by Jiang and Qi.

In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric ω such that (T_X,ω) is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on T_X .

The case of dimension two was proved by Andreotti-Frankel [-7] and the case of dimension three by Mabuchi [18] using the result of Kobayashi-Ochiai [12]. Our method of proof uses harmonic maps and the characterization of the complex projective space obtained by Kobayashi-Ochiai [15]. According to the result of Kobayashi-Ochiai the complex projective space of dimension n is characterized by the fact that its first Chern class equals 2c1 (F) for some 2> n+ 1 and some positive holomorphic line bundle F over it. Since by the result of Bishop-Goldberg [-2] the second Betti number of a compact Kiihler manifold M of positive bisectional curvature is 1, for the Main Theorem it suffices to show that ca (M) is 2 times a generator of H2 (M, 7Z.) for some 2> 1+ dim M. This can be done by proving that a generator of the free part of H2 (M, Z) can be represented by a rational curve, because the tangent bundle of M restricted to the rational curve splits into a direct sum of holomorphic line bundles over the rational curve according to the result of Grothendieck [-11]. The existence of the rational curve is obtained in the following way. According to the result of Sacks-Uhlenbeck [22] and its improved formulation by Meeks-Yau [-19], the infimum of the energies of maps from S 2 to M representing the generator of rcz (M) can be achieved by a sum of stable harmonic maps f~ from S 2 to M (1< i< m). The key step in our proof is to show that each f~ is either holomorphic or conjugate holomorphic. The known methods of proving the complex-analyticity of a harmonic map use the formula for the Laplacian of the energy function [23, 25, 26] or a variation of it [24]. Here we use

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

Building on [BB1] we prove a general criterion for convergence of (possibly singular) Bergman measures towards pluripotential-theoretic equilibrium measures on complex manifolds. The criterion may be formulated in terms of the growth properties of the unit-balls of certain norms on holomorphic sections, or equivalently as an asymptotic minimization property for generalized Donaldson$L$-functionals. Our result settles in particular a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points and it gives the convergence of Bergman measures towards the equilibrium measure for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.