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.

We show the intersection of a compact almost complex subvariety of dimension 4 and a compact almost complex submanifold of codimension 2 is a J-holomorphic curve. This is a generalization of positivity of intersections for J-holomorphic curves in almost complex 4-manifolds to higher dimensions. As an application, we discuss pseudoholomorphic sections of a complex line bundle. We introduce a method to produce J-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset of a pseudoholomorphic map between almost complex 4-manifolds is J-holomorphic. Building on this, we show degree one pseudoholomorphic maps between almost complex 4-manifolds are actually birational morphisms in pseudoholomorphic category.

In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.

We consider$Thurston maps$, i.e., branched covering maps$f:S$^{2}→$S$^{2}that are$post-critically finite$. In addition, we assume that$f$is$expanding$in a suitable sense. It is shown that each sufficiently high iterate$F$=$f$^{$n$}of$f$is$semi-conjugate$to$z$^{$d$}:$S$^{1}→$S$^{1}, where$d$= deg F. More precisely, for such an$F$we construct a$Peano curve γ$:$S$^{1}→$S$^{2}(onto), such that$F$∘$γ$($z$) =$γ$($z$^{$d$}) (for all$z$∈$S$^{1}).

In this paper, we prove some results on bi-Hölder extensions not only for biholomorphisms but also for more general Kobayashi metric quasi-isometries between the domains. Furthermore, we establish the Gehring–Hayman type theorems on certain complex domains which play an important role through the paper. Then by applying the above results, we show the bi-Hölder equivalence between the Euclidean boundary and the Gromov boundary of bounded convex domains which are Gromov hyperbolic with respect to their Kobayashi metrics.