Related papers: Uniqueness in Rough Almost Complex Structures and …

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

We give in \mathbb{R}^6 a real analytic almost complex structure J, a real analytic hypersurface M and a vector v in the Levi null set at 0 of M, such that there is no germ of J-holomorphic disc f included in M with f(0)=0 and…

We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J-holomorphic discs with boundaries attached to real tori.

In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, \Delta^m J]=0$. We prove a very…

We consider some second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at…

We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the…

In this article we introduce a class of discontinuous almost automorphic functions which appears naturally in the study of almost automorphic solutions of differential equations with piecewise constant argument. Their fundamental properties…

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular…

Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is smooth up to…

It is proved that if an almost K\"ahler manifold of dimension greater or equal to 8 is of pointwise constant antiholomorphic sectional curvature, then it is a complex space form.

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension $2n\geq 10$…

Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $$\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in \rho_{2} ( TH^{4}(M;\Z) ) \}.$$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset \rho_{2}(…

We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in…

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property…

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant {\theta}-holomorphic sectional curvature for a number {\theta} in (0,{\pi}/2) then M is of…

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is…

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that…

Let G be complex linear-algebraic group, H a subgroup, which is dense in G in the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1) G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every…

Based on recent work of T. Draghici, T.-J. Li and W. Zhang, we further investigate properties of the dimension h_J of the J-anti-invariant cohomology subgroup H_J of a closed almost Hermitian 4-manifold (M, g, J, F) using metric compatible…