Some aspects of analysis on almost complex manifolds with boundary

Partial Integrability of Almost Complex Structures on Thurston Manifolds

We prove that any left-invariant symplectic almost complex structure on a Thurston manifold which is compatible with its canonical left-invariant Riemannian metric has holomorphic type 1.

Poincaré-Lelong formula, $J$-analytic subsets and Lelong numbers of currents on almost complex manifolds

In this paper, we first establish a Poincaré-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold $(M,J)$ on a $J$-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.

Filling hypersurfaces by discs in almost complex manifolds of dimension 2

We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold of dimension 2. We prove that if the hypersurface contains no discs, then they fill a one sided neighborhood of the hypersurface.

近埃尔米特流形的受控穷竭函数与 Schwarz 型引理

Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds

针对近埃尔米特流形，论文借助 Hessian 比较定理构造特殊穷竭函数，并据此建立近全纯映射的 Schwarz 型引理及 Liouville 型定理。

On the geometry of model almost complex manifolds with boundary

We study some special almost complex structures on strictly pseudoconvex domains. They appear naturally as limits under a nonisotroping scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.

Embedding almost-complex manifolds in almost-complex Euclidean spaces

We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.

Regularized maximum of strictly plurisubharmonic functions on an almost complex manifold

We prove that the maximum of two smooth strictly plurisubharmonic functions on an almost complex manifold can be uniformly approximated by smooth strictly plurisubharmonic functions.

Wong-Rosay Theorem in almost complex manifolds

We study the compactness of sequences of diffeomorphisms in almost complex manifolds in terms of the direct images of the standard integrable structure.

The Monge-Ampère equation on almost complex manifolds

We study the Dirichlet problem for the Monge-Ampère equation on almost complex manifolds. We obtain the existence of the unique smooth solution of this problem in strictly pseudoconvex domains.

Fefferman's mapping theorem on almost complex manifolds

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds. The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds.

On the dimension of Dolbeault harmonic (1,1)-forms on almost Hermitian 4-manifolds

We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.

Gluing techniques and envelopes of disc functionals on almost complex manifolds

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic functions and to the characterization of compact psh-hulls by pseudoholomorphic discs.

Kobayashi hyperbolicity of almost complex manifolds

Our goal here is to give a simple proof of the non integrable version of Brody's characterisation theorem.

Total reality of conormal bundles of hypersurfaces in almost complex manifolds

A generalization to the almost complex setting of a well-known result by S. Webster is given. Namely, we prove that if $Γ$ is a strongly pseudoconvex hypersurface in an almost complex manifold $(M, J)$, then the conormal bundle of $Γ$ is a totally real submanifold of $(T^*M, \J)$, where $\J$ is the lifted almost complex structure on $T^*M$ defined by Ishihara and Yano.

Liouville Theorems for holomorphic maps on pseudo-Hermitian manifolds

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any $(J,J^N)$ holomorphic map from a compact pseudo-Hermitian manifold $M$ with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold $N$ with negative (resp. nonpositive) holomorphic sectional c…

Bott-Chern Laplacian on almost Hermitian manifolds

Let $(M,J,g,ω)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on $(M,J,g,ω)$, proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott-Chern Laplacian coincide. We show that such a property does not hold when $(M,J,g,ω)$ is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira-Thurston manifold. Furthermore, if $(M,J,g,ω)$ is a connected compact almost Hermitian $4$-manifold, denoting by $h^{1,1}_{BC}$ the dimension of the space of Bott…

On embeddings of almost complex manifolds in almost complex Euclidean spaces

We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class of $(M, J)$, for the existence of an embedding or an immersion in an almost complex Euclidean $4m$-space. We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in $R^6$.

Riemann maps in almost complex manifolds

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.