Pluri-polarity in almost complex structures

具有不连续边界行为的多重次调和函数

Plurisubharmonic functions with discontinuous boundary behavior

本文研究带不连续边界数据的复 Monge–Ampère Dirichlet 问题，在 b-多重极不连续集与 B-正则域条件下证明解的存在唯一性及正则性，并说明高维中这些条件并非必要。

Monge-Ampère operator on four dimensional almost complex manifolds

We define the Monge-Ampère operator for continuous J-plurisubharmonic functions on four dimensional almost complex manifolds.

Lelong functional on almost complex manifolds

We establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds.

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.

Plurisubharmonic functions and positive currents of type (1,1) over almost complex manifolds

If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ is positive, (the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ need not be closed here). The conjecture is trivial if $u$ is of class ${\cal C}^2$. The result is elementary in the complex integrable case because the operator $i\pa…

Some aspects of analysis on almost complex manifolds with boundary

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds and the elliptic regularity of some diffeomorphisms in almost complex manifolds with boundary.

Gromov hyperbolicity of strongly pseudoconvex almost complex manifolds

Let $D=\{ρ< 0\}$ be a smooth relatively compact domain in an almost complex manifold $(M,J)$, where $ρ$ is a smooth defining function of $D$, strictly $J$-plurisubharmonic in a neighborhood of the closure $\overline{D}$ of $D$. We prove that $D$ has a connected boundary and is Gromov hyperbolic.

A note on the geometry of pseudoconvex domains of finite type in almost complex manifolds

Let $D=\{ρ<0\}$ be a smooth domain of finite type in an almost complex manifold (M,J) of real dimension four. We assume that the defining function $ρ$ is J-plurisubharmonic on a neighborhood of $\overline{D}$. We study the asymptotic behavior of pseudoholomorphic discs contained in the domain D.

关于殆复流形上Poisson泛函的包络

On the envelope of Poisson functional on almost complex manifolds

该研究证明了殆复流形上Poisson泛函的包络的多重次调和性，将复流形及复二维殆复流形的相关结果推广到了一般维数的殆复流形。

Kodaira dimensions of almost complex manifolds I

This is the first of a series of papers, in which we study the plurigenera, the Kodaira dimension and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost complex manifolds. We show that the plurigenera and the Kodaira dimension as well as the irregularity are birational invariants in almost complex category, at least in dimension $4$, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are n…

Large Harnack inequalities, Kobayashi distances and holomorphic motions

We prove some generalizations and analogies of Harnack inequalities for pluriharmonic, holomorphic and "almost holomorphic" functions. The results are applied to the proving of smoothness properties of holomorphic motions over almost complex manifolds.

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.

Pseudoconvex regions of finite D'Angelo type in four dimensional almost complex manifolds

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M,J) of real dimension four. We construct a local peak J-plurisubharmonic function at every boundary point p of finite D'Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D'Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.

Potential Theory on Almost Complex Manifolds

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. I…

Monge-Ampère exhaustions of almost homogeneous manifolds

We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous Monge-Ampère equations. This extends to a new family of mixed type examples various classical results on parabolic spaces and complexifications of symmetric spaces. Rigidity results on complex spaces modeled on such new examples are given.

Kodaira dimensions of almost complex manifolds II

This is the second of a series of papers where we study the plurigenera, the Kodaira dimension and the Iitaka dimension on compact almost complex manifolds. By using the pseudoholomorphic pluricanonical map, we define the second version of Kodaira dimension as well as Iitaka dimension on compact almost complex manifolds. We show the almost complex structures with the top Kodaira dimension are integrable. For compact almost complex 4-manifolds with Kodaira dimension one, we obtain elliptic fibration like structural description. Some vanishing theorems in complex geometry are generalized to the …

Estimates of the Kobayashi metric on almost complex manifolds

We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition for the complete hyperbolicity of a domain in $(M,J)$. Finally we obtain the regularity up to the bounday of $J$-holomorphic discs attached to a totally real submanifold in $M$.

From Smooth to Almost Complex

This article mainly aims to overview the recent efforts on developing algebraic geometry for an arbitrary compact almost complex manifold. We review the results obtained by the guiding philosophy that a statement for smooth maps between smooth manifolds in terms of René Thom's transversality should also have its counterpart in pseudoholomorphic setting without requiring genericity of the almost complex structures. These include intersection of compact almost complex submanifolds, structure of pseudoholomorphic maps, zero locus of certain harmonic forms, and eigenvalues of Laplacian. In additio…

Chern connections and Chern curvature of the tangent bundle of almost complex manifolds

The $\bar{\partial}_{_{J}}$ operator over an almost complex manifold induces canonical connections of type $(0,1)$ over the bundles of $(p,0)$-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$ we can extend the corresponding connection to all Schur powers of the bundle of $(1,0)$-forms. Moreover using the canonical $\C$-linear isomorphism betwen the bundle of $(1,0)$-forms and the complex cotangent bundle $T^*_{X,J}$ we deduce canonical connections of type $(0,1)$ over the Sc…

Ricci熵、RCD结构与Kahler空间

Ricci entropy, RCD structures and Kahler spaces

本文研究具有klt奇点的紧致正规Kahler空间上的奇异Kahler度量，证明多重位势理论中Ricci电流下界与RCD理论中合成Ricci下界等价，从而每个Kahler类具有RCD结构，并推出Calabi-Yau空间基本群几乎阿贝尔及紧致性结果。