在紧殆Hermite流形上引入了\emph{biharmonic almost complex structure}的概念，并研究了它在四维空间的正则性和存在性.首先证明了紧致几乎复四维流形上的几乎Hermite结构总存在光滑的能量最小化双调和殆复结构，并且所有的能量最小化器构成一个紧致集.然后，我们研究了当一个殆复结构的同伦类被指定时的存在性问题。我们得到了依赖于$M^4$拓扑结构的能量最小化双调和殆复结构的存在性.当$M$是单连通无旋的时，对于由第一个Chern类唯一确定的同伦类，存在能量最小的双调和殆复结构.当$M$是单连通且自旋时，对于每个第一个陈类，恰好有两个同伦类对应于第一个陈类。给定一个具有殆复结构的同伦类$[τ]$，在满足$p^2=\text{id}$的同伦类$p$上存在一个规范运算，使得$p([τ])$和$[τ]$具有相同的第一陈类.我们证明了在两个同伦类$[τ]$和$p([τ])$中的（至少）一个同伦类中存在能量最小化的双调和殆复结构.在一般情况下，如果$M$不一定是单连通的，我们证明了在两个同伦类$[τ]$和$p([τ])$中的（至少）一个中存在能量最小的双调和殆复结构.双调和殆复结构的研究有着广泛的应用，特别是对殆复四维流形的光滑拓扑的研究。简要讨论了一种考虑双调和殆复结构模空间的方法，并提出了一个猜想。
英文摘要：
We introduce the notion of \emph{biharmonic almost complex structure} on a compact almost Hermitian manifold and we study its regularity and existence in dimension four. First we show that there always exist smooth energy-minimizing biharmonic almost complex structures for any almost Hermitian structure on a compact almost complex four manifold, and all energy-minimizers form a compact set. Then we study the existence problem when the homotopy class of an almost complex structure is specified. We obtain existence of energy-minimizing biharmonic almost complex structures which depends on the topology of $M^4$. When $M$ is simply-connected and non-spin, then for each homotopy class which is uniquely determined by its first Chern class, there exists an energy-minimizing biharmonic almost complex structure. When $M$ is simply-connected and spin, for each first Chern class, there are exactly two homotopy classes corresponding to the first Chern class. Given a homotopy class $[τ]$ of an almost complex structure, there exists a canonical operation on the homotopy classes $p$ satisfying $p^2=\text{id}$ such that $p([τ])$ and $[τ]$ have the same first Chern class. We prove that there exists an energy-minimizing biharmonic almost complex structure in (at least) one of the two homotopy classes, $[τ]$ and $p([τ])$. In general if $M$ is not necessarily simply-connected, we prove that there exists an energy-minimizing biharmonic almost complex structure in (at least) one of the two homotopy classes $[τ]$ and $p([τ])$.
The study of biharmonic almost complex structures should have many applications, in particular for the smooth topology of the underlying almost complex four manifold. We briefly discuss an approach by considering the moduli space of biharmonic almost complex structures and propose a conjecture.
继T. J. Li，W.张[比较驯服和相容辛锥和殆复流形的上同调性质，Comm. Anal. Geom.]，我们继续研究殆复流形的上同调与它的殆复结构之间的联系。特别地，我们在[T.-]中应用相同的论证。J. Li，W.张，比较驯服和相容辛锥和殆复流形的上同调性质，Comm. Anal. Geom.]和D. Sullivan在[Cycles for the dynamic study of foliated manifolds and复流形，Invent.数学。]研究紧致半Kähler流形上的半Kähler结构的锥.
Comments on "On Semi-Invariant Submanifolds of Almost Complex Contact Metric Manifolds" (Facta Universitatis, Series: Mathematics and Informatics v. 31, no. 4, 851-862, 2016)
On Cohomological Decomposition of Almost-Complex Manifolds and Deformations
当Kähler流形的小变形也是Kähler流形时，我们证明了上同调性质为$\mathcal{C}^\infty$ -pure-and-full在小变形下不是稳定的条件.这个属性，最近已经被T.- J. Li和W. Zhang在[Comparing tamed and compatible辛锥and cohomological properties of殆复流形，Comm. Anal. Geom.]和T. Draghghici，T.- J. Li和W. Zhang在[辛形式和几乎复四流形的上同调分解，Int.Math.Res.Not.]中，[On几乎复4-流形J-反不变上同调，出现在Q. J. Math.]，弱于Kähler的，刻画了诱导上同调分解的殆复结构.我们还研究了这一性质的稳定性沿曲线的殆复结构从特殊上同调类的调和代表。
