论文标题
Biharmonic几乎复杂的结构
Biharmonic almost complex structure
论文作者
论文摘要
我们介绍了\ emph {Biharmonic几乎复杂的结构}的概念，几乎是荒诞歧管，我们研究了它在第四维度中的规律性和存在。首先，我们表明，在紧凑的几乎复杂的四个歧管上，几乎任何赫尔米尼亚结构都始终存在平滑的能量最小化的几乎荒野结构，并且所有能量最少的均构成器构成紧凑的集合。然后，当指定了几乎复杂的结构的同质类别时，我们研究存在问题。我们获得了能量最小化的Biharmonic几乎复杂结构的存在，该结构取决于$ M^4 $的拓扑结构。当$ m $简单地连接且非自旋时，对于由Chern类别唯一决定的每个同型类别时，就会存在一个能量最小化的Biharmonic几乎复杂的结构。当$ m $是简单的连接和旋转时，对于每个Chern班，都有两个同型类别对应于第一班级。给定一个几乎复杂的结构的同型类$ [τ] $，在同型类别上存在一个规范操作$ p $满足$ p^2 = \ text {id {id {id} $，因此$ p（[τ]）$和$ [τ] $具有相同的第一个Chern类。我们证明，在（至少）两个同型类之一中，有一个能量最小化的几乎复杂的结构，即$ [τ] $和$ p（[τ]）$。总的来说，如果不一定简单地连接$ m $，我们证明存在一个能量最小化的Biharmonic几乎复杂的结构（至少）两个同型类别$ [τ] $和$ p（[τ]）$之一。 对Biharmonic几乎复杂结构的研究应该具有许多应用，特别是对于基础几乎复杂的四种歧管的平滑拓扑。我们通过考虑Biharmonic几乎复杂结构的模量空间并提出猜想来简要讨论一种方法。
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.
