The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S^1 X B^3
The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S^1 X B^3
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S^1 X B^3 上简并几乎复形结构和辛形式的伪全纯子簇的结构
DOI：
10.2140/gt.1998.2.221
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
C. Taubes
中科院分区：
文献类型：
--
作者：
C. Taubes
4 维黎曼流形上的自对偶调和 2 形式在不消失的地方是辛的。此外，远离形式的零集，2-形式的度量给出了兼容的几乎复杂的结构，从而给出了伪全纯子类型。当给定自对偶 2-形式的簇上的积分是有限的时，这样的子簇被称为具有有限能量。本文证明了当度量在形式零集附近特别简单时，这种有限能量子类型的正则定理。更准确地说，本文的主要结果断言如下：假设形式的零集是非简并的，并且零集附近的度量具有一定的规范形式。那么，除了零集上可能存在的有限点集之外，零集上的每个点都有一个球邻域，它作为有限分量集与子簇相交，并且每个分量的闭包是一个真正的解析嵌入半圆盘，其边界与形式的零集重合。
A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the integral over the variety of the given self-dual 2-form is finite. This article proves a regularity theorem for such finite energy subvarieties when the metric is particularly simple near the form's zero set. To be more precise, this article's main result asserts the following: Assume that the zero set of the form is non-degenerate and that the metric near the zero set has a certain canonical form. Then, except possibly for a finite set of points on the zero set, each point on the zero set has a ball neighborhood which intersects the subvariety as a finite set of components, and the closure of each component is a real analytically embedded half disk whose boundary coincides with the zero set of the form.