论文标题
同构性一个Kähler-ricci solitons下的Heisenberg Group Action和相关指标
Cohomogeneity one Kähler-Ricci solitons under a Heisenberg group action and related metrics
论文作者
论文摘要
我们表明，在复杂尺寸$ m $中，几乎复杂的结构的整合性在几乎荒谬的度量标准的情况下与$ m（m-1）$方程式在涉及我们所谓的剪切操作员的情况下相当。受此启发的启发，我们为Kähler指标提供了ANSATZ $ M> 1 $，其中最多只能使用这些剪切方程的$ M-1 $。 ANSATZ中梯度Kähler-ricci孤子的方程是框架依赖性PDE，它专门针对额外假设下的ODE。解决后一个系统的指标包括一类限制类别的指标，我们发现在$（2M-1）$ - 维度的Heisenberg Group的作用下，它们完全扩展了梯度Kähler-ricci solitons，以及一些不完整的稳定孤子。我们检查了以前的Ricci孤子子的曲率特性和渐近性。在我们提出的另一个特殊情况下，我们以$ M = 2 $为例，这是一类更通用类型的完整指标，我们称之为渐变的Kähler-ricci偏斜词，这是Euclidean Plane Group Action下的同时性。本文继续研究[MR，AM2]。
We show that integrability of an almost complex structure in complex dimension $m$ is equivalent, in the presence of an almost hermitian metric, to $m(m-1)$ equations involving what we call shear operators. Inspired by this, we give an ansatz for Kähler metrics in dimension $m>1$, for which at most $m-1$ of these shear equations are non-trivial. The equations for gradient Kähler-Ricci solitons in this ansatz are frame dependent PDEs, which specialize to ODEs under extra assumptions. Metrics solving the latter system include a restricted class of cohomogeneity one metrics, and we find among them complete expanding gradient Kähler-Ricci solitons under the action of the $(2m-1)$-dimensional Heisenberg group, and some incomplete steady solitons. We examine curvature properties and asymptotics for the former Ricci solitons. In another special case of the ansatz we present, for $m=2$, a class of complete metrics of a more general type which we call gradient Kähler-Ricci skew-solitons, which are cohomogeneity one under the Euclidean plane group action. This paper continues research started in [MR, AM2].
