Ivey Thomas A, Karigiannis Spiro
Department of Mathematics, College of Charleston, Charleston, SC USA.
Department of Pure Mathematics, University of Waterloo, Waterloo, Canada.
Geom Dedic. 2024;218(5):102. doi: 10.1007/s10711-024-00954-8. Epub 2024 Sep 30.
We consider the existence of cohomogeneity one solitons for the isometric flow of -structures on the following classes of torsion-free -manifolds: the Euclidean with its standard -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon -manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.
我们考虑在以下几类无挠(G)-流形上(G)-结构的等距流的上同调齐性一孤子的存在性：具有标准(G)-结构的欧几里得空间(\mathbb{R}^n)、卡拉比 - 丘三维流形上的度量柱面、近凯勒六维流形上的度量锥以及布莱恩特 - 萨拉蒙(G)-流形。在所有情况下，我们都建立了等距孤子方程的全局解的存在性，并确定了挠率的渐近行为。特别地，证明了在(\mathbb{R}^n)上存在收缩等距孤子，这为等距流可能存在I型奇点提供了支持。在每种情况下，孤子方程的研究都归结为一个具有正则奇点的特定非线性常微分方程，我们对此进行了仔细分析。最后，为了简化每种情况下相关方程的推导，我们首先为向量丛全空间上的一类一般的上同调齐性一度量建立了几个有用的黎曼几何公式，这类度量作为特殊全纯性度量的明确例子经常出现，因此应该有更广泛的应用。
