基于检测的新齐性爱因斯坦度量
New Homogeneous Einstein Metrics from Detection
AI总结：
本研究采用不变爱因斯坦方程的检测方法，在多族紧齐性空间上构造区分非对称齐性爱因斯坦度量，获多个尖锐阈值，在四因子空间$M_9$上实现两种对称类型的非对称度量并证明其性质。
AI中文摘要：
我们采用一种针对不变爱因斯坦方程的检测方法，在若干族紧齐性空间上构造并区分非对称齐性爱因斯坦度量。该方法保留一个具有几何意义的对称破缺参数，消除剩余的爱因斯坦方程，随后从所得检测器重构正度量。我们在一个双因子辛族中证明了一个尖锐阈值，在每个容许空间$\text{SU}(n)^3/\text{ΔSO}(n)$上构造出两个非对称爱因斯坦度量，为$\text{SU}(2n)^3/\text{ΔSp}(n)$得到一个尖锐阈值，并在三个特殊的三因子族上构造出非对称度量。四因子情形呈现出进一步的现象：在$M_9=\text{SU}(18)^4/\text{ΔSp}(9)$上，$3+1$和$2+2$两种对称类型均得以实现；在同缩比及内部块置换下，对应的固定族恰好包含三个非对称爱因斯坦度量，其中两个具有三个相等的水平尺度和一个不同的尺度（$3+1$型），一个具有两对相等的水平尺度（$2+2$型）；它们两两非等距、黎曼不可约且非自然约化，$2+2$型度量在整个正$2+2$固定族中，除块置换外是全局唯一的。
英文摘要：
We use a detection method for invariant Einstein equations to construct and distinguish asymmetric homogeneous Einstein metrics on several families of compact homogeneous spaces. The method retains a geometrically meaningful symmetry-breaking parameter, eliminates the remaining Einstein equations, and then reconstructs positive metrics from the resulting detector. We prove a sharp threshold in a two-factor symplectic family, construct two asymmetric Einstein metrics on each admissible space $\SU(n)^3/Δ\SO(n)$, obtain a sharp threshold for $\SU(2n)^3/Δ\Sp(n)$, and construct asymmetric metrics on three exceptional three-factor families. The four-factor case exhibits a further phenomenon. On \[ M_9=\SU(18)^4/Δ\Sp(9) \] the two symmetry types $3+1$ and $2+2$ are both realized: up to homothety and internal block permutations, the corresponding fixed families contain exactly three asymmetric Einstein metrics, two with three equal horizontal scales and one distinct scale (type $3+1$), and one with two equal pairs of horizontal scales (type $2+2$). They are pairwise non-isometric, Riemannian irreducible, and not naturally reductive. The $2+2$ metric is globally unique in the full positive $2+2$ fixed family up to block interchange.
