复射影空间上的非齐次爱因斯坦度量
Inhomogeneous Einstein metrics on complex projective spaces
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中文总结 AI 辅助
该研究针对复维数3≤n≤7的复射影空间，构造出首批非齐次爱因斯坦度量，对复偶维n=4、n=6的情形，肯定解答了伯杰关于是否存在非齐次爱因斯坦度量的问题。
中文摘要 AI 辅助
目前已知的复射影空间上的爱因斯坦度量均为齐次的：一是通过霍普夫纤维化得到的富比尼-施图迪度量；二是在奇数复维情形下，通过四元数射影空间上的扭量纤维化的典范变分得到的齐勒度量。1965年，伯杰证明富比尼-施图迪度量是复射影空间上唯一的凯勒-爱因斯坦度量，并提出是否存在其他爱因斯坦度量的问题。我们针对3≤n≤7的复维数n，构造了复射影空间上首批非齐次爱因斯坦度量，对复偶维情形n=4和n=6，肯定地回答了伯杰的问题。
英文摘要
The only Einstein metrics currently known on complex projective spaces are homogeneous: the Fubini-Study metric, arising via the Hopf fibration; and, in odd complex dimensions, Ziller's metric, obtained as a canonical variation along the twistor fibration over the quaternionic projective space. In 1965, Berger proved that the Fubini-Study metric is the unique Kähler-Einstein metric on the complex projective space, and posed the question of whether other Einstein metrics exist. We construct the first inhomogeneous Einstein metrics on complex projective spaces of complex dimension $n$ for $3\leq n \leq 7$, answering Berger's question affirmatively for the complex even dimensional cases $n=4$ and $n=6$.
