AI 中文总结
本文在 Berger 空间上构造首个非齐次近 G2 度量，其锥具有 Spin(7) 全和乐，并通过计算机辅助证明存在性，同时用 C7 气泡实现正弦锥商群的局部去奇化。
AI 中文摘要
近 $\nathrm{G}_2$ 流形是正爱因斯坦七维流形，其关联锥度量的特殊和乐群等于或包含于 $\nathrm{Spin}(7)$。我们研究在 $\nathrm{SO}(4)$ 的余齐一作用下不变的非平行 $\nathrm{G}_2$ 结构，并在 Berger 空间 $\nathrm{SO}(5)/\nathrm{SO}(3)$ 上构造了第一个非齐次非平行 $\nathrm{G}_2$ 结构。该度量上的锥具有全和乐群 $\nathrm{Spin}(7)$。解的存在性通过严格的计算机辅助打靶论证确立。我们还通过粘合 Foscolo--Haskins--Nordström 构造的渐近锥 $C_7$ 气泡，建立了 $S^3\ imes S^3$ 上正弦锥的有限商群的局部去奇化。我们证明，当一族非平行 $\nathrm{G}_2$ 流形的奇异轨道坍缩时，在奇异轨道之外，去奇化族收敛到正弦锥商群，而自然的爆破收敛到 $C_7$ 的无挠 $\nathrm{G}_2$ 度量。
英文摘要
Nearly $\mathrm{G}_2$ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, $\mathrm{Spin}(7)$. We study nearly parallel $\mathrm{G}_2$-structures invariant under a cohomogeneity-one action of $\mathrm{SO}(4)$, and construct the first inhomogeneous nearly parallel $\mathrm{G}_2$-structure on the Berger space $\mathrm{SO}(5)/\mathrm{SO}(3)$. The cone over this metric has full holonomy $\mathrm{Spin}(7)$. The existence of the solution is established by a rigorous computer-assisted shooting argument. We also establish a local desingularisation of a finite quotient of the sine cone over $S^3\times S^3$ by gluing in the asymptotically conical $C_7$ bubble constructed by Foscolo--Haskins--Nordström. We show that as the singular orbit of a family of nearly parallel $\mathrm{G}_2$ manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free $\mathrm{G}_2$-metric of $C_7$.
Comments50 pages. All comments welcome!
