AI 中文总结
研究非塌缩凯勒 - 里奇流有限时间奇点及里奇收缩子无穷远处几何，证明凯勒 - 里奇流时间切片收敛等结论，还将其与法诺纤维化联系，应用于证明特定收缩子无穷远处切空间唯一性。
AI 中文摘要
我们研究了体积非塌缩凯勒 - 里奇流的有限时间奇点的度量几何以及里奇收缩子在无穷远处的几何。对于一个接近其首个奇异时间的紧致体积非塌缩凯勒 - 里奇流，我们证明时间切片收敛到一个唯一的格罗莫夫 - 豪斯多夫极限，与里奇流时空完备化的时间零切片典范地等同。其正则部分与零轨迹的补集等同，且奇异集的豪斯多夫余维数至少为四。对于凯勒 - 里奇收缩子，我们将无穷远处的几何与相关的法诺纤维化联系起来。我们证明在纤维化双全纯的轨迹上局部光滑收敛到一个凯勒锥度量，并通过在紧致集外的双全纯性来刻画渐近锥性。对于具有有界标量曲率的里奇收缩子，我们表明时间零切片的局部紧致性意味着全自相似流到该切片的带点格罗莫夫 - 豪斯多夫收敛。作为应用，我们证明了具有有界标量曲率和欧几里得体积增长且孤子向量场生成\(S^1\)作用的凯勒 - 里奇收缩子以及所有四维有界标量曲率里奇收缩子在无穷远处切空间的唯一性。
英文摘要
We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an $S^1$-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.
Comments62 pages. Comments are welcome
