正$τ$-双Ricci曲率与双曲空间中带手术的平均曲率流
Positive $τ$-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space
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中文总结 AI 辅助
该研究证明双曲空间中满足正$τ$-双Ricci曲率的光滑闭连通浸入超曲面存在带有限次手术且可终止的平均曲率流，并推导了相应流形的微分同胚类型与紧区域的拓扑性质。
中文摘要 AI 辅助
设$n\geq3$且$0\leqτ\leq2$。我们证明，双曲空间中每一个诱导度量具有正$τ$-双Ricci曲率的光滑闭连通浸入超曲面，都存在一个带手术的平均曲率流，该流仅有有限个手术时刻且会终止。在与柱形颈相容的参数范围内，关键要素包括一个被保持的定量谱夹紧条件（它将内蕴假设转化为一致二凸性），以及在双曲标准颈替换过程中仍然有效的柱形估计和导数估计。在端点情形$n=3$、$τ=2$下，正$τ$-双Ricci曲率即为正Ricci曲率，且会迫使超曲面严格凸，因此普通平均曲率流会收敛到一个圆点。由此可得，底层流形微分同胚于球面，或者微分同胚于若干个$\mathbb{S}^{n-1}\times\mathbb{S}^1$的有限连通和。若初始超曲面是嵌入的且界定一个紧区域，则该区域是一个一维柄体，即附着了有限个一维柄的球体。
英文摘要
Let $n\geq3$ and $0\leqτ\leq2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $τ$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint $n=3$, $τ=2$, positive $τ$-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of $\mathbb{S}^{n-1}\times\mathbb{S}^1$. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.
