毛细管棱柱、考克斯特拼接与具有正标量曲率的三维流形的π₂-收缩
Capillary prisms, Coxeter gluing, and the $π_2$-systole of 3-manifolds with positive scalar curvature
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中文总结 AI 辅助
研究具有正标量曲率三维流形的π₂-收缩，用局部到全局方法，通过毛细管棱柱及考克斯特拼接，获角度敏感收缩估计，证明相关拼接和平滑定理，构造特定拓扑的大π₂-收缩度量并讨论估计的刚性与非刚性。
中文摘要 AI 辅助
我们通过局部到全局的方法研究具有正标量曲率的三维流形的π₂-收缩。镶嵌的局部构建块是毛细管棱柱M：由以规定二面角相交的平均凸面所包围的黎曼圆柱。对于一类称为能量基本棱柱的此类棱柱，我们获得了角度敏感的相对π₂-收缩估计。特别地，如果上毛细管角至多为α∈(0,π/2]，那么当α→0时，sys₂(M,∂₀M,g)·infR_g≤|logα|⁻¹。证明依赖于一种新的拟局部质量的单调性。局部到全局的步骤通过毛细管棱柱副本的考克斯特拼接来实现。我们证明了一个关于标量曲率的一般考克斯特拼接和平滑定理：在角层的考克斯特兼容性以及沿拼接面的非负平均曲率跳跃条件下，我们证明所得的分段光滑流形可以被平滑，同时保持标量曲率下界。该定理给出了，据我们所知，关于标量曲率下界的考克斯特多面体平滑原理的第一个一般严格表述和证明，该原理在正标量曲率几何中一直被启发式地使用。作为应用，我们在S²×S¹和透镜空间副本的连通和上构造具有大π₂-收缩的光滑正标量曲率度量，这些空间不被S²×R覆盖。这些例子给出了在这些拓扑上具有定量大π₂-收缩的第一个显式度量。我们还讨论了局部π₂-收缩估计的刚性和非刚性现象。
英文摘要
We study the $π_2$-systole of positive scalar curvature $3$-manifolds via a local-to-global approach. The tessellated local building blocks are capillary prisms $M$: Riemannian cylinders enclosed by mean convex surfaces meeting at prescribed dihedral angles. For a class of such prisms, called energy-essential prisms, we obtain angle-sensitive relative $π_2$-systole estimates. In particular, if the upper capillary angle is at most $α\in (0,\tfracπ{2}]$, then \[\mathrm{sys}_2(M,\partial_0 M, g)\cdot \inf R_g \lesssim |\log α|^{-1},\qquad α\to 0.\] The proof relies on the monotonicity of a new quasi-local mass. The local-to-global step is achieved via Coxeter gluing of copies of capillary prisms. We prove a general Coxeter gluing and smoothing theorem for scalar curvature: under Coxeter compatibility of the corner strata and nonnegative mean curvature jump conditions along the glued facets, we prove that the resulting piecewise smooth manifold can be smoothened while preserving the scalar curvature lower bound. This theorem gives, to our knowledge, the first general rigorous formulation and proof of the Coxeter-polyhedral smoothing principle for scalar curvature lower bounds, a principle that has long been used heuristically in positive scalar curvature geometry. As an application, we construct smooth positive scalar curvature metrics with large $π_2$-systole on connected sums of copies of $S^2\times S^1$ and lens spaces that are not covered by $S^2\times R$. These examples give the first explicit metrics with quantitatively large $π_2$-systole on these topologies. We also discuss rigidity and non-rigidity phenomena for the local $π_2$-systolic estimates.
