毛细最小切片与标量曲率刚性
Capillary minimal slicing and scalar curvature rigidity
AI总结：
本文通过毛细超曲面方法，在四维流形中证明了正标量曲率和边界条件下的刚性结论。
摘要：
我们通过毛细超曲面发展出最小切片，以理解带有边界的流形上的正标量曲率度量。该方法在毛细面积泛函的最小化器正则性成立时提供刚性陈述。特别地，在四维情况下，我们证明了如下比较和刚性陈述：给定一个紧致的黎曼4流形(M^4,g)，其边界是与R^4中连通凸域边界的微分同胚，若标量曲率非负且边界上的缩放平均曲率比较成立，则M与欧几里得域等距。
英文摘要：
We develop minimal slicing via capillary hypersurfaces to understand scalar curvature metric on manifolds bounded below with boundary. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain. Moreover, we prove a comparison and rigidity result of manifolds with Listing type scaling invariant scalar and mean curvature comparison : $R g \ge -12 g_{\mathbb{H}^{4}}$ and $\hat{H}^{2} g \ge \hat{H}_{0} g_{\mathbb{H}^{4}}$, where $\hat{H}$ is a shifted mean curvature depending on the scale of manifold. This proves the traced version of Gromov's conjecture on comparison and rigidity of hyperbolic convex domains
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