具有几乎非负数量曲率的图环面的稳定性
Stability of graphical tori with almost nonnegative scalar curvature
摘要：
根据Schoen-Yau和Gromov-Lawson的工作，任何具有非负数量曲率且与环面同构的黎曼流形都等距于平坦环面.当数量曲率达到$0$时环面关于弱Sobolev型度量的Gromov约束次收敛.我们证明了平坦和内在平坦的次收敛到一个平坦环面的非塌陷序列$3$维环面$M_j$，可以实现为图形的某些功能定义在平坦环面满足一个统一的上直径界和数量曲率界的形式$R_{g_{M_j}} \geq -1/j$。我们还证明了收敛子序列的流形的体积收敛于极限空间的体积。我们这样做，以适应黄-李，黄-李-Sormani和Allen-Perales-Sormani的结果。此外，我们的结果也成立时的条件下的数量曲率的环面$(M, g_M)$被取代的数量$-\int_T \min\{R_{g_M},0\} d{\mbox{vol}_{g_T}}$，其中$M=\mbox{graph}(f)$，$f: T \to \mathbb R$和$(T,g_T)$是一个平坦的环面上的边界。使用参数开发的Alaee，麦考密克和第一个命名的作者完成这项工作后，我们的结果保持尺寸$n \geq 4$以及。
英文摘要：
By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Allen-Perales-Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus $(M, g_M)$ is replaced by a bound on the quantity $-\int_T \min\{R_{g_M},0\} d{\mbox{vol}_{g_T}}$, where $M=\mbox{graph}(f)$, $f: T \to \mathbb R$ and $(T,g_T)$ is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions $n \geq 4$ as well.
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