高于距离低于体积
Volume Above Distance Below
摘要：
已知黎曼流形上的一对度量张量$g_1 \ge g_0$，$M$，$\operatorname{Vol}_1(M) \ge \operatorname{Vol}_0(M)$。此外还有刚性：体积相等当且仅当度量张量相同$g_1=g_0$。在这里，我们证明了，如果$g_j \ge g_0$和$\operatorname{Vol}_1(M)\to \operatorname{Vol}_0(M)$，那么$(M,g_j)$收敛到$(M,g_0)$在体积保持内在平坦的意义。众所周知的例子表明，在这种情况下，不需要获得光滑、$C^0$、Lipschitz或甚至格罗莫夫-豪斯多夫收敛。我们的定理也可以作为证明黎曼几何中各种刚性定理的几何稳定性的其他公开命题的工具。为了完成我们的证明，我们提供了一种新的方法来估计黎曼流形之间的内在平坦距离，这本身就很有趣。
英文摘要：
Given a pair of metric tensors $g_1 \ge g_0$ on a Riemannian manifold, $M$, it is well known that $\operatorname{Vol}_1(M) \ge \operatorname{Vol}_0(M)$. Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same $g_1=g_0$. Here we prove that if $g_j \ge g_0$ and $\operatorname{Vol}_1(M)\to \operatorname{Vol}_0(M)$ then $(M,g_j)$ converge to $(M,g_0)$ in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, $C^0$, Lipschitz, or even Gromov-Hausdorff convergence in this setting. Our theorem may also be applied as a tool towards proving other open conjectures concerning the geometric stability of a variety of rigidity theorems in Riemannian geometry. To complete our proof, we provide a novel way of estimating the intrinsic flat distance between Riemannian manifolds which is interesting in its own right.
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