论文标题
低于距离以下的体积
Volume Above Distance Below
论文作者
论文摘要
给定一对度量张量$ g_1 \ ge g_0 $在riemannian歧管上，$ m $，众所周知，$ \ operatatorName {vol} _1（m）\ ge \ ge \ ge \ operatotorname {vol} _0 _0（m）$。此外，一个人具有刚度：当且仅当公制张量与$ g_1 = g_0 $时，卷是相等的。在这里，我们证明，如果$ g_j \ ge g_0 $和$ \ operatorName {vol} _1（m）\ to \ operatatorName {vol} _0（m）$ then $（m，g_j）$收敛到$（m，g_0）$（m，g_0）$保存本质上的固有平面感。众所周知的例子表明，在这种情况下，不需要获得光滑，$ c^0 $，Lipschitz，甚至Gromov-Hausdorff收敛。我们的定理也可以用作证明有关riemannian几何形状各种刚性定理的几何稳定性的其他开放猜想的工具。为了完成我们的证明，我们提供了一种新颖的方式来估计里曼尼亚歧管之间的内在平坦距离，这本身就是有趣的。
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.
