Let $M$ be a compact $n$-manifold of $\mathrm{Ric}_M \geq (n - 1) H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $ \rho \gt 0$ such that for any $x \in M$, the open $ \rho $-ball at $x^{\ast}$ in the (local) Riemannian universal covering space, $ (U^{\ast}_{\rho} , x^{\ast}) \to (B_{\rho} (x) , x)$, has the maximal volume, i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H = -1$, the volume entropy of $M$ is maximal, i.e., $n - 1$ ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that $M$ is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H = 1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

中文翻译：

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较低Ricci曲率边界I下的定量体积空间形式刚度

假设$ M $是$ \ mathrm {Ric} _M \ geq（n-1）H $的紧凑$ n $流形（$ H $是常数）。我们关心以下空间形式的刚性：在以下任一情况下，$ M $与等曲率$ H $的空间形式等距：（i）存在$ \ rho \ gt 0 $使得任何$ x \ in M $，在（局部）黎曼通用覆盖空间$（U ^ {\ ast} _ {\ rho}，x ^ {\ ast}）\ to（B _ {\ rho}（x），x）$具有最大体积，即曲率$ H的简单连接$ n $-空间形式中$ \ rho $球的体积$。（ii）对于$ H = -1 $，$ M $的体积熵最大，即$ n-1 $（[LW1]）。本文的主要结果是定量的空间形式刚度，即，陈述$ M $是亚同构的并且在Gromov–Hausdorff拓扑中接近于恒定曲率$ H $的空间形式，如果$ M $在其他条件下几乎满足上述最大交易量条件。对于$ H = 1 $，在[CC2]中，改进的球面空间定形的刚性得到改善和推广。