可见n-流形和紧3-流形的极小体积和单纯范数
Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds
摘要：
定理A.设$M^n$表示具有非正截面曲率的闭黎曼流形，$\tilde M^n$表示$M^n$在提升度量下的泛覆盖.假设通用覆盖$\tilde M^n$不包含完全测地线嵌入的欧几里得平面$\mathbb{R}^2$（即，$M^n$是可见性流形）。那么Gromov的单纯体积$\| M^n \|$是非零的。因此，$M^n$是不可折叠的，同时保持Ricci曲率从下面有界。更精确地说，如果$Ric_g \ge -(n-1)$，则$vol(M^n, g) \ge \frac{1}{(n-1)^n n!} \| M^n \| > 0. Theorem B. (Perelman) Let $ M^3 $ be a closed a-spherical 3-manifold ($ K（π，1）$-space) with the fundamental group $ <$$. Suppose that $ <$$ contains no subgroups isomorphic to $ \mathbb{Z} \oplus \mathbb{Z} $. Then $ M^3 $ is diffeomorphic to a compact quotient of real hyperbolic space $ \mathbb{H}^3 $, i.e., $ M^3 \equiv \mathbb{H}^3 /<$$. Consequently, $ Minimum（M^3）\ge {1/24} M^3> 0$。我们也将讨论所有其它无边界紧致三维流形和{ \it奇异}空间的极小体积和单纯范数。
英文摘要：
Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally geodesic embedded Euclidean plane $\mathbb{R}^2$ (i.e., $M^n$ is a visibility manifold). Then Gromov's simplicial volume $\| M^n \|$ is non-zero. Consequently, $M^n$ is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if $Ric_g \ge -(n-1)$, then $vol(M^n, g) \ge \frac{1}{(n-1)^n n!} \| M^n \| > 0. Theorem B. (Perelman) Let $M^3$ be a closed a-spherical 3-manifold ($K(π, 1)$-space) with the fundamental group $Γ$. Suppose that $Γ$ contains no subgroups isomorphic to $\mathbb{Z}\oplus \mathbb{Z}$. Then $M^3$ is diffeomorphic to a compact quotient of real hyperbolic space $\mathbb{H}^3$, i.e., $M^3 \equiv \mathbb{H}^3/Γ$. Consequently, $MinVol(M^3) \ge {1/24}\| M^3 \| > 0$. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and {\it singular} spaces will also be discussed.
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