Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds

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dc.creatorCao, Jianguo
dc.creatorChen, Xiaoyang
dc.date2008-12-17
dc.date2009-02-16
dc.date.accessioned2026-07-07T12:41:50Z
dc.date.available2026-07-07T12:41:50Z
dc.descriptionTheorem 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.
dc.descriptionIn this updated version, we were able to extend our results on smooth visibility manifolds to possibly singular visibility spaces by using a result of Martin Bridson
dc.identifierhttps://arxiv.org/abs/0812.3353
dc.identifierhttp://arxiv.org/abs/0812.3353
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/219877
dc.subjectDifferential Geometry
dc.subject53C20, 53C42
dc.titleMinimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds
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