Minimal volume
In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This diffeomorphism invariant was introduced by Mikhael Gromov.
Given a smooth Riemannian manifold (M, g), one may consider its volume vol(M, g) and sectional curvature Kg. The minimal volume of a smooth manifold M is defined to be:
- [math]\displaystyle{ \operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}. }[/math]
Any closed manifold can be given an arbitrarily small volume by scaling any choice of a Riemannian metric. The minimal volume removes the possibility of such scaling by the constraint on sectional curvatures. So, if the minimal volume of M is zero, then a certain kind of nontrivial collapsing phenomena can be exhibited by Riemannian metrics on M. A trivial example, the only in which the possibility of scaling is present, is a closed flat manifold. The Berger spheres show that the minimal volume of the three-dimensional sphere is also zero. Gromov has conjectured that every closed simply connected odd-dimensional manifold has zero minimal volume.
By contrast, a positive lower bound for the minimal volume of M amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on M in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if M is a closed and connected two-dimensional manifold, then MinVol(M) = 2π|χ(M)|. The infimum in the definition of minimal volume is realized by the metrics appearing from the uniformization theorem. More generally, according to the Chern-Gauss-Bonnet formula, if M is a closed and connected manifold then:
- [math]\displaystyle{ \operatorname{MinVol}(M)\geq c(n)\big|\chi(M)\big|. }[/math]
Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality:
- [math]\displaystyle{ \operatorname{MinVol}(M)\geq\frac{\|M\|}{(n-1)^nn!}. }[/math]
References
- Misha Gromov. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. ISBN:0-8176-3898-9. doi:10.1007/978-0-8176-4583-0
- Michael Gromov. Volume and bounded cohomology.
Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
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