Minimal volume

Template:No footnotes 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 Template:Math, one may consider its volume Template:Math and sectional curvature Template:Math. The minimal volume of a smooth manifold Template:Mvar is defined to be:

$\mathrm{MinVol}(M):=\inf \{\mathrm{vol}(M,g):g\text{ a complete Riemannian metric with }|K_g|\le 1\}.$

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 Template:Mvar is zero, then a certain kind of nontrivial collapsing phenomena can be exhibited by Riemannian metrics on Template:Mvar. 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 Template:Mvar amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on Template:Mvar in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if Template:Mvar is a closed and connected two-dimensional manifold, then Template:Math. 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 Template:Mvar is a closed and connected manifold then:

$\mathrm{MinVol}(M)\ge c(n)|\chi (M)|.$

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:

$\mathrm{MinVol}(M)\ge \frac{\Vert M\Vert }{(n-1{)}^{n}n!}.$

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. Template:ISBN. Template:Doi Template:Closed access
• Michael Gromov. Volume and bounded cohomology. Template:Free access Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.