Minimal volume

In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a Riemannian manifold's topology. This invariant was introduced by Mikhail Gromov.

Definition

Consider a closed orientable connected smooth manifold Mⁿ with a smooth Riemannian metric g, and define Vol(M,g) to be the volume of a manifold M with the metric g. Let K_g represent the sectional curvature.

The minimal volume of M is a smooth invariant defined as

$MinVol(M):=\inf_{g}\{Vol(M,g) : |K_{g}|\leq 1\}$

that is, the infimum of the volume of M over all metrics with bounded sectional curvature.

Clearly, any manifold M may be given an arbitrarily small volume by selecting a Riemannian metric g and scaling it down to λg, as Vol(M,λg) = λ^{n / 2}Vol(M,g). For a meaningful definition of minimal volume, it is thus necessary to prevent such scaling. The inclusion of bounds on sectional curvature suffices, as $\textstyle K_{\lambda g} = \frac{1}{\lambda} K_g$. If MinVol(M) = 0, then Mⁿ can be "collapsed" to a manifold of lower dimension (and thus one with n-dimensional volume zero) by a series of appropriate metrics; this manifold may be considered the Hausdorff limit of the related sequence, and the bounds on sectional curvature ensure that this convergence takes place in a topologically meaningful fashion.

Related topological invariants

The minimal volume invariant is connected to other topological invariants in a fundamental way; via Chern-Weil theory, there are many topological invariants which can be described by integrating polynomials in the curvature over M. In particular, the Chern classes and Pontryagin classes are bounded above by the minimal volume.

Properties

Gromov has conjectured that every closed simply connected odd-dimensional manifold has zero minimal volume. This conjecture clearly does not hold for even-dimensional manifolds.

References

• Gromov, M. Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser (1999) ISBN 0-8176-3898-9.

• Gromov, M. Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 1—99.