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 topological invariant was introduced by Mikhail Gromov.

Definition

Consider a closed orientable connected smooth manifold $M^n$ with a smooth Riemannian metric $g$, and define $Vol({\it M,g})$ 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 $\lambda g$, as $Vol(M, \lambda g) =\lambda^{n/2}Vol(M, g)$. 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 $M^n$ 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.