# 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 ${\displaystyle M^{n}}$ with a smooth Riemannian metric ${\displaystyle g}$, and define ${\displaystyle Vol({\it {M,g}})}$ to be the volume of a manifold ${\displaystyle M}$ with the metric ${\displaystyle g}$. Let ${\displaystyle K_{g}}$ represent the sectional curvature.

The minimal volume of ${\displaystyle M}$ is a smooth invariant defined as

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

that is, the infimum of the volume of ${\displaystyle M}$ over all metrics with bounded sectional curvature.

Clearly, any manifold ${\displaystyle M}$ may be given an arbitrarily small volume by selecting a Riemannian metric ${\displaystyle g}$ and scaling it down to ${\displaystyle \lambda g}$, as ${\displaystyle Vol(M,\lambda g)=\lambda ^{n/2}Vol(M,g)}$. For a meaningful definition of minimal volume, it is thus necessary to prevent such scaling. The inclusion of bounds on sectional curvature suffices, as ${\displaystyle \textstyle K_{\lambda g}={\frac {1}{\lambda }}K_{g}}$. If ${\displaystyle MinVol(M)=0}$, then ${\displaystyle M^{n}}$ can be "collapsed" to a manifold of lower dimension (and thus one with ${\displaystyle n}$-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 ${\displaystyle M}$. 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.