Minimal volume

Definition

Consider a closed orientable connected smooth manifold M n with a smooth Riemannian metric g , and define V o l ( 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

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 λ g , as V o l ( M , λ g ) = λ n / 2 V o l ( 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 K λ g = 1 λ K g . If M i n V o l ( 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.