Minimal volume
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In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
's
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. This diffeomorphism invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold , one may consider its volume and
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
. The minimal volume of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is defined to be: :\operatorname(M):=\inf\. 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 is zero, then a certain kind of nontrivial collapsing phenomena can be exhibited by Riemannian metrics on . 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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
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 amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if is a closed and connected two-dimensional manifold, then . The infimum in the definition of minimal volume is realized by the metrics appearing from the
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
. More generally, according to the Chern-Gauss-Bonnet formula, if is a closed and connected manifold then: :\operatorname(M)\geq c(n)\big, \chi(M)\big, . 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 In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a certain measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes. Give ...
of the manifold, via the inequality: :\operatorname(M)\geq\frac.


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. . * Michael Gromov
''Volume and bounded cohomology.''
{{free access Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99. Riemannian geometry