Systolic geometry

In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations.

In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of X.

[1] Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student Pao Ming Pu.

This line of research was, apparently, given further impetus by a remark of René Thom, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter.

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the March 2008 issue of the Notices of the American Mathematical Society (see reference below).

Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals.

Every convex centrally symmetric polyhedron P in R3 admits a pair of opposite (antipodal) points and a path of length L joining them and lying on the boundary ∂P of P, satisfying An alternative formulation is as follows.

Roughly speaking, there is an integral identity relating area on the one hand, and an average of energies of a suitable family of loops on the other.

for the torus, where the case of equality is attained by the flat torus whose deck transformations form the lattice of Eisenstein integers, and for Pu's inequality for the real projective plane P2(R): with equality characterizing a metric of constant Gaussian curvature.

A number of new inequalities of this type have recently been discovered, including universal volume lower bounds.

The deepest result in the field is Gromov's inequality for the homotopy 1-systole of an essential n-manifold M: where Cn is a universal constant only depending on the dimension of M. Here the homotopy systole sysπ1 is by definition the least length of a noncontractible loop in M. A manifold is called essential if its fundamental class [M] represents a nontrivial class in the homology of its fundamental group.

We then set E = L∞(M) in the formula above, and define Namely, Gromov proved a sharp inequality relating the systole and the filling radius, valid for all essential manifolds M; as well as an inequality valid for all closed manifolds M. A summary of a proof, based on recent results in geometric measure theory by S. Wenger, building upon earlier work by L. Ambrosio and B. Kirchheim, appears in Section 12.2 of the book "Systolic geometry and topology" referenced below.

A completely different approach to the proof of Gromov's inequality was recently proposed by Larry Guth.

Similarly, just about the only nontrivial lower bound for a k-systole with k = 2, results from recent work in gauge theory and J-holomorphic curves.

The study of lower bounds for the conformal 2-systole of 4-manifolds has led to a simplified proof of the density of the image of the period map, by Jake Solomon.

Once the connection is established, the influence is mutual: known results about LS category stimulate systolic questions, and vice versa.

Given a manifold M, one looks for the longest product of systoles which give a "curvature-free" lower bound for the total volume of M (with a constant independent of the metric).

The number of factors in such a "longest product" is by definition the systolic category of M. For example, Gromov showed that an essential n-manifold admits a volume lower bound in terms of the n'th power of the homotopy 1-systole (see section above).

The study of the asymptotic behavior for large genus g of the systole of hyperbolic surfaces reveals some interesting constants.

A family of optimal systolic inequalities is obtained as an application of the techniques of Burago and Ivanov, exploiting suitable Abel–Jacobi maps, defined as follows.

of the manifold M corresponding the subgroup Ker(φ) ⊂ π is called the universal (or maximal) free abelian cover.Now assume M has a Riemannian metric.

The Abel–Jacobi map is unique up to translations of the Jacobi torus.As an example one can cite the following inequality, due to D. Burago, S. Ivanov and M. Gromov.

Let M be an n-dimensional Riemannian manifold with first Betti number n, such that the map from M to its Jacobi torus has nonzero degree.

It was discovered recently (see paper by Katz and Sabourau below) that the volume entropy h, together with A. Katok's optimal inequality for h, is the "right" intermediary in a transparent proof of M. Gromov's asymptotic bound for the systolic ratio of surfaces of large genus.

The classical result of A. Katok states that every metric on a closed surface M with negative Euler characteristic satisfies an optimal inequality relating the entropy and the area.

As an application, this method implies that every metric on a surface of genus at least 20 satisfies Loewner's torus inequality.

Gromov's filling area conjecture has been proved in a hyperelliptic setting (see reference by Bangert et al. below).

This property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane, where a pair of opposite points are at distance 2, not π.

Hersch's formula expresses the area of a metric in the conformal class of the football, as an average of the energies of the figure-8 loops from the family.

An application of Hersch's formula to the hyperelliptic quotient of the Riemann surface proves the filling area conjecture in this case.