I am interested in isoperimetric inequalities. The aim is to find relations between the growth of isoperimetric functions and the large scale geometry of the underlying space. I will use tools from geometric measure theory.
Today, I will give an overview of results, to be explained later on.
For the sake of exposition, I will stick to Riemannian manifolds, where isoperimetry is easy to define. Later on, I will turn to more general spaces, and this will require some geometric measure theory.
1.1. Filling area
Let denote the unit disk in the plane. In a Riemannian manifold, the area of a Lipschitz map is obtained by integrating the -Jacobian, i.e. the norm of the exterior square of the differential ,
Definition 1 Let be a Riemannian manifold. For a closed Lipschitz curve, define
The Filling area function in is
Example 1 In Euclidean plane, .
Example 2 In a simply connected Riemannian manifold with sectional curvature ,
- if , .
- if , .
This reflects the fact that if , long nearby geodesics diverge exponentially. We see that the large scale geometry (divergence of geodesics) strongly influences the growth type of the filling function.
- Conversely, how does influence the large scale geometry of ?
- What are the possible growth types for ?
1.2. Growth types
Notation: Write if there exists a constant such that for all large enough ,
Write if and .
Note that we do not distinguish linear from sublinear growths. Neither do we distinguish various exponential growths. But we distinguish various polynomial growths, poly versus poly times polylog.
Theorem 2 (Gromov, Bridson) Let , be universal coverings of closed Riemannian manifolds. If and are quasiisometric then .
1.3. Gromov’s gap theorem
Theorem 3 (Gromov 1987) If is Gromov hyperbolic and , then .
Proof: Long closed curves in admits shortcuts.
There is a converse. In fact, even quadratic Filling area function, with a small constant, implies hyperbolicity.
Theorem 4 (Gromov 1987) Let be Riemannian manifold. Assume that there exists such that for alla ,
Then is Gromov hyperbolic.
The constant can be replaced by for universal coverings of closed Riemannian manifolds.
Corollary 5 Nothing between quadratic and linear growth.
Apart from this gap, in coarse terms, there is hardly any restriction on Filling area functions.
Theorem 6 (Grimaldi, Pansu 2003) Let be smooth, with and such that
- for all .
Then there exists a surface of revolution with .
Note that the second assumption is necessary for to be the filling function of a surface of revolution.
1.4. A sharp gap theorem
Theorem 7 (Wenger 2008) Let be Riemannian manifold. Assume that there exist and such that
Then is Gromov hyperbolic.
This is not a large scale result, since the assumption is sensitive to small scale modification of a metric.
1.5. Nilpotent Lie groups and their filling functions
Definition 8 A Lie group is nilpotent if the lower central series terminates. Here, is the Lie algebra and span of . Say has step if and .
is a Carnot group if the Lie algebra has a stratification
such that .
Question: Let be a Carnot group. Equip it with a Riemannian metric. What is ?
Example 3 The -th Heisenberg group has , and bracket is nondegenerate.
Claim: If , .
Indeed, let be the curve in made of line segments of length , travelling first along the direction, then , then , then , then , then , then , then . It is a closed curve of length . The projection onto the vertical -plane is area decreasing. The projection of is a triangle of basis and height , thus area . This shows that the filling area of is .
Instead of a projection, we can use the differential form and . Let be a surface spanning . By Stoke’s theorem, since everywhere,
This is sharp.
Theorem 9 (Gromov, Pittet, Gersten, Holt, Riley) If is a nilpotent Lie group of step . Then in the following cases,
- is Carnot,
- contains a lattice.
Cornulier: I believe the theorem holds for all nilpotent Lie groups.
On the other hand,
Theorem 10 (Gromov, Allcock) The higher () Heisenberg groups have quadratic filling area functions.
The case of the first Heisenberg group generalizes as follows.
Theorem 11 (Baumslag, Miller, Short) If is free nilpotent of step , then .
Not much is known beyond these results.
1.6. Non polynomial filling area functions
Question: For nilpotent groups, is filling area function always exactly a polynomial ?
Theorem 12 (Wenger 2010) There exist step Carnot groups whose filling area function satisfies
where is an (unknown) function that tends to infinity.
2. Digression: filling area in finitely presented groups
Let be a group with a finite presentation with generators and relators . If a word (the free group on ) is trivial in , there exist and such that
The minimum arising in such an expression of as a product of conjugates of relators is denoted by . This is an analogue of the filling area in Riemannian geometry.
Definition 13 The Dehn function of the presentation is
Example 4 .
A trivial word gives rise to a closed circuit in the grid in , of length . Each conjugate of relators corresponds to a square in the grid, and expressing as a product of conjugates of relators amounts to filling with a collection of squares. This gives .
Theorem 14 (Gromov, Bridson) If is the universal cover of a closed manifold with fundamental group , then, for any finite presentation of ,
3. The sharp gap theorem
3.1. Filling in metric spaces
In a metric space , the area of a Lipschitz map is defined by
where denotes -dimensional Hausdorff measure and is the multiplicity function
If is injective, Area. If is a Riemannian manifold, this coincides with the previously defined area (area formula).
Definition 15 Let , be metric spaces, an isometric embedding. For a Lipschitz closed curve , let
Remark 1 Let denote the injective hull of . For all isometric embeddings ,
where the first inequality becomes an equality in case is injective.
Definition 16 For every metric space , we denote by
the absolute filling function.
Proposition 17 For every metric space ,
Proof: Let be a closed curve parametrized by arc-length, so that length. The circle embeds isometrically in a round half sphere . injective implies that there exists a -Lipschitz extension . Then
Example 5 Let be a group with a finite presentation. Let be the corresponding Cayley graph, with length edges. Then
Definition 18 Call a thickening of if is contained in a bounded neighborhood of .
Example 6 (Lang 2004) If is a geodesic hyperbolic metric space, the injective hull is a geodesic thickening of .
3.2. Gromov’s coarse filling function
In our general gap theorem, there will be a mysterious looking quadratic filling assumption. This assumption is often satisfied thanks to the following considerations. Here is how Gromov defined filling in metric spaces. He first defined Filling area as follows,
Then he let
Proposition 19 Let be a geodesic metric space, . Assume for all . Then there exists a geodesic thickening of in which
3.3. Generalized sharp gap theorem
Theorem 20 (Wenger 2008) Let be a geodesic metric space. Suppose
- There exists a geodesic thickening of with ;
- There exists and such that
Then is Gromov hyperbolic.