Notes of Stefan Wenger’s lecture nr 1

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.

 

1. Introduction

 

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 {D^2} denote the unit disk in the plane. In a Riemannian manifold, the area of a Lipschitz map {\phi:D^2 \rightarrow M} is obtained by integrating the {2}-Jacobian, i.e. the norm of the exterior square of the differential {d\phi},

\displaystyle  \begin{array}{rcl}  \mathrm{Area}(\phi):=\int_{D^2}J_2 (d\phi). \end{array}

Definition 1 Let {M} be a Riemannian manifold. For {c:S^1 \rightarrow M} a closed Lipschitz curve, define

\displaystyle  \begin{array}{rcl}  \mathrm{Fillarea}_0 (c):=\inf\{\mathrm{Area}(\phi)\,;\,\phi:D^2 \rightarrow M,\,\phi_{|S^1}=c\}. \end{array}

The Filling area function in {M} is

\displaystyle  \begin{array}{rcl}  FA_0^M (r)=\sup\{\mathrm{Fillarea}_0 (c)\,;\,c:S^1 \rightarrow M,\,\mathrm{length}(c)\leq r\}. \end{array}

 

Example 1 In Euclidean plane, {FA_0^M (r)=\frac{1}{4\pi}r^2}.

 

Example 2 In a simply connected Riemannian manifold with sectional curvature {\leq\kappa},

\displaystyle  \begin{array}{rcl}  4\pi FA_0^M (r)-\kappa FA_0^M (r)^2 \leq r^2 . \end{array}

In particular,

  • if {\kappa=0}, {FA_0^M (r)\leq \frac{1}{4\pi}r^2}.
  • if {\kappa<0}, {FA_0^M (r)\leq \frac{1}{\sqrt{|\kappa|}}r}.

This reflects the fact that if {\kappa<0}, long nearby geodesics diverge exponentially. We see that the large scale geometry (divergence of geodesics) strongly influences the growth type of the filling function.

Questions:

  1. Conversely, how does {FA_0^M (r)} influence the large scale geometry of {M} ?
  2. What are the possible growth types for {FA_0^M (r)} ?

 

1.2. Growth types

 

Notation: Write {f\preceq g} if there exists a constant {C} such that for all large enough {r},

\displaystyle  \begin{array}{rcl}  f(r)\leq Cg(Cr+C)+Cr+C. \end{array}

Write {f\sim g} if {f\preceq g} and {g\preceq f}.

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 {M}, {N} be universal coverings of closed Riemannian manifolds. If {M} and {N} are quasiisometric then {FA_0^M\sim FA_0^N}.

 

1.3. Gromov’s gap theorem

 

Theorem 3 (Gromov 1987) If {M} is Gromov hyperbolic and {FA_0^M (r)<\infty}, then {FA_0^M (r)\preceq r}.

 

Proof: Long closed curves in {M} admits shortcuts. \Box

There is a converse. In fact, even quadratic Filling area function, with a small constant, implies hyperbolicity.

Theorem 4 (Gromov 1987) Let {M} be Riemannian manifold. Assume that there exists {r_0} such that for alla {r\geq r_0},

\displaystyle  \begin{array}{rcl}  FA_0^M (r)\leq \frac{1}{4000}r^2 . \end{array}

Then {M} is Gromov hyperbolic.

The constant {\frac{1}{4000}} can be replaced by {\frac{1}{16\pi}} 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 {f:{\mathbb R}_+ \rightarrow{\mathbb R}_+} be smooth, with {f'>0} and such that

  • {f(r) \succeq r^2};
  • {f(kr)\geq kf(r)} for all {k\in{\mathbb N}}.

Then there exists a surface of revolution with {FA_0^M \sim f}.

 

Note that the second assumption is necessary for {f} to be the filling function of a surface of revolution.

 

1.4. A sharp gap theorem

 

Theorem 7 (Wenger 2008) Let {M} be Riemannian manifold. Assume that there exist {\epsilon>0} and {r_0} such that

\displaystyle  \begin{array}{rcl}  FA_0^M (r)\leq \frac{1-\epsilon}{4\pi}r^2 \quad\textrm{ for all }r\geq r_0 . \end{array}

Then {M} 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 {G} is nilpotent if the lower central series {\mathfrak{g}^{i}} terminates. Here, {\mathfrak{g}^{1}=\mathfrak{g}} is the Lie algebra and {\mathfrak{g}^{i+1}={\mathbb R}-}span of {\{[v,w]\,;\, v\in \mathfrak{g},\,w\in \mathfrak{g}^{i}\}}. Say {G} has step {k} if {\mathfrak{g}^{k+1}=0} and {\mathfrak{g}^{k}\not=0}.

{G} is a Carnot group if the Lie algebra has a stratification

\displaystyle  \begin{array}{rcl}  \mathfrak{g}=V_1 \oplus \cdots \oplus V_k \end{array}

such that {[V_i ,V_j]=V_{i+j}}.

 

Question: Let {G} be a Carnot group. Equip it with a Riemannian metric. What is {FA_0^G} ?

Example 3 The {n}-th Heisenberg group {H^n} has {V_1 ={\mathbb R}^{2n}}, {V_2 ={\mathbb R}} and bracket {V_1 \otimes V_1 \rightarrow V_2} is nondegenerate.

 

Claim: If {n=1}, {FA_0^{H^1} (r) \succeq r^3}.

Indeed, let {c_L} be the curve in {H^1} made of {8} line segments of length {L}, travelling first along the {x} direction, then {y}, then {-x}, then {-y}, then {-y}, then {-x}, then {y}, then {x}. It is a closed curve of length {8L}. The projection onto the vertical {y,z}-plane is area decreasing. The projection of {c} is a triangle of basis {2L} and height {L^2}, thus area {L^3}. This shows that the filling area of {c} is {\geq L^3}.

Instead of a projection, we can use the differential form {\alpha=y\,dz} and {d\alpha=dy\wedge dz}. Let {\Sigma} be a surface spanning {c}. By Stoke’s theorem, since {|d\alpha\|\leq 1} everywhere,

\displaystyle  \begin{array}{rcl}  L^3 =\int_{c}\alpha=\int_{\Sigma}d\alpha\leq \mathrm{Area}(\Sigma). \end{array}

This is sharp.

Theorem 9 (Gromov, Pittet, Gersten, Holt, Riley) If {G} is a nilpotent Lie group of step {k}. Then {FA_0^G (r)\preceq r^{k+1}} in the following cases,

  • {G} is Carnot,
  • {G} contains a lattice.

Cornulier: I believe the theorem holds for all nilpotent Lie groups.

On the other hand,

Theorem 10 (Gromov, Allcock) The higher ({n\geq 2}) Heisenberg groups have quadratic filling area functions.

 

The case of the first Heisenberg group generalizes as follows.

Theorem 11 (Baumslag, Miller, Short) If {G} is free nilpotent of step {k}, then {FA_0^G (r)\sim r^{k+1}}.

 

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 {2} Carnot groups {G} whose filling area function satisfies

\displaystyle  \begin{array}{rcl}  r^{2}\rho(r)\preceq FA_0^G (r)\preceq r^2 \,\log r, \end{array}

where {\rho} is an (unknown) function that tends to infinity.

 

2. Digression: filling area in finitely presented groups

 

Let {\Gamma=\langle S\,|\,R\rangle} be a group with a finite presentation with generators {S} and relators {R}. If a word {w=s_1 \cdots s_n \in F_S} (the free group on {S}) is trivial in {\Gamma}, there exist {g_i \in F_S} and {r_i \in R} such that

\displaystyle  \begin{array}{rcl}  w=\prod_{i=1}^{k}g_{i}r_{i}^{\pm 1}g_{i}^{-1}. \end{array}

The minimum {k} arising in such an expression of {w} as a product of conjugates of relators is denoted by {A(w)}. This is an analogue of the filling area in Riemannian geometry.

Definition 13 The Dehn function of the presentation is

\displaystyle  \begin{array}{rcl}  \delta_{\Gamma}(n):=\max\{A(w)\,;\,w \textrm{ trivial word, }|w|\leq n\}. \end{array}

 

Example 4 {{\mathbb Z}\oplus{\mathbb Z}=\langle a,b\,|\,aba^{-1}b^{-1}\rangle}.

A trivial word {w} gives rise to a closed circuit in the grid in {{\mathbb R}^2}, of length {|w|}. Each conjugate of relators corresponds to a square in the grid, and expressing {w} as a product of conjugates of relators amounts to filling {c} with a collection of squares. This gives {\delta_{{\mathbb Z}^2}(n)\sim n^2}.

Theorem 14 (Gromov, Bridson) If {M} is the universal cover of a closed manifold with fundamental group {\Gamma}, then, for any finite presentation of {\Gamma},

\displaystyle  \begin{array}{rcl}  \delta_{\Gamma}\sim FA_0^M . \end{array}

 

3. The sharp gap theorem

 

3.1. Filling in metric spaces

 

In a metric space {Y}, the area of a Lipschitz map {\phi:D^2 \rightarrow Y} is defined by

\displaystyle  \begin{array}{rcl}  \mathrm{Area}(\phi)=\int_{Y}N(\phi,y)\,d\mathcal{H}^2 (y), \end{array}

where {\mathcal{H}^d} denotes {d}-dimensional Hausdorff measure and {N(\phi,\cdot)} is the multiplicity function

\displaystyle  \begin{array}{rcl}  N(\phi,y):=|\{z\in D^2 \,;\,\phi(z)=y\}|. \end{array}

If {\phi} is injective, Area{(\phi)=\mathcal{H}^2 (\phi(D^2))}. If {Y} is a Riemannian manifold, this coincides with the previously defined area (area formula).

Definition 15 Let {X}, {Y} be metric spaces, {\iota:X\rightarrow Y} an isometric embedding. For a Lipschitz closed curve {c:S^1 \rightarrow X}, let

\displaystyle  \begin{array}{rcl}  Fillarea_0^Y (c)=\inf\{\mathrm{Area}(\phi)\,;\,\phi:D^2 \rightarrow Y \textrm{ Lipschitz},\,\phi_{|S^1}=\iota\circ c\}. \end{array}

\displaystyle  \begin{array}{rcl}  FA_{0}^{X,Y}(r)=\sup\{Fillarea_0^Y (c)\,;\,c:S^1 \rightarrow X \textrm{ Lipschitz},\,\mathrm{length}(c)\leq r\}. \end{array}

 

Remark 1 Let {E(X)} denote the injective hull of {X}. For all isometric embeddings {\iota:X\rightarrow Y},

\displaystyle  \begin{array}{rcl}  FA_{0}^{X,E(X)}\leq FA_{0}^{X,Y}\leq FA_{0}^{X,X}. \end{array}

where the first inequality becomes an equality in case {Y} is injective.

 

Definition 16 For every metric space {X}, we denote by

\displaystyle  \begin{array}{rcl}  FA_{0}^{X,\infty}=FA_{0}^{X,E(X)} \end{array}

the absolute filling function.

 

Proposition 17 For every metric space {X},

\displaystyle  \begin{array}{rcl}  FA_{0}^{X,\infty}\leq \frac{1}{2\pi}r^2. \end{array}

 

Proof: Let {c} be a closed curve parametrized by arc-length, so that length{(c)=2\pi Lip(c)}. The circle {S^1} embeds isometrically in a round half sphere {S^{2}_{+}}. {E(X)} injective implies that there exists a {Lip(c)}-Lipschitz extension {\phi:S^{2}_{+}\rightarrow E(X)}. Then

\displaystyle  \begin{array}{rcl}  \mathrm{Area}(\phi)\leq 2\pi \,Lip(c)^2 =\frac{1}{2\pi}\mathrm{length}(c)^2 . \end{array}

\Box

Example 5 Let {\Gamma=\langle S\,|\,R\rangle} be a group with a finite presentation. Let {X=Cay(\Gamma,S)} be the corresponding Cayley graph, with length {1} edges. Then

\displaystyle  \begin{array}{rcl}  FA_{0}^{X,\infty}(n)\leq \frac{1}{2\pi}\max\{|r|^2 \,;\,r\in R\}\delta_{\gamma}(n). \end{array}

 

Definition 18 Call {Y} a thickening of {X} if {Y} is contained in a bounded neighborhood of {X}.

 

Example 6 (Lang 2004) If {X} is a geodesic hyperbolic metric space, the injective hull {E(X)} is a geodesic thickening of {X}.

 

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,

\displaystyle  \begin{array}{rcl}  A_{\delta}(c)&=&\inf\{|P^{(2)}|\,;\,P \textrm{ triangulation of }D^2 ,\,\exists \phi:P^{(1)}\rightarrow X \textrm{ continuous extension of }c,\\ &&\textrm{such that }\forall \Delta\in P^{(2)},\,\textrm{diameter}(\phi(\partial\Delta))<\delta\}. \end{array}

Then he let

\displaystyle  \begin{array}{rcl}  FA_{\delta}^X (r)=\sup\{A_{\delta}(c)\,;\,c:S^1 \rightarrow X,\,\mathrm{length}(c)\leq r\}, \end{array}

as usual.

Proposition 19 Let {X} be a geodesic metric space, {\delta>0}. Assume {A_{\delta}(r)>0} for all {r}. Then there exists a geodesic thickening {Y} of {X} in which

\displaystyle  \begin{array}{rcl}  FA_0^Y \preceq FA_{\delta}^X \quad\textrm{ and }\quad FA_0^Y (r) \leq C\,r^2 \textrm{ for }r\leq 1. \end{array}

 

3.3. Generalized sharp gap theorem

 

Theorem 20 (Wenger 2008) Let {X} be a geodesic metric space. Suppose

  1. There exists a geodesic thickening {Y} of {X} with {FA_0^{X,Y}\preceq r^2};
  2. There exists {\epsilon>0} and {r_0 >0} such that\displaystyle  \begin{array}{rcl}  FA_0^{X,\infty}(r)\leq \frac{1-\epsilon}{4\pi}r^2 \quad\forall r\geq r_0 . \end{array}

    Then {X} is Gromov hyperbolic.

 

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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2 Responses to Notes of Stefan Wenger’s lecture nr 1

  1. Riikka says:

    Hi,

    just a small typo in 3.1 in the definition of the multiplicity function N: the demand phi(z)=y is missing from the formula.

    Thanks for the excellent notes, I was (again) able to fill out some missing parts of my own notes with the help of these.

    – Riikka

  2. Stefan Wenger says:

    Dear Pierre,

    Thank you for taking all the notes. There were only two minor details that I found which could maybe be corrected:

    – In Gromov’s second theorem in Section 1.3 one should add “for all r >r_0” after the inequality.
    – In Remark 1 in Section 3.1 one could replace “with equality if Y is injective” with “where the first inequality becomes an equality in case Y is injective.” One could furthermore add “In the above, E(X) denotes the injective hull of X.”

    Thanks again.
    Stefan

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