Notes of Pierre Pansu’s lecture nr 1

Differential forms and the Hölder homeomorphism problem, after Gromov and Rumin

1. Gromov’s Hölder homeomorphism problem

1.1. The problem

Question (Gromov 1993). Let {M} be an {n}-dimensional sub-Riemannian manifold. For which {\alpha\in(0,1)} does there exist locally a homeomorphism {{\mathbb R}^n \rightarrow M} which is {C^{\alpha}}-Hölder continuous ? Motivation: We have some understanding of Lipschitz maps. On the other hand, we have no structure theory of Hölder maps, and rather few Hölder invariants. So it is a challenge.

Definition 1 Let {\alpha(M)=\sup\{\alpha\in(0,1)\,|\,\exists} locally a homeomorphism {{\mathbb R}^n \rightarrow M\}}.

Example 1 If {G} is a {r}-step Carnot group, the exponential map {\mathfrak{g}=Lie(G)\rightarrow G} is locally {C^{1/r}}-Hölder continuous. Thus {\alpha(M)\geq 1/r}.

Theorem 2 (Gromov 1993) Let {M} be sub-Riemannian, with Hausdorff dimension {Q}. Then {\alpha(M)\leq\frac{n-1}{Q-1}}.

Let {M} be a {2m+1}-dimensional contact manifold. Then {\alpha(M)\leq\frac{m+1}{m+2}} ({<\frac{n-1}{Q-1}=\frac{2m}{2m+1}} if {m>1}).

1.2. Hausdorff dimensions of subsets

Gromov’s proof uses Hausdorff dimension of subset of given topological dimension: if all subsets of topological dimension {k} have Hausdorff dimension {\geq k'}, then {\alpha(M)\leq\frac{k}{k'}}.

To get lower bounds on Hausdorff dimension of subsets, Gromov constructs local foliations by horizontal submanifolds. If there are enough such dimension {k} foliations, all subsets of topological dimension {n-k} have Hausdorff dimension {\geq Q-k}, therefore {\alpha(M)\leq\frac{n-k}{Q-k}}.

Constructing horizontal submanifolds amounts to solving a system of PDE’s. If {k=1}, it is an ODE, the method applies to all (equiregular) sub-Riemannian manifolds. Gromov solves the relevant PDE for contact {2m+1}-manifolds and {k=m}, and, more generally, for generic {h}-dimensional distributions, and {k} such that {h-k\geq(n-h)k}.

Today, I describe an alternative method, due again to Gromov, but based on Rumin’s theory of differential forms on sub-Riemannian manifolds. A motivation to further study this theory in this seminar.

2. Cochains

2.1. Definition

Definition 3 On a metric space {X}, a (straight) {q}-cochain of size {\epsilon} is a function {c} on {q+1}-uples of diameter {\leq \epsilon}. Its {\epsilon}-absolute value is

\displaystyle  \begin{array}{rcl}  |c|_{\epsilon}=\sup\{c(\Delta)\,;\,diam(\Delta)\leq \epsilon\}. \end{array}

In other words, straight cochains of size {\epsilon} coincide with simplicial cochains on the simplicial complex whose vertices are points of {X} and a {q}-face joins {q+1} vertices as soon as all pairwise distances are {\leq \epsilon}. Therefore, they form a complex {\mathcal{C}_{\epsilon}^{.}}. There is a dual complex of chains {\mathcal{C}_{.,\epsilon}}.

Lemma 4 Assume {X} is a manifold with boundary, or bi-Hölder homeomorphic to such, then the inductive limit complex {\underrightarrow{\lim}} {\mathcal{C}_{\epsilon}^{\cdot}} computes cohomology.

Definition 5 Given a cohomology class {\kappa} and a number {\nu>0}, one can define the {\nu}-norm

\displaystyle  \begin{array}{rcl}  ||\kappa||_{\nu}=\liminf_{\epsilon\rightarrow 0}\epsilon^{-\nu}\inf\{|c|_{\epsilon}\,|\,\textrm{ cochains } c \textrm{ of size } \epsilon \textrm{ representing } \kappa\}. \end{array}

2.2. Metric weights

Definition 6 Let {X} be a metric space, let {q\in\mathbb{N}}. Define the metric weight {MW_q (X)} as the supremum of numbers {\nu} such that there exist arbitrarily small open sets {U\subset M} and nonzero straight cohomology classes {\kappa\in H^q (U,{\mathbb R})} with finite {\nu}-norm {||\kappa||_{\nu}<+\infty}.

Proposition 7 In a Riemannian manifold with boundary, all straight cocycles {c} representing a nonzero class {\kappa} of degree {q} satisfy {|c|_{\epsilon}\geq \textrm{const.}(\kappa)\,\epsilon^q}. In other words, {||\kappa||_{q}>0}.

Proof. Fix a cycle {c'} such that {\kappa(c')>0}. Subdivide it as follows : fill simplices with geodesic singular simplices, subdivide them and keep only their vertices. This does not change the homology class. The number of simplices of size {\epsilon} thus generated is {\leq \textrm{const.}(c')\,\epsilon^{-q}}. For any representative {c} of size {\epsilon} of {\kappa},

\displaystyle  \begin{array}{rcl}  \kappa(c')=c(c')\leq \textrm{const.}\,\epsilon^{-q} |c|_{\epsilon}. \end{array}

Corollary 8 Euclidean {n}-space has {MW_q \leq q} for all {q=1,\ldots,n-1} (later, we shall see that {MW_q =q}).

2.3. Hölder covariance

Proposition 9 Let {f:X\rightarrow Y} be a {C^{\alpha}}-Hölder continuous homeomorphism. Let {\kappa\in H^q (Y,{\mathbb R})}. Then

\displaystyle  \begin{array}{rcl}  ||\kappa||_{\nu}<+\infty\Rightarrow||f^{*}\kappa||_{\nu\alpha}<+\infty. \end{array}

In particular, {MW_q (X)\geq \alpha MW_q (Y)}.

Consequence: {\alpha(M)\leq\frac{q}{MW_q (M)}} for all {q}.

Proof. If {\sigma} is a straight simplex of size {\epsilon} in {X}, {f(\sigma)} has size {\epsilon' \leq ||f||_{C^{\alpha}}\,\epsilon^{\alpha}} in {Y}. If {c} is a representative of {\kappa}, {f^* c} is a representative of {f^{*}\kappa}, and

\displaystyle  \begin{array}{rcl}  \epsilon'^{-\nu}|c|_{\epsilon'} &\geq&\epsilon'^{-\nu}|c(f(\sigma))|\\ &=&\epsilon'^{-\nu}|f^* c(\sigma)|\\ &\geq&||f||_{C^{\alpha}}^{-\nu}\,\epsilon^{-\nu\alpha}|f^* c(\sigma)|. \end{array}

Therefore

\displaystyle  \begin{array}{rcl}  \epsilon^{-\nu\alpha}|f^* c|_{\epsilon} \leq ||f||_{C^{\alpha}}^{\nu}\,\epsilon'^{-\nu}|c|_{\epsilon'}. \end{array}

This leads to

\displaystyle  \begin{array}{rcl}  ||f^* \kappa||_{\nu\alpha}\leq ||f||_{C^{\alpha}}^{\nu}\,||\kappa||_{\nu}. \end{array}

3. Differential forms

3.1. Weights of differential forms

Let {G} be a Carnot group with Lie algebra {\mathfrak{g}}. Left-invariant differential forms on {G} split into homogeneous components under the dilations {\delta_\epsilon},

\displaystyle  \begin{array}{rcl}  \Lambda^* \mathfrak{g}^* =\bigoplus_{w}\Lambda^{*,w} \quad \textrm{where}\quad \Lambda^{*,w} =\{\alpha\,|\,\delta_{\epsilon}^{*}\alpha=\epsilon^{w}\alpha\}. \end{array}

Therefore Lie algebra cohomology splits {H^{q}(\mathfrak{g})=\bigoplus_{w}H^{q,w}(\mathfrak{g})}.

Example 2 If {G=Heis^{2m+1}} is the Heisenberg group, for each degree {q\not=0}, {2m+1},

\displaystyle  \begin{array}{rcl}  \Lambda^q \mathcal{G}^* =\Lambda^{q,q}\oplus\Lambda^{q,q+1}, \end{array}

where {\Lambda^{q,q}=\Lambda^{q}(V^1 )^*} and {\Lambda^{q,q+1}=\Lambda^{q-1}(V^1 )^* \otimes (V^2 )^*}.

This gradation by weight depends on the group structure. What remains for general sub-Riemannian manifolds is a filtration.

Definition 10 Let {(M,\Delta)} be a sub-Riemannian manifold, {m\in M}. Say a {q}-form {\alpha} on {T_m M} has weight {\geq w} if it vanishes on {q}-vectors of {\Delta^{i_1}\otimes\cdots \otimes \Delta^{i_q}} whenever {i_1 + \cdots +i_q <w}. If {(M,\Delta)} is equiregular, such forms constitute a subbundle {\Lambda^{q,\geq w}T^* M}. The space of its smooth sections is denoted by {\Omega^{*,\geq w}}.

Note that each {\Omega^{*,\geq w}} is a differential ideal in {\Omega^*}.

3.2. Algebraic versus metric weights

Proposition 11 Let {M} be an equiregular sub-Riemannian manifold. Let {U\subset M} be a bounded open set with smooth boundary. Let {\omega} be a closed differential form on {U} of weight {\geq w}. Then, for every {\epsilon} small enough, the cohomology class {\kappa\in H^q (U,{\mathbb R})} of {\omega} can be represented by a straight cocycle {c_{\epsilon}} (maybe defined on a slightly smaller homotopy equivalent open set) such that {|c_{\epsilon}|_{\epsilon}\leq \textrm{const.}\,\epsilon^w }. In other words, {||\kappa||_{w}<+\infty}.

Proof. In the case of a Carnot group {G}. Use the exponential map to push affine simplices in the Lie algebra to the group. Fill in all straight simplices in {G} of unit Carnot-Carathéodory size with such affine singular simplices. Apply {\delta_{\epsilon}} and obtain a filling {\sigma_{\epsilon}} for each straight simplex {\sigma} in {G} of Carnot-Carathéodory size {\epsilon}. Define a straight cochain {c_{\epsilon}} of size {\epsilon} on {U} by

\displaystyle  \begin{array}{rcl}  c_{\epsilon}(\sigma)=\int_{\sigma_{\epsilon}}\omega. \end{array}

Since {\omega} is closed, Stokes theorem shows that {c_{\epsilon}} is a cocycle. Its cohomology class in {H^q (U',{\mathbb R})\simeq H^q (U,{\mathbb R})} is the same as {\omega}‘s. Furthermore,

\displaystyle  \begin{array}{rcl}  |c_{\epsilon}(\sigma)| =\int_{\sigma_1}\delta_{\epsilon}^{*}\omega \leq V\,||\delta_{\epsilon}^{*}\omega||_{\infty} \leq\textrm{const.}(\omega)\,\epsilon^{w}. \end{array}

3.3. Algebraic weights

Definition 12 Let {M} be a sub-Riemannian manifold. Define the algebraic weight {AW_q (M)} as the largest {w} such that there exists arbitrarily small open sets with smooth boundary {U\subset M} and nonzero classes in {H^q (U,{\mathbb R})} which can be represented by closed differential forms of weight {\geq w}.

Remark 1 Equiregular sub-Riemannian manifolds satisfy {MW_q \geq AW_q}.

Corollary 13 Let {M} be a sub-Riemannian manifold. Then for all {q=1,\ldots,n-1}, {\alpha(M)\leq\frac{q}{AW_q}}.

So our goal now is to show that for certain sub-Riemannian manifolds, for certain degrees {q}, in every open set, every closed differential {q}-form is cohomologous to a form of high weight.

4. Estimates on algebraic weights

4.1. Rumin’s complex

Rumin’s complex is a subcomplex of the de Rham complex, homotopic to it, consisting of differential forms of preferably high weights. The construction requires to invert the weight {0} component {d_0} of {d}. {d_0} identifies with the exterior differential on left-invariant forms on tangent Lie algebras {\mathfrak{g}_m}. So one needs that the cohomology {m\mapsto H^{q,w}(\mathfrak{g}_m)} be constant, whence the word equihomological. It turns out that the obstruction for cohomologing {q}-forms towards weight {>w} is {H^{q,w}(\mathfrak{g}_m)}.

Theorem 14 (Rumin 2005) Let {M} be a equihomological sub-Riemannian manifold. Assume that there exists a point {m\in M} such that, in the cohomology of the tangent Lie algebra {\mathfrak{g}_m}, {H^{q,w'}(\mathfrak{g}_m)=0} for all {w'<w}. Then {AW_q (M)\geq w}.

On Carnot groups, the grading of cohomology is compatible with Poincaré duality, {H^{q,w}(\mathfrak{g})=H^{n-q,Q-w}(\mathfrak{g})}. So

\displaystyle  \begin{array}{rcl}  \exists m~H^{n-q}(\mathfrak{g}_m)=H^{n-q,\leq Q-w}(\mathfrak{g}_m)\quad\Rightarrow\quad AW_q (M)\geq w. \end{array}

Example 3 Degree {n-1}. On any Carnot Lie algebra {\mathfrak{g}}, closed {1}-forms belong to {(V^1)^* =\Lambda^{1,1}}, so {H^1 (\mathfrak{g})=H^{1,1}(\mathfrak{g})}, and {AW_{n-1}(M)\geq Q-1}.

4.2. More examples

Example 4 Contact case. Closed {m}-forms belong to {\Lambda^{m,m}}. Therefore {H^m (\mathfrak{g})=H^{m,m}(\mathfrak{g})}, and {AW_{m+1}(M)\geq m+2}.

Indeed, if {\omega\in\Lambda^{m,m+1}}, {\omega=\theta\wedge\phi} where {\theta\in (V^2)^*}, {\phi\in\Lambda^{m-1,m-1}}, {(d\omega)^{m+1,m+1}=(d\theta)\wedge\phi\not=0} since {d\theta} is symplectic on {\Delta}.

Example 5 Generic sub-Riemannian case. Let {h=dim(\Delta)} and {k} be such that {h-k\geq(n-h)k}. Then {H^{k}(\mathfrak{g}_m)=H^{k,k}(\mathfrak{g}_m)}, thus {AW(M)\geq Q-k}.

Let {\theta} be a {{\mathbb R}^{n-h}}-valued {1}-form defining {\Delta}. Say a {k}-plane {S\subset \Delta_m} is isotropic if {d\theta_{|S}=0}. Say {S} is regular if the map {\Delta_m \rightarrow Hom(S,{\mathbb R}^{n-h})}, {u\mapsto (\iota_u d\theta)_{|S}} is onto. {h-k\geq(n-h)k} is a necessary condition for existence of regular isotropic horizontal {k}-planes. It is generically sufficient. When it holds, closed left-invariant {k}-forms have to be of weight {k}, so {H^{k}(\mathfrak{g}_m)=H^{k,k}(\mathfrak{g}_m)} for {m\in M}.

Remark 2 The method just exposed seems to cover all presently known results on the Hölder homeomorphism problem.

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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