## 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

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

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

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 . 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'. 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.