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 be an -dimensional sub-Riemannian manifold. For which does there exist locally a homeomorphism which is -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 locally a homeomorphism .
Example 1 If is a -step Carnot group, the exponential map is locally -Hölder continuous. Thus .
Theorem 2 (Gromov 1993) Let be sub-Riemannian, with Hausdorff dimension . Then .
Let be a -dimensional contact manifold. Then ( if ).
1.2. Hausdorff dimensions of subsets
Gromov’s proof uses Hausdorff dimension of subset of given topological dimension: if all subsets of topological dimension have Hausdorff dimension , then .
To get lower bounds on Hausdorff dimension of subsets, Gromov constructs local foliations by horizontal submanifolds. If there are enough such dimension foliations, all subsets of topological dimension have Hausdorff dimension , therefore .
Constructing horizontal submanifolds amounts to solving a system of PDE’s. If , it is an ODE, the method applies to all (equiregular) sub-Riemannian manifolds. Gromov solves the relevant PDE for contact -manifolds and , and, more generally, for generic -dimensional distributions, and such that .
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.
Definition 3 On a metric space , a (straight) -cochain of size is a function on -uples of diameter . Its -absolute value is
In other words, straight cochains of size coincide with simplicial cochains on the simplicial complex whose vertices are points of and a -face joins vertices as soon as all pairwise distances are . Therefore, they form a complex . There is a dual complex of chains .
Lemma 4 Assume is a manifold with boundary, or bi-Hölder homeomorphic to such, then the inductive limit complex computes cohomology.
Definition 5 Given a cohomology class and a number , one can define the -norm
2.2. Metric weights
Definition 6 Let be a metric space, let . Define the metric weight as the supremum of numbers such that there exist arbitrarily small open sets and nonzero straight cohomology classes with finite -norm .
Proposition 7 In a Riemannian manifold with boundary, all straight cocycles representing a nonzero class of degree satisfy . In other words, .
Proof. Fix a cycle such that . 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 thus generated is . For any representative of size of ,
Corollary 8 Euclidean -space has for all (later, we shall see that ).
2.3. Hölder covariance
In particular, .
Consequence: for all .
Proof. If is a straight simplex of size in , has size in . If is a representative of , is a representative of , and
This leads to
3. Differential forms
3.1. Weights of differential forms
Let be a Carnot group with Lie algebra . Left-invariant differential forms on split into homogeneous components under the dilations ,
Therefore Lie algebra cohomology splits .
Example 2 If is the Heisenberg group, for each degree , ,
where and .
This gradation by weight depends on the group structure. What remains for general sub-Riemannian manifolds is a filtration.
Definition 10 Let be a sub-Riemannian manifold, . Say a -form on has weight if it vanishes on -vectors of whenever . If is equiregular, such forms constitute a subbundle . The space of its smooth sections is denoted by .
Note that each is a differential ideal in .
3.2. Algebraic versus metric weights
Proposition 11 Let be an equiregular sub-Riemannian manifold. Let be a bounded open set with smooth boundary. Let be a closed differential form on of weight . Then, for every small enough, the cohomology class of can be represented by a straight cocycle (maybe defined on a slightly smaller homotopy equivalent open set) such that . In other words, .
Proof. In the case of a Carnot group . Use the exponential map to push affine simplices in the Lie algebra to the group. Fill in all straight simplices in of unit Carnot-Carathéodory size with such affine singular simplices. Apply and obtain a filling for each straight simplex in of Carnot-Carathéodory size . Define a straight cochain of size on by
Since is closed, Stokes theorem shows that is a cocycle. Its cohomology class in is the same as ‘s. Furthermore,
3.3. Algebraic weights
Definition 12 Let be a sub-Riemannian manifold. Define the algebraic weight as the largest such that there exists arbitrarily small open sets with smooth boundary and nonzero classes in which can be represented by closed differential forms of weight .
Remark 1 Equiregular sub-Riemannian manifolds satisfy .
Corollary 13 Let be a sub-Riemannian manifold. Then for all , .
So our goal now is to show that for certain sub-Riemannian manifolds, for certain degrees , in every open set, every closed differential -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 component of . identifies with the exterior differential on left-invariant forms on tangent Lie algebras . So one needs that the cohomology be constant, whence the word equihomological. It turns out that the obstruction for cohomologing -forms towards weight is .
Theorem 14 (Rumin 2005) Let be a equihomological sub-Riemannian manifold. Assume that there exists a point such that, in the cohomology of the tangent Lie algebra , for all . Then .
On Carnot groups, the grading of cohomology is compatible with Poincaré duality, . So
Example 3 Degree . On any Carnot Lie algebra , closed -forms belong to , so , and .
4.2. More examples
Example 4 Contact case. Closed -forms belong to . Therefore , and .
Indeed, if , where , , since is symplectic on .
Example 5 Generic sub-Riemannian case. Let and be such that . Then , thus .
Let be a -valued -form defining . Say a -plane is isotropic if . Say is regular if the map , is onto. is a necessary condition for existence of regular isotropic horizontal -planes. It is generically sufficient. When it holds, closed left-invariant -forms have to be of weight , so for .
Remark 2 The method just exposed seems to cover all presently known results on the Hölder homeomorphism problem.