# Are Hölder manifolds a thing?

We know topological manifolds and we know Lipschitz manifolds. It seems that "Hölder manifolds" should be somewhere in between but not much seems published about them.

In the context of this question, a Hölder manifold is a topological manifold equipped with an atlas whose transition functions are locally Hölder continuous (that is, belong to the Hölder class $$C^\alpha$$ for some fixed $$\alpha \in (0,1)$$.

The definition above just mimics the definition of Lipschitz manifolds.

What is known about Hölder manifolds, and is there any particular reason less has been published about them than other classes of manifolds?

• Not quite what you're asking for but let me comment that $C^{1+\alpha}$ manifolds were considered in the context of topological dynamics by Sullivan in the 80ies. math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0079.pdf Dec 22, 2019 at 10:26
• Hölder structures play a role in the study of maps between hyperbolic spaces because the boundary of a hyperbolic space has a canonical Hölder structure, albeit not a canonical $C^\infty$-structure: any two visual metrics on the boundary of a hyperbolic space are Hölder-equivalent. Quasi-isometries between spaces induce bi-Hölder maps between the boundaries. See I.Kapovich-N.Benakli, Section 3. Dec 22, 2019 at 10:53
• For instance, Lipschitz perturbations of a hyperbolic operator are Hölder-conjugate, (and in general not Lipschitz-conjugate, even in smooth cases), by the Hartman-Grobman thm. Dec 22, 2019 at 12:26

I don't think this has been studied but I would note two things. First, Sullivan proved that in dimensions $$n\ne 4$$ any topological manifold $$M^n$$ has a Lipschitz structure and any two such structures are Lipschitz equivalent. Therefore the same is true for Hölder structures. So your question is only interesting in dimension 4. In this case AFAIK nothing is known but I would guess that the same is true. Note that this would be different from the Lipschitz case where Sullivan and Donaldson showed that there are 4-manifolds without Lipschitz structures and there are homeomorphic 4-manifolds with nonequivalent Lipschitz structures. But I don't believe this can happen with Hölder structures because it seems to me that it should be possible to make Casson handles Hölder so Freedman's theory should work in the Hölder category for example. This is pure speculation though and one would need to look at the details of the construction and see if it works.