Bi-H"older invariants in o-minimal structures
An V. Q. Huynh, Minh B. Nguyen, Nhan X. V. Nguyen, Minh Q. Vu
TL;DR
The paper investigates which metric-invariant properties of definable germs are preserved under bi-α-Hölder equivalence in polynomially bounded o-minimal structures. It proves the existence of an α0 ∈ (0,1) so that, for α≥α0, bi-α-Hölder equivalence preserves (i) Lipschitz normal embedding, (ii) tangent cone dimension, and (iii) the homotopy type of the links of tangent cones. It leverages cone- and ST-equivalence techniques along with o-minimal tools to extend bi-Lipschitz rigidity results to the near-Lipschitz regime, with an application yielding a stronger smoothness criterion for complex analytic germs bi-α-Hölder near 1 to Euclidean spaces. The work also demonstrates that these rigidity phenomena fail in non-polynomially bounded o-minimal structures via explicit counterexamples. These results deepen our understanding of how metric and topological tangent data behave under intermediate equivalences between definable germs.
Abstract
We prove that for any two definable germs in a polynomially bounded o-minimal structure, there exists a critical threshold $α_0 \in (0,1)$ such that if these germs are bi-$α$-H"older equivalent for some $α\ge α_0$, then they satisfy the following: \begin{itemize}[label=$\circ$] \item The Lipschitz normal embedding (LNE) property is preserved; that is, if one germ is LNE then so is the other; \item Their tangent cones have the same dimension; \item The links of their tangent cones have isomorphic homotopy groups. \end{itemize} As an application, we show that a complex analytic germ that is bi-$α$-H"older homeomorphic to the germ of a Euclidean space for some $α$ sufficiently close to $1$ must be smooth. This provides a slightly stronger version of Sampaio's smoothness theorem, in which the germs are assumed to be bi-$α$-H"older homeomorphic for every $α\in (0,1)$.