2025/11/23 by Andrew Huynh, Huynh, An V. Q., Minh B. Nguyen +4
Mathematics · #14B05 #32C05 #32C15 #32S50 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2511.18402
openalex publication_date 2025/11/23 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28
We prove that for any two definable germs in a polynomially bounded o-minimal structure, there exists a critical threshold α0 ∈ (0,1) such that if these germs are bi-α-H"older equivalent for some α≥ α0, then they satisfy the following: \beginitemize[label=∘] \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. \enditemize As an application, we give an simple proof that a complex analytic germ which 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∈ (0,1).