Sampaio's Smoothness Theorem

• Sampaio's Smoothness Theorem is a fundamental rigidity result asserting that if a singular germ is bi-Hölder equivalent to a Euclidean germ, then it is analytically smooth.
• It employs key metric invariants like Lipschitz Normal Embedding, tangent cone dimensions, and homotopy groups of links to establish and diagnose smoothness properties.
• Recent extensions using o-minimal geometry, including Hardt triviality and Lipschitz-conic retraction, generalize the theorem to a broader class of definable and analytic germs.

Sampaio’s Smoothness Theorem establishes a fundamental rigidity phenomenon for singularities in metric geometry, especially within the context of complex analytic and definable set-germs. It asserts that substantial “metric regularity”—expressed as bi-Hölder equivalence between a singular germ and the standard Euclidean germ—implies analytic smoothness. Recent refinements utilize tools from o-minimal geometry and confirm that analytic smoothness can be deduced from the existence of a single homeomorphism with Hölder exponent sufficiently close to $1$, greatly strengthening and generalizing Sampaio’s original result.

1. Formal Statement and Context

Let $(X,0)\subset\mathbb{C}^n$ be the germ of a complex analytic set at $0$. Sampaio’s theorem states that if $(X,0)$ is bi-$\alpha$-Hölder homeomorphic to the standard Euclidean germ $(\mathbb{R}^k,0)$ for every $\alpha\in(0,1)$, then $(X,0)$ is in fact smooth—i.e., analytically isomorphic to $(\mathbb{C}^k,0)$. This equivalence means there exists a homeomorphism $\varphi:(X,0)\to(\mathbb{R}^k,0)$ and $L\geq 1$ such that for all $x,x'$ near $0$: $$\frac{1}{L} \|x-x'\|^{1/\alpha} \leq \|\varphi(x) - \varphi(x')\| \leq L \|x-x'\|^\alpha.$$ Recently, as shown in "Bi-Hölder invariants in o-minimal structures" (Huynh et al., 23 Nov 2025), the hypothesis can be weakened: it suffices to have such a homeomorphism for some $\alpha \geq \alpha_0$, where $0 < \alpha_0 < 1$ is a threshold depending only on $X$. The result generalizes to any definable germ in a polynomially bounded o-minimal structure.

2. Key Metric and Topological Invariants

The theorem leverages invariants that are preserved under bi-$\alpha$-Hölder equivalence (for suitable $\alpha$):

• Lipschitz Normal Embedding (LNE): $(X,0)$ is LNE if the intrinsic (inner) metric is bilipschitz equivalent to the outer Euclidean metric. This rules out cusp-like pathologies and ensures metric regularity.
• Tangent Cone Dimension: The tangent cone $C_0(X)$ at $0$ reflects the first-order metric geometry. Bi-$\alpha$-Hölder equivalence with $\alpha$ sufficiently close to $1$ forces equality of dimensions of tangent cones ($\mathrm{dim}\,C_0(X) = \mathrm{dim}\,C_0(Y)$).
• Homotopy Groups of the Link: The link $\mathrm{Link}(C_0(X)) = C_0(X) \cap S^{n-1}$ determines the topological type of the singularity via its homotopy groups $\pi_i$. The theorem ensures $$\pi_i(\mathrm{Link}(C_0(X))) \cong \pi_i(\mathrm{Link}(C_0(Y)))$$.

3. Proof Strategy and the Role of the Hölder Exponent Threshold

The proof utilizes advanced metric and topological techniques from o-minimal geometry and singularity theory:

• Preservation of LNE: If $(X,0)$ is LNE but $(Y,0)$ is not, the intrinsic metric on $(Y,0)$ exhibits degenerate scaling (like $\|x-y\|^\beta$, $\beta<1$). Under a bi-$\alpha$-Hölder map, the contradiction arises unless $\alpha^2 > \beta$, so for sufficiently large $\alpha$ (i.e., $\alpha$ close to $1$) LNE must be preserved.
• Dimension of Tangent Cones: Sea-tangle neighborhoods $ST_d(X,C)=\{x:\mathrm{dist}(x,X) \leq C\|x\|^d\}$ are used; a bi-$\alpha$-Hölder map distorts the exponent by $\alpha^2$. The volume asymptotics of these neighborhoods are sensitive to dimension; inconsistency in scaling reveals dimension mismatch unless $\alpha$ exceeds a threshold $\alpha_0$.
• Homotopy Groups of Links: Hardt triviality and Lipschitz-conic retraction in o-minimal geometry guarantee stability of link homotopy under bi-$\alpha$-Hölder maps for $\alpha\geq\alpha_0$. Isomorphisms of homotopy groups between the links then follow.

Once these invariants are stable under the equivalence, classical rigidity results (Prill’s theorem for cones, and Birbrair-Fernandes-Lê-Sampaio's smoothness criterion for LNE germs with linear tangent cones) yield analytic smoothness.

4. Extension to O-minimal Structures and New Techniques

Where Sampaio’s original theorem was proved for subanalytic or complex analytic germs, the recent generalization (Huynh et al., 23 Nov 2025) replaces analytic arguments with systematic o-minimal tools, including:

• Łojasiewicz Inequality: Ensures control over metric degeneracy in polynomially bounded o-minimal settings.
• Hardt Triviality: Provides deformation retracts of parameterized neighborhoods, crucial for link invariance.
• Lipschitz-conic Retraction: Facilitates control of metric neighborhoods in definable sets.

This extension covers germs definable in semialgebraic, globally subanalytic, and Pfaffian categories, with the rigidity threshold $\alpha_0$ determined by the geometry of the germ.

5. Comparative Perspective and Relationships to Related Rigidity Theorems

Sampaio’s theorem can be viewed as a metric analogue of the classical Myers–Steenrod theorem, in that sufficiently strong metric regularity determines the underlying smooth structure. Notably, the theorem does not require the full strength of bi-Lipschitz equivalence (i.e., $\alpha=1$), but instead exploits the rigidity of metric invariants under bi-$\alpha$-Hölder equivalence for thresholds $\alpha \to 1$.

This stands in contrast to smoothness results for subRiemannian isometries (Capogna et al., 2013), which require equiregularity and distance-preserving homeomorphisms, and to orbit space isometry smoothness theorems in Riemannian geometry (Alexandrino et al., 2011). Sampaio's theorem is uniquely focused on the interplay between analytic smoothness and strong metric equivalence in the singular setting.

6. Consequences, Applications, and Open Questions

• Rigidity of Germs: Sampaio’s theorem and its o-minimal extension show that complex analytic and definable germs with sufficiently strong metric equivalence to Euclidean germs cannot exhibit singularities.
• Metric Invariants: The identification and stability of LNE, tangent cone dimension, and link homotopy groups provide diagnostic criteria for analytic smoothness in the presence of metric regularity.
• Threshold Phenomenon: The existence of a critical exponent $\alpha_0$ below which rigidity breaks down is a central insight, inviting further exploration of sharp thresholds, dependence on singularity type, and generalization to larger classes of definable germs.
• Methodological Impact: Transition from analytic to o-minimal methods indicates the broad relevance of tame geometry for singularity theory and metric geometry.

A plausible implication is that similar smoothness rigidity phenomena may be discoverable in other geometric categories under appropriate bi-Hölder equivalence hypotheses, contingent on the preservation of key metric-topological invariants.