A generalization of Reifenberg's theorem in R^N for flat cones

Abstract: In this paper we prove that if a closed set in R^N is close to a cone over a simplicial complex at each point and at each scale, then it is locally bi-Hölder equivalent to such a cone. This generalizes Reifenberg's Topological Disk Theorem in 1960 and G. David, T. De Pauw and T. Toro's result in 2008.

• The paper introduces a generalization of Reifenberg's theorem by parameterizing closed sets in R^N as approximations of flat cones over simplicial complexes.
• It develops a hierarchical, recursive method to construct local bi-Hölder homeomorphisms with explicit quantitative distortion and displacement estimates.
• The framework unifies previous Reifenberg-type results, extending applications to geometric measure theory and regularity analyses in minimal surfaces and elliptic PDEs.

Generalization of Reifenberg's Theorem for Flat Cones in $\mathbb{R}^N$

This paper "A generalization of Reifenberg's theorem in $\mathbb{R}^N$ for flat cones" (2604.11418) proposes a broad extension of Reifenberg's topological disk theorem to a large class of closed sets which resemble cones over simplicial complexes at all locations and scales in $\mathbb{R}^N$. The results unify and generalize classical Reifenberg-type regularity theorems and recent bi-Hölder parametrizations for singular sets, laying a comprehensive foundation for the description and analysis of sets approximable by multifaceted conical geometries.

Background and Motivation

The classical Reifenberg theorem characterizes closed sets quantitatively close (in Hausdorff sense) to $n$-planes in $\mathbb{R}^N$ at every point and scale, ensuring that the set is homeomorphic via a bi-Hölder mapping to a standard $n$-disk. Efforts since have divided between more quantitative rectifiability measures (using Jones' $\beta$-numbers, e.g., [NV17]) and generalization to sets modeled on more singular geometries—particularly minimal cones. Previous results in this latter direction include parametrizations in $\mathbb{R}^3$ for sets modeled on classical minimal cones (planes, $\mathbb{Y}$, and $\mathbb{T}$ sets) [DDT08], and frameworks for tangent set stratification in singular measure theory [BL15], but parameterizations for higher and more general cone types remained open.

This work addresses the classification and characterization problem for arbitrary cones over finite simplicial complexes and their products with Euclidean factors, thus subsuming all previously considered models, and yielding a complete regularity and parametrization theory for this large geometric setting.

Statement of Main Results

The principal contribution is a parametrization theorem for closed sets $\mathbb{R}^N$0 that, at every point $\mathbb{R}^N$1 and scale $\mathbb{R}^N$2, can be approximated (to accuracy $\mathbb{R}^N$3) by an element $\mathbb{R}^N$4 from a prescribed finite family (up to isometries) $\mathbb{R}^N$5 of "model" cones of some type $\mathbb{R}^N$6 ($\mathbb{R}^N$7). Each such set is locally bi-Hölder homeomorphic (via a controlled, canonical mapping $\mathbb{R}^N$8) to a corresponding cone, with optimal uniform estimates on both the map and the approximation.

Structure of Model Cones

These "types" are defined inductively as products of an $\mathbb{R}^N$9-dimensional cone over a simplicial complex and $\mathbb{R}^N$0, equipped with a nonflatness condition to prevent reducible or degenerate geometries (e.g., planes arising as unions of co-planar faces). Model cones admit a natural hierarchy of spines $\mathbb{R}^N$1, corresponding to the union of faces of dimension $\mathbb{R}^N$2, and every type is stratified accordingly.

Main Parametrization Theorem

Let $\mathbb{R}^N$3 be a finite collection (modulo isometry) of such models. If $\mathbb{R}^N$4 admits at every $\mathbb{R}^N$5, $\mathbb{R}^N$6 an approximant $\mathbb{R}^N$7 (the iterated blow-up closure of $\mathbb{R}^N$8) such that the normalized Hausdorff distance $\mathbb{R}^N$9, then there exist $n$0 and a homeomorphism $n$1 such that $n$2 is bi-Hölder on $n$3, parameterizes $n$4 locally, and satisfies explicit quantitative bounds:

• For all $n$5,

$n$6

• $n$7 (pointwise displacement)
• $n$8

This theorem implies that $n$9 is locally bi-Hölder equivalent (with dimension and geometry specified by the type) to a model cone, for arbitrarily small approximation error.

Inductive Construction and Stratification

A core methodological advance of this work is the systematic, inductive treatment of the parameterization—built recursively on the stratification of the model cone and of $\mathbb{R}^N$0 itself.

Figure 1

Figure 2: Hierarchical decomposition of spines and their surrounding neighborhoods for a 3-cone, visualizing open neighborhoods ($\mathbb{R}^N$1) associated to binary words which classify the region relative to spines of various dimensions.

Each cone and its spines are organized so that, at each scale, the set is partitioned into regions modeled on neighborhoods of different spine dimensions. Open sets $\mathbb{R}^N$2 parameterized by binary words encode the relative position to these spines, supporting a recursive partition of unity and map construction. This hierarchical approach circumvents the need for an enumeration of local geometries—an issue in previous literature when only finitely many cases were known.

Stepwise Parametrization

The map $\mathbb{R}^N$3 is defined on each stratum of the spine as a limit of homeomorphisms $\mathbb{R}^N$4 (with $\mathbb{R}^N$5), each $\mathbb{R}^N$6 deforms $\mathbb{R}^N$7 by an explicit, localized motion controlled by the partition of unity. Inside conical regions between spines, the map is constructed as the (piecewise) orthogonal projection onto the "nearest" spine, and is extended across strata using a tree-structured extension procedure (as shown in Figure 2).

At each step, regularity, injectivity, and consistency conditions across intersections of neighborhoods and spines are maintained with quantitative control.

Structural Lemmas and Stratification Properties

The argument establishes:

• Existence and uniqueness of type for nonflat cones (Ambiguity is precluded by the nonflatness condition; see Proposition 2.7 and associated remark).
• Complete stratification of any set close to models into layers $\mathbb{R}^N$8 consisting of those points whose small-scale type is $\mathbb{R}^N$9 (Proposition 3.7).
• Stability of strata and cones under blow-ups; closure and containment relations analogous to polyhedral skeleta.
• Quantitative separation of branches and nontrivial lower bounds for distances across different cone types.
• For each $n$0, $n$1 is locally close to an $n$2-plane in the sense of the original Reifenberg theorem, except possibly in small neighborhoods of lower strata.

Figure 2

Figure 2: Regions around the base point $n$3 of a 3-cone. Solid lines and colored branches represent 1/2/3-dimensional spines; gray zones illustrate conical open sets $n$4 and their complements $n$5, supporting the local parameterization and inductive extension.

Numerical and Theoretical Strength

The main theorem yields uniform constants of regularity, and the construction explicitly produces the bi-Hölder map $n$6. Quantitative bounds on distortion, displacement, branching separation (controlled by $n$7), and layer transitions ($n$8) are central to the construction.

This framework subsumes all previously known Reifenberg and De Pauw-David-Toro-type parameterizations: for $n$9-planes, for minimal cones in lower dimensions, and for products of these with Euclidean factors. The approach is also fully constructive, avoiding non-quantitative global arguments.

Implications and Future Development

Practical implications include:

• Applicability to geometric measure theory problems involving rectifiability and parameterization near singularities, such as analysis of minimal surfaces, free boundary problems, or zero sets of analytic functions where tangent cones are not necessarily smooth.
• The establishment of a complete local classification of geometric types for sets approximable by cones over simplicial complexes, which can be translated directly to questions of regularity for solutions to elliptic PDEs or for limiting objects in geometric flows.

Theoretical ramifications:

• Provides a unifying formalism for blow-up analysis and singular set stratification, in line with the tangent measure and $\beta$0-number approaches for quantitative rectifiability.
• The recursive, tree-based local cover and map extension method is likely to be adaptable to more general stratified and singular settings, including possibly to non-Euclidean or metric spaces with cone-type tangent structures.

Future directions may include:

• Quantitative versions (in measure or density) via the adaptation of Jones' $\beta$1-numbers and higher order geometry, bridging toward analytic regularity (see [NV17], [ENV19]).
• Extension toward an analogue in Riemannian or metric space settings, or in the presence of further singularities.

Conclusion

This paper provides a sharp and highly general parametrization theorem for sets close to arbitrary cones over finite simplicial complexes in $\beta$2. The new hierarchical, stratified, and inductive techniques bridge the gap between classical Reifenberg theory and modern singular blow-up analysis, providing a canonical description of the geometry of multi-scale, multi-type singular sets. The explicit, constructive nature of the parametrization and the general model class set a new standard for regularity theorems in geometric analysis and geometric measure theory.

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References:

• [NV17] Naber, A.; Valtorta, D.: Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps, Annals of Mathematics (2017).
• [DDT08] David, G.; De Pauw, T.; Toro, T.: A generalization of Reifenberg’s theorem in $\beta$3 for sets close to minimal cones, Annales de l'Institut Fourier, 58(6):1971–2068, 2008.
• [BL15] Badger, M.; Lewis, S.: Local set approximation: Geometry and measure, Transactions of the AMS, 2015.