[Paper Review] $C^{1+α}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$

Guy David|ArXiv.org|Jun 12, 2008

Point processes and geometric inequalities7 references4 citations

TL;DR

This paper presents a new, elementary proof of Jean Taylor's result on the $C^{1+\alpha}$-regularity of two-dimensional almost-minimal sets in $\mathbb{R}^n$, extending it partially to higher codimensions. By using harmonic replacement on conical surfaces over small Lipschitz graphs and leveraging a local separation result and a generalized Reifenberg parameterization theorem, the authors show that such sets are locally $C^{1+\alpha}$-equivalent to minimal cones, with decay estimates on density excess controlling the regularity. The key contribution is a quantitative decay of the density excess $f(r) = \theta(r) - d(x)$, which implies $C^{1+\alpha}$ regularity under suitable conditions.

ABSTRACT

We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\R^3$ are locally $C^{1+α}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in $\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point.

Motivation & Objective

• To provide a new, more elementary proof of Taylor's result on the $C^{1+\alpha}$-regularity of 2D almost-minimal sets in $\mathbb{R}^3$.
• To extend this regularity result partially to two-dimensional almost-minimal sets in $\mathbb{R}^n$ for $n > 3$.
• To establish quantitative decay estimates for the density excess $f(r) = \theta(r) - d(x)$, which governs the regularity of the set.
• To show that the final regularity result depends on the list of minimal cones obtained as blow-up limits at a point in $\mathbb{R}^n$.

Proposed method

• Use of harmonic replacement on conical surfaces over small Lipschitz graphs in the sphere to reduce area, exploiting the fact that homogeneous degree-one Lipschitz functions are far from harmonic.
• Application of a local separation result from [D3] to reduce the problem to unions of cones over arcs of Lipschitz graphs.
• Generalization of Reifenberg’s topological disk theorem via [DDT] to establish bi-Hölder and ultimately $C^{1+\alpha}$-regularity.
• Introduction of the density excess $f(r) = \theta(r) - \lim_{t \to 0^+} \theta(t)$ as a key quantity to measure deviation from minimal cones.
• Derivation of a differential inequality for $f(r)$ in Theorem 4.5, showing its decay like a power of $r$, which controls the Hausdorff distance to minimal cones.
• Use of Taylor expansion and geometric control (e.g., in Lemma 14.27) to estimate the length of deformed curves on the sphere and verify the full length property.

Experimental results

Research questions

• RQ1 Can the $C^{1+\alpha}$-regularity of 2D almost-minimal sets in $\mathbb{R}^3$ be re-proven with a more elementary method that avoids currents and compactness arguments?
• RQ2 To what extent can the regularity result for 2D almost-minimal sets in $\mathbb{R}^3$ be extended to $\mathbb{R}^n$ for $n > 3$?
• RQ3 How does the decay rate of the density excess $f(r)$ relate to the $C^{1+\alpha}$-regularity of the set?
• RQ4 What role does the list of minimal cone blow-up limits play in determining the final regularity of the set in higher codimensions?
• RQ5 Can the full length property (Definition 4.10) be verified independently on connected components of the set on the sphere?

Key findings

• The density excess $f(r) = \theta(r) - d(x)$ decays at a definite power rate, which is essential for proving $C^{1+\alpha}$-regularity.
• The decay of $f(r)$ is established via a differential inequality in Theorem 4.5 and a main example in Lemma 5.11, showing that $f(r)$ decays like $r^\beta$ for some $\beta > 0$.
• The Hausdorff distance from $E$ to a minimal cone in small balls is controlled by $f(r)$, as shown in Theorem 11.4.
• The final $C^{1+\alpha}$-regularity result is deduced from a variant of Reifenberg’s theorem [DDT], which provides a bi-Hölder equivalence that improves to $C^{1+\alpha}$ under the decay of $f(r)$.
• The full length property (Definition 4.10) is shown to be verifiable component-wise on $K = E \cap \partial B(x,r)$, and is preserved under small deformations.
• In the case $n > 3$, the $C^{1+\alpha}$-regularity of $E$ may depend on the specific list of minimal cones obtained as blow-up limits at a point, due to the complexity of possible cone configurations.