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Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces

Marcos Agnoletto, Julio C. Correa Hoyos, Márcio Fabiano da Silva, Stefano Nardulli

TL;DR

This work develops an intrinsic theory for $m$-dimensional varifolds in Alexandrov spaces with double-sided curvature bounds, culminating in Allard's Interior $\varepsilon$-Regularity Theorem in a nonsmooth ambient setting. By building an intrinsic varifold framework (Definitions, First Variation, Monotonicity, Constancy, and Tangent/Rectifiability) and then deriving Caccioppoli inequalities, density estimates, and affine-approximation results, the paper achieves a Euclidean-like regularity theory that crucially avoids Nash embeddings. The results provide explicit constants depending only on dimension, curvature, and injectivity radius, enabling a density-dominated approach to represent varifolds locally as $\mathcal{C}^{1,\alpha}$ graphs over intrinsic submanifolds. The methodology leverages harmonic submanifolds and harmonic coordinates to manage curvature terms and delivers a robust regularity toolkit for nonsmooth metric-measure spaces with controlled geometry.

Abstract

In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, for $p \in \mathbb{R}$ such that $p>m$, in Alexandrov spaces of dimension $n$ with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension $n$ whose metric tensor is at least of class $\mathcal{C}^2$, without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on $n$, $m$, the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the $\mathcal{C}^{1,α}$ topology, for every $α\in ]0,1[$ (in fact, it is enough with the $W^{2,q}$ topology for some suitable $q$ large enough).

Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces

TL;DR

This work develops an intrinsic theory for -dimensional varifolds in Alexandrov spaces with double-sided curvature bounds, culminating in Allard's Interior -Regularity Theorem in a nonsmooth ambient setting. By building an intrinsic varifold framework (Definitions, First Variation, Monotonicity, Constancy, and Tangent/Rectifiability) and then deriving Caccioppoli inequalities, density estimates, and affine-approximation results, the paper achieves a Euclidean-like regularity theory that crucially avoids Nash embeddings. The results provide explicit constants depending only on dimension, curvature, and injectivity radius, enabling a density-dominated approach to represent varifolds locally as graphs over intrinsic submanifolds. The methodology leverages harmonic submanifolds and harmonic coordinates to manage curvature terms and delivers a robust regularity toolkit for nonsmooth metric-measure spaces with controlled geometry.

Abstract

In this paper, we prove Allard's Interior -Regularity Theorem for -dimensional varifolds with generalized mean curvature in , for such that , in Alexandrov spaces of dimension with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension whose metric tensor is at least of class , without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on , , the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the topology, for every (in fact, it is enough with the topology for some suitable large enough).

Paper Structure

This paper contains 17 sections, 53 theorems, 619 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theory of general varifolds in abstract Riemannian manifolds
  3. Intrinsic definition of varifolds in Riemannian manifolds
  4. First variation of a varifold
  5. Monotonicity formula
  6. Constancy theorem
  7. Varifold tangent and rectifiability theorem
  8. Allard's Interior E-Regularity Theorem and its consequences
  9. Preliminaries on submanifolds of Riemannian manifolds
  10. Caccioppoli inequality
  11. Density estimates
  12. Affine approximation
  13. Lipschitz approximation
  14. Harmonic approximation
  15. Tilt-excess decay
  16. ...and 2 more sections

Key Result

Theorem A

Let $m,n \in \mathbb{N}$ and $p \in \mathbb{R}$ such that $1 \leq m < n$ and $m < p < \infty$; in case $m=1$, we require that $p \geq 2$, $(X,g)$ be an $n$-dimensional Alexandrov space with double-sided bounded intrinsic sectional curvature, $\xi \in X$, and $\varepsilon \in ]0,1[$. Then there exist for $||V||$-a.e. $x \in B_{d_g}(\xi,\rho)$, and for all $Y \in \mathfrak{X}^1_c(X)$ such that $\m

Figures (5)

  • Figure 1: The function $\overline{\phi}$ (in blue) is a continuous extension of $\phi$ to $\mathrm{Gr}_m\left(\overline{B_{d_g}(x,\rho)}\right)$. The red regions represent the smooth decay added for the construction of the function $\Phi$ as described above.
  • Figure 2: The function $c(t,b)$.
  • Figure 3: The function $\frac{c(t,b)-1}{t}$.
  • Figure 4: Illustration of two different intersection scenarios between the $m$-rectifiable set $\mathrm{supp}(|V|)$ (in purple) and the submanifold $\Sigma$ (in blue). Points where the tangent spaces coincide are highlighted in green, otherwise are highlighted in yellow, which are excluded from the integral calculation. Additionally, by \ref{['equa3proof:lemm:excess:section:DensityEstimatesApp']}, the regions highlighted in purple have null integrating factor.
  • Figure 5: The point $x$ in the submanifold $\Sigma$ (in blue) and the point $y$ in $\mathrm{supp}(||V||)$ (in red). In the square on the right side of the figure, we illustrate how the distance in the Riemannian setting encompasses both the sectional curvature of the ambient Riemannian manifold and the curvature of the submanifold $\Sigma$, as can be seen in \ref{['equa5proof:coro:poslemm:section:DensityEstimatesApp']}.

Theorems & Definitions (156)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Corollary A
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Example 2.1
  • Definition 2.4
  • ...and 146 more