Compactness of Rectifiable Varifolds

• Rectifiable varifolds are generalized submanifolds defined as Radon measures on the Grassmannian bundle, with their structure ensured by two-sided density and energy bounds.
• Uniform Ahlfors regularity and averaged height-excess energy control guarantee that sequences of varifolds converge in the weak-* topology to a d-rectifiable limit.
• These compactness results underpin variational methods in geometric measure theory and support applications in Plateau problems, anisotropic energies, and mean-curvature analyses.

A $d$-rectifiable varifold is a generalization of a smooth submanifold, defined as a Radon measure on the Grassmannian bundle of $d$-planes, whose support is contained in a countably $d$-rectifiable set and admits a multiplicity function. The compactness of rectifiable varifolds addresses the following question: under what quantitative or qualitative hypotheses does a sequence (or net) of varifolds—possibly arising as discrete, diffuse, or non-smooth approximations—converge (in the varifold, or weak-$*$ topology) to a $d$-rectifiable varifold, possibly with additional structure? This theme underlies both foundational geometric measure theory and applied settings where approximations by non-smooth, discrete, or computational representations are crucial.

1. Core Compactness Theorems for Rectifiable Varifolds

The archetype of compactness for rectifiable varifolds is the result that under two-sided density bounds and a uniform upper bound on an averaged height-excess energy, weak* limits of sequences of $d$-varifolds must be $d$-rectifiable. For $\Omega\subset\mathbb R^n$ open and $1\le d<n$, consider a sequence $V_i$ of $d$-varifolds in $\Omega$ satisfying:

• Uniform mass bound: $\sup_i \|V_i\|(\Omega)<+\infty$.
• Uniform Ahlfors regularity (density bounds):

$$C_1 r^d \le \|V_i\|(B_r(x)) \le C_2 r^d \quad \text{for } r\in(\beta_i, d(x,\Omega^c)),\ \|V_i\|\text{-a.e. } x,$$

for some sequences $\beta_i\downarrow 0$ and constants $0<C_1<C_2<\infty$.

• Uniform energy bound:

$$\sup_i \int_{\Omega\times G_{d,n}} E_{\alpha_i}(x,P, V_i)\, dV_i(x,P) < \infty,$$

where $E_{\alpha}(x,P, V)$ is the $\alpha$-truncated averaged height-excess defined by

$$E_\alpha(x,P,V) = \int_{r=\alpha}^1 \frac{1}{r^d}\int_{B_r(x)\cap \Omega} \left(\frac{d(y-x, P)}{r}\right)^2 d\|V\|(y) \frac{dr}{r},$$

and $d(v,P)$ is the Euclidean distance from $v$ to the plane $P$.

Under these conditions, any weak$^*$ limit $V$ of the $V_i$ is a $d$-rectifiable varifold in $\Omega$ (Buet, 2014). Integral rectifiability results or stronger regularity statements may require further structural or variational assumptions, such as integrality of the sequence, or mass/energy bounds on their first variation.

2. Quantitative Energy and Density Conditions

The conditions ensuring compactness and rectifiability involve two quantitative elements:

• Density Bounds (Uniform Ahlfors Regularity): The two-sided estimate

$C_1 r^d \le \|V_i\|(B_r(x)) \le C_2 r^d$

prevents both mass concentration (blow-up) and mass loss (vanishing) locally, ensuring that limiting measures are supported on sets of correct dimension and regularity.

• Averaged Height-Excess Energy: The functional $E_\alpha(x,P,V)$ measures the mean squared distance of $V$ from the plane $P$ at scale $r$ in a neighborhood of $x$, averaged over scales from $\alpha$ to 1. Uniform bound on its integral implies that $V$ locally "remains close" to some $d$-plane in an $L^2$ average sense, which is a quantitative proxy for rectifiability (Buet, 2014).

Discrete varifold approximations (e.g., pixelizations, point clouds, volumetric discretizations on a mesh) can be handled within this framework by ensuring $\alpha_i$ dominates the discretization scale, yielding the necessary energy control and allowing recovery of rectifiable limits.

3. Extension to Variational and Anisotropic Problems

The compactness framework for rectifiable varifolds underlies direct methods in geometric variational problems, including anisotropic area, Plateau-type problems, and mean-curvature functionals:

• Anisotropic Energies: For a $C^1$ anisotropic integrand $F(x,T)$ on the Grassmann bundle $G(\mathbb R^n)$ bounded by $0<\lambda\le F(x,T)\le \Lambda<\infty$ and satisfying an atomic ellipticity condition, any minimizing sequence for the energy

$\mathscr F(V) = \int F(x,T)\,dV(x,T)$

in a class of $d$-rectifiable varifolds closed under Lipschitz deformations, with a uniform lower density bound, converges (up to subsequence) to a $d$-rectifiable varifold. If the competitors are integral varifolds with uniformly locally bounded anisotropic first variation, the limit is integral (Rosa, 2016).

• Mean Curvature and Prescribed-MC Varifolds: For codimension-1 integral varifolds with bounded first variation, $L^p$ mean-curvature ($p>n$), and structural hypotheses precluding classical singularities and controlling touching singularities, similar compactness holds; the limit is a smooth $C^2$ hypersurface away from a singular set of Hausdorff dimension at most $n-7$ (Bellettini et al., 2019, Bellettini et al., 2018).

4. Methodology and Proof Outline

The argument for compactness of rectifiable varifolds proceeds through several technical steps:

• Compactness via Banach-Alaoglu: Uniform mass bounds yield tightness for the sequence of associated Radon measures, so by Banach-Alaoglu there exists a weak$^*$ convergent subsequence $V_i \rightharpoonup^* V$.
• Passage of Density Estimates: The two-sided density bounds on $V_i$ pass to the limit, ensuring $V$ is Ahlfors regular of dimension $d$.
• Limits for Energy Functionals: Uniform convergence of $E_{\alpha_i}(x,P,V_i)$ to $E_0(x,P,V)$ as $i\to\infty$ is established using uniform bounds, metric properties of $G_{d,n}$, and properties of weak convergence of measures.
• Static Rectifiability Criterion: If the limit $V$ satisfies two-sided density bounds and an integrated height-excess (or anisotropic first variation) bound, it is $d$-rectifiable (Buet, 2014, Rosa, 2016).
• Discrete Approximation Compatibility: For discretized or “diffuse” varifolds, appropriate control on $\alpha_i$ and mesh size ensures compatibility with these quantitative conditions (Buet, 2014).

5. Structural and Variational Hypotheses: Role and Necessity

In more general varifold compactness results with additional structure (e.g., prescribed mean curvature, anisotropic functionals, or stability constraints), compactness depends crucially on:

• Structural Hypotheses: Excluding “classical singularities” (three-fold or higher order intersections of sheets) and quantifying the regularity of “touching singularities” (forcing the coincidence set to have zero $n$-measure off the $g=0$ locus) maintains regularity in the limit (Bellettini et al., 2019, Bellettini et al., 2018).
• Variational Hypotheses: Uniform bound on first variation in $L^p$, weak (or finite-index) stability of immersed components, and stationarity of the associated functional ensure persistence of geometric and variational properties in the limit.
• Curvature Control ($L^q$-bounds): For varifolds with prescribed mean curvature, a locally uniform $L^q$-bound ($q>1$) on the weak second fundamental form is necessary to eliminate “hidden boundaries,” i.e., formation of zero mean curvature components in the limit, and guarantees that prescribed mean curvature persists (Bellettini, 2022).

6. Implications for Discrete and Diffuse Approximations

The compactness principles for rectifiable varifolds make it possible to rigorously analyze the convergence of discrete, computational, or non-smooth approximations to classical geometric objects. In particular:

• Volumetric and Point Cloud Varifolds: By properly calibrating the scale $\alpha_i$ in the energy functional relative to mesh size or sampling density, $d$-rectifiable limits can be obtained, meaning that minimizing sequences of discrete energies converge in the varifold sense to classical rectifiable sets or surfaces (Buet, 2014).
• Plateau and Minimal Surface Problems: The direct method for existence of minimizers of anisotropic or Plateau-type functionals leverages these compactness theorems, since closure under varifold limits and rectifiability of the limit are both guaranteed under mild density and energy conditions (Rosa, 2016).
• CMC and Prescribed-MC Hypersurfaces: In the context of constant or prescribed mean curvature, uniform control on curvature and mass ensures that direct limits of boundaries, or immersed hypersurfaces, remain smooth away from a possibly small (codimension at least 7) singular locus, and that prescribed geometric data is preserved in the limit (Bellettini et al., 2019, Bellettini et al., 2018, Bellettini, 2022).

7. Extensions, Limitations, and Classical Context

The compactness-rectifiability results synthesize, extend, and generalize classical compactness theorems for integral varifolds (e.g., Allard’s compactness and regularity), adapting to settings without monotonicity formulas (anisotropic functionals), or where only quantitative energy/density control is available.

Key limitations and necessary hypotheses include:

• Necessity of Two-Sided Density and Energy Bounds: Without upper and lower density bounds and suitable energy or first variation control, non-rectifiable or “diffuse” limits can arise.
• Exclusion of Classical Singularities: Permitting higher-fold intersections generically causes failure of compactness in the class of regular submanifolds.
• Control of Touching and Multiplicity: For codimension-1 varifolds, weak stability and strict structural assumptions on the nature of sheet intersection are required for compactness and regularity to be preserved.

When $F(x,T)\equiv1$ and only area and first variation are involved, these results recover Allard’s compactness for integral varifolds, showing smoothness away from a dimensionally controlled singular set, and integrality in the limit under appropriate hypotheses (Rosa, 2016, Bellettini et al., 2018). For anisotropic functionals, absence of a monotonicity formula makes the deformation lemma and associated quantitative density bounds indispensable (Rosa, 2016).

The enduring theme is that paired quantitative (energy/density) and qualitative (structural/variational) hypotheses provide a robust pathway for establishing compactness of rectifiable varifolds in both classical geometric analysis and contemporary discrete or computational applications.