VADB Theorem in Geometric Analysis

• VADB Theorem is a unifying principle that quantitatively links global volume with local distance data, triggering rigidity phenomena when equality holds.
• It provides a framework to analyze stability and convergence in geometric analysis across Riemannian, Finsler, and Lorentzian settings.
• Applications include controlling metric convergence, filling volume issues, and establishing measure inequalities in convex and causal geometries.

The Volume Above Distance Below (VADB) Theorem is a unifying stability principle in geometric analysis that relates volume and measure inequalities to control over distances or sectional lower bounds, providing a quantitative and often rigid link between local boundary data and global geometric structure. It has become a central tool in areas spanning convex geometry, Riemannian and Finsler geometry, and recent developments in Lorentzian geometry and metric space convergence.

1. Foundational Formulations and Definitions

The VADB principle states that the “volume above”—the global volume or total measure of a geometric object or manifold—can be quantitatively controlled (bounded below or above) by the “distance below”—the minimal distance function, sectional or projection measures, or boundary metric data. In stability contexts, the loss or preservation of global features is forced by the local distance measures: equality often triggers a rigidity phenomenon, implying the metrics coincide.

For closed Riemannian manifolds, if $(M, g_j)$ is a sequence of metrics that satisfy

• $g_j(v,v) \ge g_0(v,v)$ for all $v$,
• diameter $\leq D$,
• volumes converge, $$\operatorname{Vol}(M, g_j) \to \operatorname{Vol}(M, g_0)$$,

then $(M, g_j)$ converges to $(M, g_0)$ in the volume preserving Sormani-Wenger Intrinsic Flat sense (Allen et al., 2020).

For convex bodies and measures, the inequality

$$|K|^{\frac{n-k}{n}} - |L|^{\frac{n-k}{n}} \geq c_{n,k} \frac{1}{d_{\mathrm{ovr}}(K, \mathcal{BP}_k^n)^k} \max_{F \in \mathrm{Gr}_{n-k}} \left( |K \cap F| - |L \cap F| \right)$$

relates the volume difference to the maximal sectional difference, modulated by the outer volume ratio to a model class (Giannopoulos et al., 2016).

With boundaries, it suffices to impose a uniform bound on the area of the boundary rather than stronger regularity (Allen et al., 17 Jun 2024).

2. Geometric Manifestations: Riemannian, Finsler, and Banach Space Perspective

In Riemannian geometry, the classical filling volume problem is characterized by Gromov’s definition

$$\operatorname{FillVol}(N, f) = \inf \{ \operatorname{Vol}(M) : \partial M = N,\ \mathrm{bd}_M \geq f \}$$

where $N$ is a closed $(n-1)$-manifold and $f$ is a boundary distance function (Ivanov, 2010). Minimal fillings satisfy a VADB-type inequality: if $d_{M'}(x, y) \geq d_M(x, y)$ for all $x,y$, then $\operatorname{Vol}(M') \geq \operatorname{Vol}(M)$.

By embedding the manifold via its boundary distance map $\Phi: M \to L^\infty(S)$, with $\Phi(x) = d_M(x, \cdot)$, the filling volume problem is reframed as an area minimization among Lipschitz surfaces in Banach space. Area minimizers in $L^\infty$ correspond precisely to minimal fillings.

Extending to Finsler geometry, multiple non-equivalent notions of area (e.g., Busemann, Holmes–Thompson, Loewner) arise, but the VADB philosophy persists: boundary distances constrain the minimal filling area, with rigidity present in simple Finsler-2D discs (Ivanov, 2010).

3. Convex Bodies, Sections, Projections, and Measure Inequalities

In convex geometry, the VADB theorem generalizes classical problems by establishing that the volume difference between two bodies ($K \supset L$) is controlled by the maximal difference of their lower-dimensional sections, weighted by the outer volume ratio from model intersection (or projection) bodies (Giannopoulos et al., 2016). For arbitrary measures $\mu$,

$$\mu(K) - \mu(L) \leq C_{k}\,d_{\mathrm{ovr}}(K, \mathcal{BP}_k^n)^k\, \max_{F} \left( \mu(K \cap F) - \mu(L \cap F) \right)$$

quantifies the same principle for generalized densities.

These results underpin stability and rigidity phenomena. For example, the sharp volume gap ensures that if sectional or projection differences vanish, the global bodies must coincide up to model class approximation.

4. Intrinsic Flat Convergence, Boundary Effects, and Quantitative Rigidity

The intrinsic flat distance, introduced by Sormani and Wenger, allows for weak geometric convergence when more restrictive versions (Gromov–Hausdorff, Lipschitz) fail (Allen et al., 2020, Allen et al., 2020, Allen et al., 17 Jun 2024). The VADB theorem ensures that under volume convergence, uniform diameter bounds, and pointwise lower bounds on the metric tensor, the sequence converges in the intrinsic flat sense.

For manifolds with boundary, it is sufficient to control the area of the boundary. Earlier approaches (doubling with necks) required excessively strong hypotheses, but refined estimates on the SWIF distance now show only area bounds are needed (Allen et al., 17 Jun 2024). The convergence of distance functions almost everywhere enables effective embedding into a common metric space and quantitative control of the flat distance between induced currents.

Concrete counterexamples in these works demonstrate that failure to control the boundary area (e.g., boundary area diverging while volume and diameter converge) leads to loss of convergence, highlighting the sharpness of the VADB hypothesis.

5. Extensions to Lorentzian and Causal Geometries: Null Distance and Spacetimes

Recent work has adapted the VADB theorem for static and globally hyperbolic Lorentzian spacetimes equipped with the null distance—a metric space structure compatible with the causal structure (Allen, 2 Oct 2025). In this setting, the spacetime metric is of form $g_j = - h_j^2 dt^2 + \sigma_j$ on $M$, and VADB-type conditions are imposed on the spatial part $\sigma_j$ and associated volume and area quantities.

If spatial metrics are controlled from below, volume and boundary area bounds are satisfied,

• $\sigma_j(v,v) \geq (1 - 1/j)\sigma_\infty(v,v)$,
• $$\operatorname{Vol}(M, \sigma_j) \to \operatorname{Vol}(M, \sigma_\infty)$$,
• $\operatorname{Area}(\partial M, \sigma_j) \leq A$,

then pointwise and uniform convergence of null distances follows, implying metric convergence in uniform, Gromov–Hausdorff, and intrinsic flat senses.

For time-dependent globally hyperbolic spacetimes, conjectural generalizations are formulated where all hypotheses are “integrated” in time, for instance,

$$\int_{t_0}^{t_1}\operatorname{Vol}(M, \sigma_j(t)/h_j(t))\,dt \to \int_{t_0}^{t_1}\operatorname{Vol}(M, \sigma_\infty(t)/h_\infty(t))\,dt$$

with appropriate lower bounds. The conformal invariance of null distance facilitates gauge-fixing for these comparisons.

6. Applications, Examples, and Future Directions

The VADB theorem has been pivotal in recent advances in scalar curvature stability (e.g., Positive Mass Theorem, Llarull’s Rigidity), geometric analysis with weak regularity, and the systematic understanding of metric convergence in various geometric and causal contexts (Allen et al., 2020, Allen et al., 2020, Allen et al., 17 Jun 2024, Allen, 2 Oct 2025).

Examples in the literature illustrate both necessity and sufficiency of the hypotheses: warped product metrics, metrics with divergent boundary area, and cinching phenomena point to failures of convergence in the absence of VADB bounds.

Current research developments aim to refine VADB-type theorems:

• Minimizing additional boundary and area assumptions for broader manifold classes,
• Leveraging conformal changes and local control in convergence problems,
• Extending results to measured Gromov–Hausdorff or pointed intrinsic flat convergence,
• Developing further analogs in Lorentzian and non-smooth geometric settings,
• Investigating applications to norms and volume estimates in Banach and Finsler geometries.

The VADB principle thus underlies a wide spectrum of geometric rigidity and stability results, providing a quantitative framework linking local metric data (distances, sections, projections, or boundary) to global invariants (volume, measure, area) in both classical and modern geometric analysis.