Quadratic Flatness Theorem in Varifold Theory

• Quadratic flatness theorem is a second-order regularity principle in geometric measure theory that uses quadratic decay estimates to characterize codimension-one varifolds.
• It employs anisotropic integrands defined by uniformly convex norms and leverages height and tilt decay estimates to achieve C² rectifiability of the support.
• The theorem extends Brakke’s perpendicularity and establishes locality of the anisotropic mean curvature vector, linking varifold geometry with smooth hypersurface models.

The quadratic flatness theorem is a regularity principle in geometric measure theory for codimension-one varifolds with bounded anisotropic mean curvature. In its varifold formulation, the theorem concerns an $n$-varifold $V$ in an open set $\Omega\subset \mathbf{R}^{n+1}$, equipped with an anisotropic surface energy determined by a uniformly convex norm $\phi$, and asserts that under locally finite $\mathscr{H}^n$-measure of the support, the support is almost everywhere governed by a second-order flatness structure. In the companion development, this second-order structure is used to prove an anisotropic extension of Brakke’s perpendicularity theorem and a locality theorem for the anisotropic mean curvature vector (Santilli et al., 24 Jul 2025, Kolasiński et al., 23 Mar 2026).

1. Geometric and variational framework

The ambient space is an open set $\Omega\subset \mathbf{R}^{n+1}$, and the objects are codimension-one varifolds $V\in V_n(\Omega)$, with weight measure

$\|V\|(A)=V(A\times G(n+1,n)).$

The support $\operatorname{spt}\|V\|\subset \Omega$ is the geometric carrier of the varifold. In the regularity theory at issue, codimension one is fixed throughout: $\adim=n+1$ and $V$0 (Kolasiński et al., 23 Mar 2026).

The anisotropic integrand is induced by a uniformly convex $V$1 norm $V$2 on $V$3. If $V$4 is a chosen unit normal to a plane $V$5, the associated autonomous integrand is

$V$6

Uniform convexity is quantified by the lower bound

$V$7

In the codimension-one, autonomous, strictly elliptic setting, this integrand satisfies a strong ellipticity or scalar atomic condition (Kolasiński et al., 23 Mar 2026).

The anisotropic first variation is defined by

$V$8

in the autonomous case. For the norm-induced integrand, $V$9 has the explicit form

$\Omega\subset \mathbf{R}^{n+1}$0

When $\Omega\subset \mathbf{R}^{n+1}$1 is Radon, one obtains a weak anisotropic mean curvature field $\Omega\subset \mathbf{R}^{n+1}$2, and the full anisotropic mean curvature vector is

$\Omega\subset \mathbf{R}^{n+1}$3

The companion regularity theory also uses the equivalent $\Omega\subset \mathbf{R}^{n+1}$4-weighted formulation $\Omega\subset \mathbf{R}^{n+1}$5, under which $\Omega\subset \mathbf{R}^{n+1}$6 is said to have bounded mean $\Omega\subset \mathbf{R}^{n+1}$7-curvature (Santilli et al., 24 Jul 2025).

A standing hypothesis inherited from the first paper is that $\Omega\subset \mathbf{R}^{n+1}$8 has locally finite $\Omega\subset \mathbf{R}^{n+1}$9-measure. This allows rectifiability theory to be applied to the support itself, not only to $\phi$0 (Kolasiński et al., 23 Mar 2026).

2. The theorem: tangent balls, blow-ups, and quadratic decay

One formulation of the theorem is geometric. Let $\phi$1, and define the tangent-ball set

$\phi$2

Thus $\phi$3 precisely when there are two open Euclidean balls of equal radius, centered at $\phi$4, both disjoint from $\phi$5, and both tangent to $\phi$6 at $\phi$7. The blow-up set is

$\phi$8

where $\phi$9 is defined through Hausdorff-limit blow-ups $\mathscr{H}^n$0 as $\mathscr{H}^n$1. The quadratic flatness theorem states that if $\mathscr{H}^n$2 is a uniformly convex $\mathscr{H}^n$3-norm, $\mathscr{H}^n$4 has bounded anisotropic mean curvature, $\mathscr{H}^n$5 is Radon, and $\mathscr{H}^n$6, then

$\mathscr{H}^n$7

and

$\mathscr{H}^n$8

In particular, $\mathscr{H}^n$9 is $\Omega\subset \mathbf{R}^{n+1}$0-rectifiable of class $\Omega\subset \mathbf{R}^{n+1}$1 (Santilli et al., 24 Jul 2025).

This geometric statement is complemented by an analytic one. Under bounded anisotropic mean curvature and locally finite $\Omega\subset \mathbf{R}^{n+1}$2-measure of the support, the first paper proves that at $\Omega\subset \mathbf{R}^{n+1}$3-almost every point $\Omega\subset \mathbf{R}^{n+1}$4 there is an approximate tangent plane $\Omega\subset \mathbf{R}^{n+1}$5 such that

$\Omega\subset \mathbf{R}^{n+1}$6

and also

$\Omega\subset \mathbf{R}^{n+1}$7

These are the quadratic height-decay and quadratic tilt-decay estimates. The same paper concludes that $\Omega\subset \mathbf{R}^{n+1}$8 is countably $\Omega\subset \mathbf{R}^{n+1}$9-rectifiable of class $V\in V_n(\Omega)$0 (Kolasiński et al., 23 Mar 2026).

The geometric meaning is that near a typical point the support is trapped between quadratic barriers. In coordinates adapted to a tangent plane $V\in V_n(\Omega)$1, the existence of two tangent balls gives a bound of the form

$V\in V_n(\Omega)$2

for a suitable cylinder $V\in V_n(\Omega)$3. This is the second-order flatness encoded by the theorem (Santilli et al., 24 Jul 2025).

3. Anisotropic Brakke perpendicularity

A principal consequence of quadratic flatness is an anisotropic extension of Brakke’s perpendicularity theorem. The setting is no longer rotationally invariant, anisotropic monotonicity formulas are in general not available, and the standard Lipschitz approximation yields weaker control of the bad set. The key innovation is a local linear straightening of the anisotropy (Kolasiński et al., 23 Mar 2026).

For a tangent plane $V\in V_n(\Omega)$4, the paper introduces

$V\in V_n(\Omega)$5

where $V\in V_n(\Omega)$6 and $V\in V_n(\Omega)$7. One has $V\in V_n(\Omega)$8. After pushing forward the varifold by $V\in V_n(\Omega)$9 and pulling back the integrand, the new integrand $\|V\|(A)=V(A\times G(n+1,n)).$0 satisfies

$\|V\|(A)=V(A\times G(n+1,n)).$1

Thus the anisotropy becomes quadratically flat in the Grassmannian directions near $\|V\|(A)=V(A\times G(n+1,n)).$2, replacing the Euclidean Pythagorean structure in Brakke’s isotropic proof (Kolasiński et al., 23 Mar 2026).

A Caccioppoli-type inequality then relates tilt excess to height excess. Under the hypotheses of the localized perpendicularity theorem, one assumes the height-decay condition

$\|V\|(A)=V(A\times G(n+1,n)).$3

and the tilt-decay condition

$\|V\|(A)=V(A\times G(n+1,n)).$4

together with a bad-set smallness hypothesis controlling points of different density along the fiber of $\|V\|(A)=V(A\times G(n+1,n)).$5. The conclusion is

$\|V\|(A)=V(A\times G(n+1,n)).$6

so the anisotropic mean curvature vector is orthogonal to the tangent plane (Kolasiński et al., 23 Mar 2026).

On the unit-density layer

$\|V\|(A)=V(A\times G(n+1,n)).$7

the regularity from the first paper supplies the required height and tilt decay, and the local single-sheet $\|V\|(A)=V(A\times G(n+1,n)).$8 structure makes the bad set empty for small radii. This is how quadratic flatness is converted into perpendicularity on $\|V\|(A)=V(A\times G(n+1,n)).$9 (Kolasiński et al., 23 Mar 2026).

4. Locality of the anisotropic mean curvature vector

After perpendicularity, the second principal consequence is locality. The general locality theorem considers an integral varifold $\operatorname{spt}\|V\|\subset \Omega$0 with $\operatorname{spt}\|V\|\subset \Omega$1 Radon, together with a properly embedded $\operatorname{spt}\|V\|\subset \Omega$2 hypersurface $\operatorname{spt}\|V\|\subset \Omega$3 with $\operatorname{spt}\|V\|\subset \Omega$4. If $\operatorname{spt}\|V\|\subset \Omega$5 denotes the set of points $\operatorname{spt}\|V\|\subset \Omega$6 such that

$\operatorname{spt}\|V\|\subset \Omega$7

and

$\operatorname{spt}\|V\|\subset \Omega$8

then

$\operatorname{spt}\|V\|\subset \Omega$9

Here $\adim=n+1$0 is the classical anisotropic mean curvature vector of the smooth hypersurface $\adim=n+1$1 (Kolasiński et al., 23 Mar 2026).

The main locality theorem specializes this to the regular unit-density layer. If $\adim=n+1$2 is a uniformly convex $\adim=n+1$3 norm, $\adim=n+1$4 is an integral $\adim=n+1$5-varifold with

$\adim=n+1$6

then for every embedded $\adim=n+1$7-hypersurface $\adim=n+1$8 with $\adim=n+1$9,

$V$00

This identifies the anisotropic mean curvature vector of the varifold with the classical anisotropic mean curvature of any $V$01 model hypersurface that represents the support near the point (Kolasiński et al., 23 Mar 2026).

A plausible implication is that, on the unit-density layer, anisotropic mean curvature is determined by the approximate second-order structure of $V$02 rather than by the particular varifold representation. The paper states this point in local terms: the mean curvature is a local geometric quantity determined by the approximate second-order structure of the support (Kolasiński et al., 23 Mar 2026).

5. Proof architecture and regularity mechanism

The proof strategy replaces isotropic monotonicity with a combination of blow-up analysis, Caccioppoli inequalities, ellipticity estimates, and curvature theory for closed sets. One component is blow-up analysis around almost every point: $V$03 Bounded anisotropic mean curvature gives compactness, and under codimension one and ellipticity the blow-ups are multiplicity-$V$04 anisotropic minimal cones that are planes. This is the starting point for both height and tilt flatness (Kolasiński et al., 23 Mar 2026).

A second component is an elliptic estimate for the anisotropic projections $V$05 and $V$06: $V$07 These inequalities encode the scalar atomic condition and make it possible to relate angular deviations of tangent planes to first-variation quantities (Kolasiński et al., 23 Mar 2026).

The first paper also develops a closed-set curvature theory on the support. The support $V$08 is shown to be an $V$09-set, so viscosity barriers touching $V$10 from one side satisfy anisotropic mean-curvature inequalities. Through the $V$11-normal bundle, the $V$12-reach, and the $V$13-principal curvatures $V$14, one derives one-sided curvature bounds and then proves that points with two distinct normals occur at $V$15-almost every point of the nontrivial blow-up set. This yields the tangent-ball formulation of quadratic flatness and the $V$16-rectifiability of $V$17 (Santilli et al., 24 Jul 2025).

Once quadratic flatness is available, Allard’s anisotropic regularity theorem can be applied. Under the additional unit-density and absolute continuity hypotheses, the support contains an embedded $V$18-hypersurface $V$19 that is relatively open in $V$20 and satisfies

$V$21

Moreover,

$V$22

Thus the $V$23 regular part is open and dense in the support and agrees up to $V$24-null sets with the nontrivial blow-up set (Santilli et al., 24 Jul 2025).

6. Terminology, related results, and distinct uses of the phrase

The phrase “quadratic flatness” is used in more than one mathematical sense. In the varifold theory described above, it refers to second-order flatness of the support of a codimension-one varifold with bounded anisotropic mean curvature, expressed either through quadratic height and tilt decay or through the existence of two mutually tangent balls and $V$25-rectifiability (Santilli et al., 24 Jul 2025, Kolasiński et al., 23 Mar 2026).

In convex geometry, the same phrase is used for localized characterizations of quadrics via flat shadow boundaries or flat grazes. A central localized Blaschke theorem states that if a strictly convex body $V$26, $V$27, has the property that for every $V$28 in an open set $V$29 there is a hyperplane $V$30 with

$V$31

then there exists

$V$32

for a non-negative definite quadratic form $V$33 such that

$V$34

The same paper proves that if $V$35, with $V$36 a polyhedron, and every graze $V$37 for $V$38 is flat, then $V$39 is an ellipsoid (Jerónimo-Castro et al., 28 Feb 2025).

In the PDE theory of surfaces, a “quadratic flatness theorem” refers to a differential characterization of quadrics. For $V$40 on a connected open $V$41, if the Hessian determinant is somewhere positive, then the graph of $V$42 is contained in a quadratic surface if and only if $V$43 is a weak solution of the third-order system

$V$44

$V$45

In that usage, quadratic flatness is a PDE criterion for the graph to lie in a quadric (Zawalski, 2023).

A common terminological confusion is with the “quadric flat torus theorem,” which is unrelated. That theorem concerns metrically proper actions of free abelian groups on quadric complexes and proves that any non-cyclic free abelian group acting metrically properly is isomorphic to $V$46 and preserves a $V$47-invariant flat in the standard square tiling sense (Munro et al., 2024).

In the varifold context, however, the term has a specific technical role: it names the second-order flatness mechanism that makes anisotropic perpendicularity and locality accessible without an anisotropic monotonicity formula.