Affine-Invariant Measures on Surfaces

• Affine-Invariant Measures on Surfaces is defined as a canonical method to assign volumes and area elements that remain unchanged under volume-preserving affine transformations.
• They are constructed using affine differential geometry techniques, including the Blaschke metric and curvature tensors, to derive invariant area forms.
• These measures have significant applications in convex geometry, moduli space dynamics, and computational shape analysis by providing robust geometric quantification.

An affine-invariant measure on a surface is a canonical, group-invariant way of assigning "volume," "area," or higher-order geometric weights to objects in affine or equiaffine geometry, such that the measure remains unchanged under the action of volume-preserving affine transformations. These measures play a central role in convex geometry, affine differential geometry, moduli spaces of flat surfaces, and analysis on manifolds, by giving intrinsic ways to quantify geometry and dynamics independently of coordinate choices.

1. Classical Constructions of Affine-Invariant Measures

The construction of affine-invariant measures originates in classical affine differential geometry of surfaces. For a regular smooth 2-surface $M$ embedded in $\mathbb R^3$, the equiaffine area element is locally given by the formula

$d\mu_A = |\det D^2 f|^{1/4}\, du\, dv$

where $z = f(u,v)$ parametrizes the surface as a graph and $D^2 f$ denotes the Hessian matrix. This area form arises from the Blaschke metric, defined via the affine normal field $\xi$ satisfying volume normalization and transversality conditions. Explicitly,

$$g = |H|^{-1/4}\, (f_{uu} du^2 + 2 f_{uv} du\, dv + f_{vv} dv^2), \qquad H = f_{uu} f_{vv} - f_{uv}^2.$$

Under any volume-preserving affine transformation, both the Hessian and Lebesgue measures transform so that $d\mu_A$ is invariant (Arnaldsson et al., 2020, Zhao et al., 2018).

In higher codimension or dimension, canonical affine-invariant measures are constructed using the so-called affine curvature tensors, derived from wedge products of all derivatives up to the appropriate order, combined with invariant-theoretic symmetrization via the infimum over $SL(d)$ acting on parameter frames. For a surface $M^2\subset \mathbb R^n$ parametrized by $f(u^1, u^2)$, the Gressman measure takes the form

$$d\mu_{\mathrm{aff}}(u) = \left|\det\left[f_u,\, f_v,\, f_{uu},\, f_{uv},\, f_{vv},\,\ldots \right]\right|^{2/Q} du\, dv$$

with $Q$ a minimum degree summing to $n$ terms (Gressman, 2017).

2. Affine Metrics and Canonical Structures

The notion of an affine-invariant metric is central to the modern treatment of surface geometry. For surfaces in $\mathbb R^3$, the "equi-affine metric" is encoded by the tensor

$$g_{ij} = \det(\mathbf x_1, \mathbf x_2, \mathbf x_{ij})\, |\det(\tilde G)|^{-1/4}$$

with $\tilde G$ constructed from the derivatives of the immersion. This metric is positive-definite on strictly convex surfaces, making $((X, g_{ij}))$ a canonical Riemannian manifold, on which every volume-preserving affine transformation acts by isometry (Raviv et al., 2010, Raviv et al., 2010).

For locally strictly convex surfaces in affine $4$-space, a one-parameter family of affine metrics is introduced via combinations of second fundamental forms $h_\lambda = h_1 + \lambda h_2$, normalized as

$$g_\lambda(X, Y) = \frac{h_\lambda(X, Y)}{\det[h_\lambda(X_i, X_j)]^{1/4}}$$

yielding area forms $$dA_{g_\lambda} = (\det h_\lambda(X_i, X_j))^{1/4} du \wedge dv$$ invariant under any unimodular affine transformation (Ballesteros et al., 2014).

3. Curvature-Driven Measures in Convex and Centro-Affine Geometry

In convex geometry, affine-invariant measures arise as surface measures for smooth convex hypersurfaces or convex bodies. The canonical example is the $L_p$-affine surface area for a convex body $K\subset \mathbb R^n$,

$$\Omega_p(K) = \int_{S^{n-1}} \left[\frac{F_K(u)}{h_K(u)^{p-1}}\right]^{n/(n+p)}\, d\mathcal H^{n-1}(u)$$

equivalently, as an infimum involving curvature measures $C_0$ and $C_{n-1}$ over the boundary of $K$: $$\Omega_p(K) = \inf_{g > 0} \Big( \int_{\partial K} g(x)^{-n}\, dC_0(K, x) \Big)^{p/(n+p)} \Big( \int_{\partial K} g(x)^p h_K(\nu_K(x))^{1-p}\, dC_{n-1}(K, x) \Big)^{n/(n+p)}$$ This construction yields an invariant sensitive to curvature and support properties and is invariant under $SL(n)$ maps (Zhao, 2015, Stancu, 2010). A centro-affine flow on smooth convex bodies generates an infinite hierarchy of such invariants, each encoding geometric information via volume derivatives under invariant flows.

4. Generalized Affine-Invariant Measures and the Oberlin Condition

Gressman provided a general unifying framework that constructs a canonical affine-invariant density on $d$-dimensional submanifolds $M^d \subset \mathbb R^n$ (for $1 \leq d \leq n-1$), defined up to normalization by the tensorial wedge product of derivatives and symmetrization under $SL(d)$ actions via the Kempf-Ness infimum. For sufficiently smooth or analytic immersions $f:M\to \mathbb R^n$, this measure $\mu_{aff}$ satisfies the Oberlin affine curvature condition with optimal exponent $\alpha = d/Q$: $\mu_{aff}(K \cap M) \leq C\, |K|^\alpha$ for all convex compact $K\subset \mathbb R^n$, where $Q$ is the minimum order of derivatives required to span $\mathbb R^n$. For hypersurfaces in $\mathbb R^3$, this specializes to the classical equiaffine area element $|\det D^2 \varphi|^{1/2} dx\,dy$ for $z=\varphi(x,y)$ (Gressman, 2017, Dendrinos et al., 2022).

The regularity (existence and uniform non-degeneracy) of such measures is guaranteed under real-analyticity or sufficient smoothness, with explicit determinantal formulas obtained from the jet expansion of $f$.

5. Affine-Invariant Measures in Moduli, Dynamics, and Ergodic Theory

Affine-invariant probability measures are fundamental in the dynamics of translation surfaces and moduli spaces. On strata of translation surfaces of genus $g$, the action of $SL(2,\mathbb R)$ preserves a class of measures on the moduli space $H_g$, which are called affine-invariant. Eskin–Mirzakhani–Mohammadi proved ergodic classification: each $SL(2,\mathbb R)$-invariant ergodic probability measure is supported on an affine submanifold in period coordinates. Avila–Matheus–Yoccoz established "regularity": for any such measure $\mu$, the mass of the locus with two non-parallel short saddle connections decays as $o(\rho^2)$ as the length bound $\rho \to 0$,

$$\mu\Big\{ M \colon \exists\,2\,\text{non-parallel saddle connections of length} \leq \rho \Big\} = o(\rho^2) \quad (\rho \to 0),$$

ensuring regularity and underpinning ergodic integrals, Lyapunov exponent formulas, and counting results (Avila et al., 2013, Sanchez, 2023).

In effective equidistribution (e.g., for the horocycle or Teichmüller geodesic flow), affine-invariant measures provide the unique invariant probabilities on orbit closures and control convergence rates of averages, as shown via leafwise measures and mixing rates in affine-invariant submanifolds.

6. Computational Affine-Invariant Surface Metrics and Applications

Affine-invariant surface metrics have practical significance in shape analysis. For $X \subset \mathbb R^3$, the equi-affine metric tensor constructed from local parameterizations provides the basis for defining geodesic distances that are exactly preserved under volume-preserving affine maps: $$g_{ij}(u) = \det(\mathbf x_1, \mathbf x_2, \mathbf x_{ij})\, |\det[\tilde g_{ij}]|^{-1/4}$$ This enables algorithmic implementations of shape signatures, Voronoi diagrams, and fast-marching geodesics invariant under affine deformations (Raviv et al., 2010, Raviv et al., 2010). For meshes, the piecewise-affine metric is used to discretize the Laplace–Beltrami operator, yielding affine-invariant heat kernels and spectral signatures for robust comparison of 3D shapes or symmetry detection.

Experimental results confirm the stability and invariance of shape retrieval, correspondence, and spectral descriptors under strong affine perturbations, underscoring the metric's utility in applied geometry.

7. Algebraic and Invariant Theoretic Foundations

Underlying all constructions of affine-invariant measures is the machinery of geometric invariant theory (GIT), the method of moving frames, and jet bundle analysis. The essential algebraic step is the construction of multilinear alternating forms (affine curvature tensors) from derivatives of surface parametrizations and symmetrization over $SL(d)$ via the Kempf-Ness criterion or Hilbert–Mumford stability. The sharpness of Oberlin-type curvature exponents and existence of canonical measures depend on non-degeneracy conditions verified using GIT techniques and volume bounds for images of infinitesimal neighborhoods.

In practice, this means all invariant surface measures in the affine geometric setting are governed by explicit tensorial formulas, regularity and integrability can be checked using determinantal or curvature conditions, and the invariance property is ultimately enforced via the symmetry group action.

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References:

• (Avila et al., 2013) Avila, Matheus, Yoccoz, "SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are regular"
• (Raviv et al., 2010) Raviv et al., "Affine-invariant geodesic geometry of deformable 3D shapes"
• (Raviv et al., 2010) Raviv et al., "Affine-invariant diffusion geometry for the analysis of deformable 3D shapes"
• (Ballesteros et al., 2014) Nuño-Ballesteros, Sánchez, "Affine metrics of locally strictly convex surfaces in affine 4-space"
• (Gressman, 2017) Gressman, "On the Oberlin affine curvature condition"
• (Dendrinos et al., 2022) Gressman, "A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$"
• (Arnaldsson et al., 2020) Arnaldsson, Valiquette, "Invariants of Surfaces in Three-Dimensional Affine Geometry"
• (Zhao et al., 2018) Zhao, Gao, "General affine differential geometry of surfaces in affine space $A^3$, I: the elliptical case"
• (Zhao, 2015) Zhao, "On $L_p$ Affine Surface Area and Curvature Measures"
• (Stancu, 2010) Stancu, "Centro-Affine Invariants for Smooth Convex Bodies"