Geometric Measure Theory

How can the existence of a minimal surface spanning a wire frame be proven without an explicit formula, and how does this connect to fractals and data geometry?

• **Plateau problem:** Joseph Plateau observed soap films on wire contours in 1873 - minimal surfaces with prescribed boundary
• **Machine learning:** high-dimensional data manifolds have complex geometry; Hausdorff estimates and intrinsic dimension underpin their quantitative analysis
• **Condensed matter physics:** grain boundaries in crystals minimize area; the Plateau problem in materials science
• **Computer graphics:** varifolds and integer currents discretize surfaces in numerical computational geometry

Предварительные знания

• Differential forms and integration on manifolds
• Hausdorff measure and Hausdorff dimension
• Functional analysis and weak convergence

• Optimal Transport

Hausdorff Measure and Fractal Dimension

In 1918, Felix Hausdorff introduced a measure that works at any size: 1-dimensional length, 2-dimensional area, and fractional dimension for fractals. This made it possible to measure the British coastline (dimension ≈ 1.25), the Koch snowflake (log4/log3 ≈ 1.26), and attractors of chaotic systems. Modern data-manifold analysis in machine learning starts with Hausdorff estimates on empirical point clouds.

The Hausdorff dimension of the Lorenz attractor is ≈ 2.06, of Brownian motion is 2, and of the Henon strange attractor is ≈ 1.26. In machine learning, intrinsic dimension is the empirical analog of Hausdorff dimension and is estimated via TwoNN or MLE methods.

Rectifiable Sets and Tangent Cones

A rectifiable set is a set that looks like a smooth manifold almost everywhere, though not necessarily everywhere. This is the right generalization of surface for irregular objects: coastlines, grain boundaries in metals, cluster boundaries in high-dimensional data. In the 1960s Federer built the full machinery for working with such sets, the foundation of geometric measure theory.

Alberti's theorem on the flow structure of rectifiable measures is a modern result (1991): every rectifiable measure admits a decomposition along integral curves of a Lebesgue direction field. An analog of the Radon-Nikodym theorem for geometric measures.

Currents and the Plateau Problem

In 1873, Joseph Plateau observed soap films on wire frames - physical minimal surfaces. 87 years later, Herbert Federer and Wendell Fleming (1960) proved the existence theorem using currents: generalized surfaces as linear functionals on differential forms. Without this generalization the Plateau problem had no solution: smooth minimizing sequences converge to singular objects.

Federer-Fleming guarantees existence but not regularity. In dimension n >= 8 singularities are possible (Bombieri-De Giorgi-Giusti, 1969). In dimensions 2-7 mass-minimizing currents are smooth outside a null set.