Geometric Measure Theory¶
Job: measure-theoretic calculus of geometric objects — sets, surfaces, and maps — with weak compactness enabling variational existence for area-minimizing and BV problems.
Core¶
GMT replaces smooth submanifolds by rectifiable sets, currents, and varifolds so that area (or mass) is lower semicontinuous and minimizing sequences have limits. Regularity theory then upgrades weak minimizers to classical surfaces almost everywhere.
| Page | Focus |
|---|---|
| Foundations | Hausdorff measure, rectifiability, BV, finite perimeter, area/coarea, currents/varifolds pointers |
| Minimal surfaces | Plateau, regularity one-liners, GMT vs classical DG |
Theorem spine¶
- Hausdorff measure \(\mathcal{H}^k\) measures \(k\)-dimensional size; Lipschitz images are \(\mathcal{H}^k\)-rectifiable.
- Sets of finite perimeter \(=\) Caccioppoli sets: characteristic functions in \(\mathrm{BV}\); reduced boundary is rectifiable.
- Area and coarea formulae link Lipschitz maps, Jacobians, and slicing.
- Compactness theorems (BV, integral currents, varifolds) \(\Rightarrow\) existence of minimizers.
- Regularity: minimizers are smooth outside a closed singular set of controlled dimension.
Interfaces¶
- Measure theory: Radon measures, differentiation, \(L^p\)
- Differential geometry: Riemannian volume, mean curvature, second fundamental form
- Functional analysis: BV as dual-type Banach structure; weak-* compactness
- PDE / calculus of variations: Euler–Lagrange, free boundary
See it¶
Plateau problem (modern). Among integral currents \(T\) with \(\partial T=\llbracket\Gamma\rrbracket\) fixed, minimize mass \(\mathbf{M}(T)\). Federer–Fleming compactness + closure \(\Rightarrow\) existence of mass-minimizing \(T\). Interior regularity: \(\operatorname{sing}T\) has Hausdorff dimension \(\le n-8\) for hypersurface minimizers in \(\mathbb{R}^n\) (Simons, Federer, …).
Check¶
Why is \(\mathcal{H}^n\llcorner E\) for arbitrary \(E\subset\mathbb{R}^n\) not a satisfactory ‘surface measure’ when minimizing area?
Without rectifiability / perimeter control, \(\mathcal{H}^n\)-mass need not interact well with boundaries (no Gauss–Green), and minimizing sequences can oscillate or cancel (need currents with orientation or varifolds without cancellation). Finite perimeter / integral currents restore a divergence theorem and compactness.
State the coarea formula for \(f\in\mathrm{Lip}(\mathbb{R}^n,\mathbb{R})\) and non-negative measurable \(g\).
\(\int_{\mathbb{R}^n} g\,|Df|\,dx=\int_{-\infty}^\infty\int_{f^{-1}(t)}g\,d\mathcal{H}^{n-1}\,dt\). Level sets of Lipschitz functions are countably rectifiable for a.e. \(t\).
Pitfalls¶
- Hausdorff dimension \(\neq\) rectifiability: purely unrectifiable sets (e.g. four-corner Cantor) have positive \(\mathcal{H}^1\) but no approximate tangents a.e.
- Varifolds vs currents: varifolds forget orientation (no cancellation, good for stationary surfaces); currents carry orientation and boundary.
- Smooth minimal surfaces theory (mean curvature zero immersions) and GMT minimizers share Euler–Lagrange intuition but differ in category and singularity allowance.
Refs¶
- Simon, Lectures on Geometric Measure Theory
- Maggi, Sets of Finite Perimeter and Geometric Variational Problems
- Federer, Geometric Measure Theory
- Giusti, Minimal Surfaces and Functions of Bounded Variation
- Mattila, Geometry of Sets and Measures in Euclidean Spaces
Next¶
Foundations · Minimal surfaces · Measure theory · Differential geometry · Functional analysis