Geometric Measure Theory: Hausdorff Measure, Rectifiability, and the Plateau Problem¶
Job: develop the Hausdorff measure / dimension / fractal, the rectifiability / Federer–Fleming structure / area formula, the currents / Plateau / compactness, the minimal surface regularity (De Giorgi / Reifenberg / Almgren), and the varifolds / Allard / monotonicity / tangent measures — with the Plateau via Federer–Fleming currents / De Giorgi regularity as the worked example.
Core¶
Hausdorff measure and the dimension¶
The Hausdorff measure \(\mathcal{H}^s(E)\) (the \(s\)-dimensional Hausdorff measure, the infimum of \(\sum(\mathrm{diam}\,U_i)^s\) over the covers, the \(s\)-dim volume). The Hausdorff dimension \(\dim_H(E)=\inf\{s:\mathcal{H}^s(E)=0\}=\sup\{s:\mathcal{H}^s(E)=\infty\}\).
| Hausdorff | Description |
|---|---|
| \(\mathcal{H}^s(E)\) | the \(s\)-dim Hausdorff measure (\(\inf\sum(\mathrm{diam}\,U_i)^s\) over covers) |
| \(\dim_H(E)\) | \(\inf\{s:\mathcal{H}^s(E)=0\}=\sup\{s:\mathcal{H}^s(E)=\infty\}\) |
| \(\mathcal{H}^n\) = Lebesgue | on \(\mathbb{R}^n\) (the Hausdorff the general, the non-integer \(s\) the fractal) |
Rectifiability and the Federer–Fleming¶
The countably \(n\)-rectifiable set (the \(E=\bigcup_i f_i(A_i)\) the countable union of Lipschitz images of subsets of \(\mathbb{R}^n\)).
| Rectifiability / Federer–Fleming | Description |
|---|---|
| Countably \(n\)-rectifiable | \(E=\bigcup_i f_i(A_i)\) (Lipschitz images of \(\mathbb{R}^n\) subsets) |
| Federer–Fleming structure | the finite-mass \(\mathcal{H}^n\)-measurable set = rectifiable + purely unrectifiable (the Besicovitch decomposition) |
| Area formula | \(\int_E J_n f\,d\mathcal{H}^n=\int\mathcal{H}^0(E\cap f^{-1}\{y\})\,d\mathcal{H}^n(y)\) (the Jacobian, the Federer) |
Currents and the Plateau problem¶
The currents (the Federer–Fleming: the \(n\)-currents the continuous linear functionals on the differential \(n\)-forms, the de Rham the pairing, the mass / the boundary).
| Currents / Plateau | Description |
|---|---|
| Currents | the \(n\)-currents (linear functionals on \(n\)-forms, the mass / the boundary) |
| Plateau problem | the least-area surface spanning a boundary \(\Gamma\) (the currents the solution) |
| Integral currents | the currents the integration over the rectifiable sets (the Federer–Fleming) |
| Compactness | the Federer–Fleming compactness / the closure (the minimizing sequence the compact, the limit the minimizer) |
The minimal surfaces and the regularity¶
| Minimal surfaces / regularity | Description |
|---|---|
| Minimal surfaces | the Plateau solution (the area-minimizing, the zero mean curvature) |
| De Giorgi / Reifenberg / Almgren | the interior regularity of the area-minimizing currents (smooth except a singular set of codim \(\ge8\)) |
| Almgren | the \(Q\)-valued functions, the regularity, the singular set |
Varifolds and the Allard regularity¶
| Varifolds / Allard / monotonicity | Description |
|---|---|
| Varifolds (Almgren / Allard) | the measures on the Grassmannian (the generalized surfaces, the first variation / the mean curvature) |
| Allard regularity | the \(\varepsilon\)-regularity (the varifold the regular, the \(L^p\) mean curvature) |
| Monotonicity formula | the mass the monotone (the density, the tangent measures, the Preiss) |
See it¶
Plateau problem via the Federer–Fleming currents and the De Giorgi regularity. The Plateau: the least-area surface spanning a wire \(\Gamma\subset\mathbb{R}^3\) (the minimal surface the boundary \(\Gamma\)). The currents: the \(2\)-current \(T\) with \(\partial T=\Gamma\) (the current the boundary the wire, the mass the area), the Federer–Fleming the compactness (the minimizing sequence the compact, the limit the minimizer). The minimizer: the area-minimizing integral current, the minimal surface. The regularity: the De Giorgi (the 2D area-minimizing the smooth except the boundary, the Reifenberg), the Almgren (the higher-D the singular set of codim \(\ge8\)). The key point: the Plateau problem the Federer–Fleming currents (the existence / the compactness), the De Giorgi / the Almgren the regularity (the smooth except the singular) — the geometric measure theory the rigorous Plateau.
Check¶
Show that the Federer–Fleming compactness theorem gives the existence of the area-minimizing current, and explain the regularity.
The Plateau: minimize \(\mathbf{M}(T)\) (the mass) over the integral currents \(T\) with \(\partial T=\Gamma\) (the boundary the wire). The Federer–Fleming compactness: the integral currents with the mass + the boundary mass uniformly bounded are the compact (the closure theorem: the limit of the integral currents the integral, the compactness the minimizing sequence the convergent subsequence). The minimizing sequence \(T_n\) (the \(\mathbf{M}(T_n)\to\inf\), the \(\partial T_n=\Gamma\)) the compact \(\Rightarrow\) the limit \(T\) (the area-minimizing, the \(\partial T=\Gamma\), the \(\mathbf{M}(T)=\inf\)). The regularity: the De Giorgi (the 2D area-minimizing current the smooth except the boundary, the Reifenberg the boundary regularity), the Almgren (the higher-D: the singular set of codim \(\ge8\), the \(Q\)-valued functions the regularity). So the Federer–Fleming the existence (the compactness), the De Giorgi / the Almgren the regularity (the smooth except the singular).
Explain the Hausdorff dimension of a self-similar fractal (e.g. the Cantor set), and the relation to the Hausdorff measure.
The Cantor set \(C\subset[0,1]\) (the self-similar, the \(C=\bigcap C_n\) the \(3^n\) intervals of length \(3^{-n}\)). The Hausdorff dimension: \(\dim_H(C)=\log2/\log3\) (the self-similar dimension: the \(2^n\) intervals of length \(3^{-n}\), the \(\dim_H=\log N/\log(1/r)=\log2/\log3\)). The Hausdorff measure: \(\mathcal{H}^s(C)\) for \(s=\log2/\log3\) (the \(\mathcal{H}^s(C)\in(0,\infty)\) the critical, the \(\mathcal{H}^s(C)=0\) for \(s>\log2/\log3\), the \(=\infty\) for \(s<\log2/\log3\)). The relation: the \(\dim_H\) the critical exponent (the \(\mathcal{H}^s\) the \(0\) / \(\infty\) / the finite), the self-similar the \(\dim_H=\log N/\log(1/r)\). The Hausdorff the fractal (the non-integer \(s=\log2/\log3\)), the Lebesgue the \(\mathcal{H}^1(C)=0\) (the Cantor the zero length).
Pitfalls¶
- The GMT the Federer–Fleming (the currents / the Plateau the existence, the regularity the De Giorgi / the Almgren the singular set the codim \(\ge8\)) — the existence the currents, the regularity the smooth except the singular (the codim \(\ge8\) the Almgren).
- The varifolds the Allard (the \(\varepsilon\)-regularity, the \(L^p\) mean curvature) — the varifolds the generalized surfaces (the measures on the Grassmannian), the Allard the \(\varepsilon\)-regularity.
- The monotonicity formula the mass the monotone (the density, the tangent measures the Preiss) — the monotonicity the density, the tangent measures the blow-up.
- The area formula the Federer (the Jacobian, the \(\int J_n f\,d\mathcal{H}^n\)) — the area / the change of variables for the Lipschitz maps.
- The rectifiability the countably \(n\)-rectifiable (the Lipschitz images), the purely unrectifiable (the Besicovitch) — the decomposition the structure.
Refs¶
- Federer, Geometric Measure Theory (Springer 1969); the Federer–Fleming currents / the Plateau
- De Giorgi, "Frontiere orientate di misura minima" (1961); Almgren, Q-valued functions / Plateau's problem
- Allard, "On the first variation of a varifold" (Annals 1972); the \(\varepsilon\)-regularity
- Preiss, "Geometry of measures in \(\mathbb{R}^n\)" (Annals 1987); the tangent measures; Simon, GMT lecture notes
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