Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Theorem

Jean Taylor's theorem on soap film singularities

Builds on Plateau's laws Hausdorff measure Hölder continuity Tetrahedron

Statement

Let $S \subseteq \R^3$ be a closed set such that $\mathcal{H}^2(S \cap K) < \infty$ for every compact set $K$ , where $\mathcal{H}^2$ is the $2$ -dimensional Hausdorff measure. Let $\delta > 0$ , $C > 0$ and $\alpha > 0$ . Suppose that $S$ is almost minimizing in the sense of Almgren: for every Lipschitz map $\phi \colon \R^3 \to \R^3$ , with $W = \set{x : \phi(x) \ne x}$ and $r = \operatorname{diam}(W \cup \phi(W)) < \delta$ , $\mathcal{H}^2(S \cap W) \le (1 + C r^\alpha) \, \mathcal{H}^2(\phi(S \cap W)).$ Then every point of $S$ has a neighbourhood $U$ with the following property. The set $S \cap U$ is the image, under a $C^{1,\beta}$ diffeomorphism with $\beta > 0$ , of an open piece of one of these three cones:

a plane;
the cone $Y$ : three closed half-planes with a common boundary line, meeting along it at angles of $120^\circ$ ;
the cone $T$ over the six edges of a regular tetrahedron centred at the origin, whose six sectors meet in threes along four rays.

Remarks

Jean Taylor proved the theorem in 1976, using the regularity theory of Almgren. The main step is to classify the minimizing cones. A tangent cone of $S$ at a point is a cone over a network of arcs of great circles on the unit sphere. The minimizing condition leaves only the three cones above. Then an epiperimetric inequality shows that $S$ approaches its tangent cone at a rate that is a power of the radius, which gives the $C^{1,\beta}$ parametrization. Soap films and clusters of soap bubbles are modelled by such almost-minimizing sets, so the theorem proves Plateau's laws within this model. For example, take a soap film on a wire frame shaped like the edges of a regular tetrahedron. It consists of six flat triangles meeting at the centre, a piece of the cone $T$ .

Going deeper

For a student

The theorem is a model case of the classification of singularities in geometric variational problems. Its ingredients, the tangent cone classification and the epiperimetric inequality, have since been used for other problems, such as the regularity of free boundaries. The almost-minimizing condition is flexible: it covers films spanning wires, bubbles enclosing fixed volumes, and films with small external forces.

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Hausdorff measure concept
Hölder continuity concept
Mean curvature concept
Minimal surface concept
Tetrahedron concept
Plateau's laws statement
Jean Taylor's theorem on soap film singularities statement

Tag	`APGRBE5DW9`
URI	`m2://APGRBE5DW9`
Kind	statement, theorem
Writing	W0 stub
MSC	49Q05, 49Q20, 53A10
Status	This copy of the site has no build data. Run `m2 status APGRBE5DW9`.
About	Plateau's laws
Uses	Plateau's laws, Hausdorff measure, Hölder continuity, Tetrahedron