For the third (and last) installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss a talk that brought me back some good memories from the time I did my PhD at IMPA when I took a course on Geometric Measure Theory (taught by Hermano Frid) based on the books of Evans-Gariepy (for the introductory part of the course) and Guisti (for the core part of the course).
In fact, our goal today is to revisit Luigi Ambrosio’s Bourbaki seminar talk entitled “The regularity theory of area-minimizing integral currents (after Almgren-DeLellis-Spadaro)”. Here, besides the original works of Almgren and DeLellis-Spadaro (in these five papers here), the main references are the video of Ambrosio’s talk and his lecture notes (both in English).
Disclaimer. As usual, all errors and mistakes are my entire responsibility.
This post is centered around solutions to the so-called Plateau’s problem.
A formulation of Plateau’s problem in dimension and codimension is the following. Given a -dimensional Riemannian manifold and a -dimensional compact embedded oriented submanifold (without boundary), find a -dimensional embedded oriented submanifold with boundary such that
for all oriented -dimensional submanifold with . (Here, denotes the -dimensional volume of ).
This formulation of Plateau’s problem allows for several variants. Moreover, the solution to Plateau’s problem is very sensitive on the precise mathematical formulation of the problem (and, in particular, on the dimension and codimension ).
Example 1 The works of Douglas and Radó (based on the conformal parametrization method and some compactness arguments) provided a solution to (a formulation of) Plateau’s problem when and the boundary is circular (i.e., is parametrized by the round circle ). Unfortunately, the techniques employed by Douglas and Radó do not work for arbitrary dimension and codimension .
The singular immersed disk
satisfies and the so-called calibration method can be applied to prove that
for all smooth oriented -dimensional submanifold with . (Here, stands for the -dimensional Hausdorff measure on .)
The example above motivates the introduction of weak solutions (including immersed submanifolds) to Plateau’s problem, so that the main question becomes the existence and regularity of weak solutions.
Remark 1 The “PDE counterpart” of this point of view is the study of existence and regularity of weak solutions of PDEs in Sobolev spaces.
In arbitrary dimensions and codimensions, De Giorgi’s regularity theory provides weak solutions to Plateau’s problem which are smooth on open and dense subsets.
As it was pointed out by Federer, this is not a satisfactory regularity statement. Indeed, De Giorgi’s regularity theory allows (in principle) weak solutions to Plateau’s problem whose singular set (i.e., the subset of points where the weak solution is not smooth) could be “large” in the sense that its (-dimensional) Hausdorff measure could be positive.
In this context, Almgren wrote two preprints which together had more than 1000 pages (published in posthumous way here) where it was shown that the singular set of weak solutions to Plateau’s problem has codimension at least:
Theorem 1 (Almgren) Let be an integral rectifiable -dimensional current in a -dimensional Riemannian manifold .If is area-minimizing (i.e., a weak solution to Plateau’s problem), then there exists a closed subset of such that:
- has codimension : the Hausdorff dimension of is , and
- is the singular set of : the subset is induced by a smooth oriented -dimensional submanifold of .
We will explain the notion of area-minimizing integral rectifiable currents (appearing in the statement above) in a moment. For now, let us just make some historical remarks. Ambrosio has the impression that some parts of Almgren’s work were not completely reviewed, even though several experts have used some of the ideas and techniques introduced by Almgren. For this reason, the simplifications (of about ) and, more importantly, improvements of Almgren’s work obtained by DeLellis-Spadaro in a series of five articles (quoted in the very beginning of the pots) were very much appreciated by the experts of this field.
Closing this introduction, let us present the plan for the remaining sections of this post:
- the next section reviews the construction of solutions to the generalized Plateau problem via Federer-Fleming compactness theorem for integral rectifiable currents;
- then, in the subsequent section, we revisit some aspects of the regularity theory of weak solutions to Plateau’s problem in codimension ; in particular, we will see in this setting a stronger version of Theorem 1;
- after that, in the last section of this post, we make some comments on the works of Almgren and DeLellis-Spadaro on the regularity theory for Plateau’s problem in codimension ; in particular, we will sketch the proof of Theorem 1 above.
Remark 2 For the sake of exposition, from now on we will restrict ourselves to the study of Plateau’s problem in the case of Euclidean ambient spaces, i.e., . In fact, this is not a great loss in generality because the statements and proofs in the Euclidean setting can be adapted to arbitrary Riemannian manifolds with almost no extra effort.