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.
2. Federer-Fleming theory of currents
Definition 2 An integral rectifiable -dimensional current in is a triple where:
- is a countably -rectifiable set, i.e., where has zero -dimensional Hausdorff measure () and for all one has that for some -dimensional oriented submanifold ;
- is an orientation of : is a measurable map such that for each and for -almost every , one has that
for an oriented orthonormal basis . In other terms, is the approximate tangent space of at ;
- is the multiplicity: is a measurable -integrable function such that intuitively describes how many copies of oriented pieces of the current has near a -typical point (for each ).
An integral rectifiable -dimensional current induces a continuous linear functional on the space of compactly supported and smooth differential -forms on via the formula:
Remark 3 In general, a -dimensional current is an element of the dual of (i.e., a continuous linear functional on ).
Example 3 The notion of integral rectifiable -dimensional current generalizes the definition of an oriented smooth compact -dimensional submanifold . Indeed, given such a , we have that the triple is an integral rectifiable -dimensional current (where is the orientation of and is the constant function.)
The notions of boundary, mass and support of an integral rectifiable -dimensional current are defined as follows.
Definition 3 The boundary is the -dimensional current satisfying Stokes formula:
where is the exterior derivative of .The mass of is .
The support of is the support of the measure , i.e., .
Example 4 For the current associated to an oriented smooth compact manifold with boundary , the boundary is
the mass is , and the support is .
These definitions motivate the following generalization of Plateau’s problem (formulated in the introduction of this post):
Generalized Plateau’s problem. Let be an integral rectifiable -dimensional current compactly supported in such that . Find an integral rectifiable -dimensional current such that and
for all integral rectifiable -dimensional currents with .
Of course, the main point in generalizing Plateau’s problem is that one can always solve the generalized Plateau’s problem!
More precisely, a classical “cone construction” shows that there are always integral rectifiable currents with . In particular, it makes sense to consider
Next, the space of currents has a weak- topology associated to the evaluation maps
for each . As it is expected from weak- topologies (cf. Banach-Alaoglu theorem), we have a compactness property allowing us to extract a convergent subsequence from any area-minimizing sequence in the sense that are integral rectifiable currents with and . In particular, .
Furthermore, it is possible to prove that the mass depends on a lower semicontinuous way on the current. Since and , this means that satisfies
This almost puts us in position to solve the generalized Plateau problem. Indeed, the weak- limit constructed above solves the generalized Plateau problem except for the fact that it is not obvious at all that the current is integral rectifiable!
Fortunately, it turns out that is an integral rectifiable current: this is a consequence of the following compactness theorem for integral rectifiable currents of Federer-Fleming.
Theorem 4 (Federer-Fleming) Let be a sequence of integral rectifiable -dimensional currents converging to . Suppose that
Then, is also an integral rectifiable -dimensional current.
In summary, Federer-Fleming’s theorem is a key result permitting us to find solutions to the generalized Plateau’s problem.
Definition 5 A solution to the generalized Plateau problem is called an area-minimizing integral rectifiable current.
Once the existence of area-minimizing currents is established, one can hope to answer the original Plateau problem by studying their regularity.
More concretely, let
be the set of (interior) regular points of , and let
be the singular set of . In this language, one wants to understand the size of .
Remark 4 By requiring to be a -dimensional submanifold in the definition of , we are implicitly skipping the interesting question of boundary regularity of ! In fact, the study of the boundary regularity of area-minimizing currents is a delicate problem (which is very sensible to the regularity of in the statement of the generalized Plateau problem): one disposes of results in the codimension case, but there is no satisfactory boundary regularity theory in arbitrary codimensions.
We divide the analysis of the size of into two parts. In the next section, we will review De Giorgi’s regularity theory in codimension : in particular, we will see that has codimension for any area-minimizing integral rectifiable current . Then, we dedicate the final section to present a rough sketch of the theorems of Almgren and DeLellis-Spadaro.
3. Plateau’s problem in codimension
In this section, we follow De Giorgi’s original approach by studying the regularity of sets with finite perimeter. In terms of currents, if we consider the -dimensional current
associated to via integration of -differential forms and we require boundary current has finite mass, then it is possible to prove that has the form
Remark 5 The fact that the integral rectifiable current has multiplicity is a very peculiar feature of the codimension case!
In this setting, is called the essential boundary of and the perimeter of is .
For later use, we denote by the approximate unit normal to , i.e., for -a.e. , we take to be the unique normal vector to the -plane associated to which is positively oriented (i.e., where stands for the canonical basis of ).
3.1. De Giorgi’s almost everywhere regularity theorem
In order to study the size of the singular set of a locally perimeter minimizing set , De Giorgi observed that the deviations (actually, variance) of the approximate unit normal from its mean are a fundamental tool.
More concretely, De Giorgi introduced the following quantity called the excess:
By definition, on , the excess is the variance of from its mean .
In this context, De Giorgi showed that a small excess permits to detect regular points of locally perimeter minimizing sets:
Theorem 6 (De Giorgi) For each , there exists a dimensional threshold with the following property.Let be a set locally minimizing perimeter in an open set (i.e., for every such that the symmetric difference is compactly contained in a ball with closure ).
Let and suppose that one has a -small excess
for some ball compactly contained in . Then, in an appropriate system of coordinates, is the graph of a smooth (and even real-analytic) function.
This theorem is sometimes called a -regularity theorem because it provides smoothness (of near ) whenever the excess is below a certain threshold . For this reason, we will refer to it as De Giorgi’s -regularity theorem in what follows.
Since -almost every is a Lebesgue point of the approximate unit normal , an immediate consequence of De Giorgi’s -regularity theorem is the following restriction on the size of the singular set:
Corollary 7 (De Giorgi) If locally minimizes the perimeter in an open set , then is almost everywhere regular in , i.e.,
The proof of De Giorgi’s -regularity theorem is based on the so-called excess decay lemma:
Lemma 8 (Excess decay lemma) For each , there exists a (small) dimensional constant with the following property.Let be a locally perimeter minimizing set (in some open set ). Then,
whenever and .
In fact, the excess decay lemma is the starting point of an iteration scheme of the open condition in showing that the approximate unit normal at scale is Hölder continuous on uniformly on the scale .
From this uniform Hölder continuity property near with , one can show that, for some (and, actually, for all ), the set is the graph (in an adequate system of coordinates) of a function in a neighborhood of .
Since minimizes the perimeter (locally in ), the function is a weak solution to the minimal surfaces equation
In particular, we can invoke the regularity theory of quasilinear elliptic PDEs to conclude that the weak solution of the previous equation is necessarily smooth (real-analytic).
In summary, we just saw the sketch of proof of DeGiorgi’s -regularity theorem modulo the excess decay lemma.
The proof of the excess decay lemma is by contradiction (and it uses an idea that is helpful for the study of Plateau problem in higher codimensions).
We start with the situation where the excess is very small (), but the excess does not decay as expected. After performing some scalings and rotations (to “blow-up” near ), one obtains a family , , of finite perimeter sets contained in the unit ball of such that and the corresponding approximate unit normals deviate very little from an arbitrarily fixed direction (independent of ).
This means that can be approximated by the graph of a function on the unit ball of with a very small Lipschitz constant. By using this information to linearize the area functional, one gets that the perimeter of has an expansion of the form
Since are obtained from by scalings and rotations, also minimizes the perimeter. By combining this fact with the previous expansion of the area functional, one can prove that as and is a harmonic function (as minimizes the Dirichlet energy ).
In this way, one gets a contradiction in the limit because any harmonic function satisfies the decay estimate
(where ), a property that is incompatible with our assumption that (and a fortiori ) do not have the expected decay of excess.
3.2. Tangent cones and the codimension of singular sets
Besides the excess, De Giorgi introduced other key tools in the analysis of area-minimizing currents such as the tangent cone.
In simple terms, the tangent cone arises from blow-ups of a current near a given point.
More precisely, for and , let us consider the scaling . Given a current , let be defined by . The family of currents corresponds to zooming in at .
Theorem 9 If is an area-minimizing current and is an “interior point”, then any weak- limit of the currents (as ) is a cone without boundary, i.e.,
which is locally area-minimizing in , i.e.,
whenever is compactly supported in a bounded open set .
Definition 10 Any cone as in the theorem above is called a tangent cone to at .
Remark 6 An important open problem is the uniqueness of tangent cones.
An important consequence of the theorem above (of existence of tangent cones) and De Giorgi’s -regularity theorem is the following characterization of the regular set of an area-minimizing currents in codimension one. Let be a area-minimizing -dimensional current in . Then:
is regular if and only if some tangent cone of at is flat
Indeed, this happens because the excess of any flat cone is zero (and, a fortiori, it is below the critical threshold in De Giorgi’s -regularity theorem).
Remark 7 As we will see later, this characterization of is particular of the codimension one case. In fact, it is simply false in higher codimensions!
Remark 8 Behind the characterization of singular points in terms of non-flat tangent cones, there is a principle of “persistence of singularities”: more precisely, if are singular points of (locally in ) area-minimizing currents , and and as , then .
This intimate relationship between flat cones and regular points (in the codimension case) gives a precise description of the singular set of area-minimizing currents. This fact is the starting point of the proof of the following (optimal) theorem on the codimension of the singular set:
Theorem 11 Let be a set of finite perimeter. Assume that locally minimizes perimeter in some open set .
- If , then ;
- If , then is discrete;
- In general, for all , so that the Hausdorff dimension of the singular set is .
The proof of this theorem goes as follows. By the principle of persistence of singularities, the tangent cone at a singular point (obtained the scalings ) is singular at the origin. Furthermore, we saw that is a minimal cone (i.e., is a cone without boundary which locally minimizes area). In other words, a tangent cone at is always a singular minimal cone.
From this, we deduce that the singular set is empty when (and it might be non-empty when ), so that the first item of the theorem is proved.
The proofs of the two remaining items of the theorem (the discreteness of the singular set when and the estimative of its codimension in general) rely on Federer’s reduction of dimension argument.
For the sake of exposition, we will illustrate this argument by proving just the discreteness of the singular set when . If is not discrete for some , we would have a sequence converging to (by the principle of persistence of singularities).
By making a blowup of (on ) at scales , we obtain a tangent cone which is singular at the origin and at some point in the unit sphere of (again by the principle of persistence of singularities).
Since is a cone, the half-line between and is contained in . By doing a new blowup at the middle point of the segment between and , we obtain a new tangent cone of the form (where the factor comes from the contribution in the limit of to the blowup) such that would be a singular minimal cone in , a contradiction with the fact that there are no singular minimal cones in when .
At this point, we dispose of sharp regularity results for solutions of the generalized Plateau problem in codimension (area-minimizing -dimensional currents in ), so that it is time to move to the case of higher codimensions.
4. Plateau’s problem in higher codimensions
The regularity theory of area-minimizing -dimensional currents in when is substantially more involved than its codimension one counterpart due to the following new phenomena:
- the presence of flat tangent cones at a point does not imply its regularity;
- the presence of branch points in the singular set;
- the necessity of non-homogenous blowups (i.e., push-forwards by are no longer sufficient to understand the local geometry of area-minimizing currents).
In fact, these difficulties already appear in the context of Example 2, i.e., the singular -disk
with boundary .
4.1. Examples of center manifolds and multivalued functions
More precisely, given an integral rectifiable current (with orientation ), we say that is calibrated by a smooth closed differential -form on an open subset with for all whenever
for -almost every .
The fundamental (and elementary) remark based on Stokes formula is that calibrated currents are locally area-minimizing. Furthermore, the Wirtinger inequality says that any complex submanifold of complex dimension induces a current that is calibrated by
In particular, these facts imply that is locally area-minimizing.
Observe that this shows that Almgren’s theorem 1 is an optimal solution to Plateau’s problem in codimension : in fact, is a integral rectifiable locally area-minimizing -dimensional current in with a singular set of Hausdorff dimension .
By considering the scalings
we see that the tangent cone to at (i.e., the weak- limit of as ) is the integral rectifiable current
associated to the horizontal plane with multiplicity two. Geometrically, this means that, after scaling (“zooming in near the origin”), the two branches of (corresponding to the two determinations of , i.e., square-root of , near ) merge together into the plane .
It is worth to notice that the tangent cone of at is flat, but, nevertheless, is a singular point of due to its branch nature. In other words, contrary to the case of Plateau’s problem in codimension , it is no longer true in higher codimensions that flat tangent cones are associated only to regular points.
Remark 9 If we have some extra information on flat tangent cones (e.g., if we know in advance that its multiplicity is one), then we can still ensure that the associated point is regular.
An important point in the example of (near the origin) is that it behaves differently along the and coordinates. This suggest that we should replace the homogenous scalings by non-homogenous scalings taking into account the distinct behaviors of the and variables along . Moreover, by analogy with the case of Plateau’s problem in codimension , we want a non-homogenous scalings such that the limit objects (multivalued functions) are harmonic.
As it turns out, the correct non-homogenous scaling of near the origin is . In fact, the main point here is that this scaling fixes (that is, ) because the functions and corresponding to the determinations of the square-root of (i.e., ) are already harmonic functions.
In general, it is an important problem to construct non-homogenous scalings leading to non-trivial limit objects (blowups), and, in fact, almost half of Almgren’s work is dedicated to address this issue.
In order to get a grasp on the tools introduced by Almgren to overcome these difficulties, let us consider another example. Let
By the calibration method, we have that is locally area-minimizing.
Also, it is not hard to check that is singular at the origin. If one tries to play with non-homogenous scalings , , then it is possible to verify that the sole scaling producing an interesting (non-trivial and “harmonic”) blowup is
A quick calculation reveals that converges (as ) to the current induced by the smooth complex curve
In particular, this non-homogenous scaling of near the origin leads to a non-flat limit object (which is certainly not a cone)!
This suggests that we should forget the idea of getting tangent cones as the blowup (limit object) under non-homogenous scalings.
In fact, the main point of the example of is that the induced current has a “regular part” given by the smooth curve and a “singular branching” part coming from the determination of the square-root of . This means that the parametrization of the current with the flat coordinates is not adequate: instead, it is a better idea to parametrize in terms of the smooth complex curve , i.e., we have a complex parameter for the smooth complex curve (“regular part of ”), and we have two functions and with parametrizing the “singular branching part of ”.
This decomposition led Almgren to introduce the notions of center manifolds (“regular parts”) and multivalued harmonic functions (“singular branching part”). In the case of , the curve is the center manifold, and the multivalued function is the multivalued harmonic function.
Almgren’s construction of center manifold (via adequate non-homogenous scalings) is very delicate. This construction was simplified by DeLellis-Spadaro, but it still consists a complicated part of the proof of Almgren’s theorem 1.
For this reason, let’s assume that we already dispose of adequate non-homogenous scalings and center manifolds, and we consider now the description of the “singular branching” using multivalued harmonic functions.
Very roughly speaking, multivalued functions are obtained by a sort of averaging procedure of the branches of the current over the center manifold.
Remark 10 It is important to ensure that the “order of contact” of the branches with the center manifold is not infinity (otherwise one would get a trivial blowup with this procedure). In this direction, Almgren introduced a certain monotonicity formula allowing him to prove that the “order of contact” is always finite.
More concretely, a multivalued function (over the center manifold ) is defined as follows. We consider (“number of branches”), and we let
be the space of unordered -tuples of points in . Here, is the equivalent relation for all and is a permutation of .
In this setting, a –valued function is simply a function .
Note that is a metric space. For example, this space possesses the (Wasserstein) metric
This metric space structure on allows us to talk about Lipschitz -valued functions, but this is not satisfactory for our purposes: recall that we want to make sense of harmonic multivalued functions!
4.2. Two approaches to harmonic multivalued functions
Almgren proposed the following extrinsic approach for the definition of -valued harmonic functions. First, one tries to isometrically embed into some Euclidean space . Then, one defines Sobolev spaces, Dirichlet energy and harmonicity (i.e., minimizers of the Dirichlet energy) of -valued functions with the aid of the nice structures of : for example, the Sobolev space is defined as the space of functions (in the usual sense) such that for almost every .
As it turns out, these definitions work well when is isometrically embedded in in such a way that it is bi-Lipschitz equivalent to a Lipschitz retract of . This led Almgren to prove that, in fact, this is always the case for some .
In their proof of Almgren’s theorem, DeLellis-Spadaro proposed an alternative intrinsic approach for the definition of -valued harmonic functions: the idea is to rely only on the metric structure of (without making any mention to ambient spaces like ). More precisely, we say that belongs to the Sobolev space if there are , such that
- for each , the function belongs to the (usual) Sobolev space ;
- for each and , for almost every .
In other words, we measure the (Sobolev) regularity of by testing the Sobolev regularity of the real-valued functions on obtained by measuring the -distance of the values of to arbitrary points .
Remark 11 One can show that there are (minimal) functions such that for any -functions satisfying the inequalities above.
In this context, the Dirichlet energy of is
where . Moreover, we say that a -valued function is harmonic if minimizes the Dirichlet energy (with Dirichlet or Neumann boundary condition).
In any event, the notion of harmonic -valued function plays a key role in the proof of Almgren’s theorem 1 because the graphs of such functions (describing the “singular branching parts” of area-minimizing currents) have the following regularity property:
Theorem 12 (Almgren-DeLellis-Spadaro) Let be a bounded open subset with Lipschitz boundary (e.g., a piece of the center manifold ), and .Then, there exists minimizing the Dirichlet energy in the class of with . Furthermore:
- any such is locally -Hölder continuous for a certain , and is locally for some ;
- there exists a closed (“singular”) subset with Hausdorff dimension such that the graph
of outside is a smooth embedded -dimensional submanifold of .
4.3. Sketch of proof of Almgren’s theorem
Closing this post, we note that the general scheme is similar to De Giorgi’s regularity theory in codimension one, even though the details are obviously different (and much more technical):
- one starts with a locally area-minimizing integral rectifiable current and one makes a blowup near a point using non-homogenous scalings;
- in this way, one obtains a center manifold (describing the regular part of the current) and Lipschitz multivalued functions (describing the singular branching part of the current) providing approximations to the branches of the scalings of the area-minimizing current;
- one shows that the quality of approximation of these Lipschitz multivalued functions depend in a superlinear way on a certain (variant of De Giorgi’s) excess; by combining this fact with an expansion of Dirichlet’s energy and (and Almgren’s monotonicity formula for the control of the “order of contact”), one obtains a -valued harmonic function as the limit of these Lipschitz multivalued functions;
- finally, one applies the regularity estimate of Theorem 12 to this -valued harmonic function to conclude the proof of Theorem 1 (i.e., to deduce that the singular set of an area-minimizing integral rectifiable -dimensional current has Hausdorff dimension ).