Posted by: matheuscmss | March 15, 2015

First Bourbaki seminar of 2015 (III): Ambrosio’s talk

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.

1. Introduction

This post is centered around solutions to the so-called Plateau’s problem.

A formulation of Plateau’s problem in dimension {m} and codimension {n} is the following. Given a {(m+n)}-dimensional Riemannian manifold and a {(m-1)}-dimensional compact embedded oriented submanifold {\Gamma\subset M} (without boundary), find a {m}-dimensional embedded oriented submanifold {\Sigma\subset M} with boundary {\partial\Sigma=\Gamma} such that

\displaystyle \textrm{vol}_m(\Sigma)\leq \textrm{vol}_m(\widetilde{\Sigma})

for all oriented {m}-dimensional submanifold {\widetilde{\Sigma}} with {\partial\widetilde{\Sigma} = \Gamma}. (Here, {\textrm{vol}_m(A)} denotes the {m}-dimensional volume of {A}).

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 {m} and codimension {n}).

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 {m=2} and the boundary {\Gamma} is circular (i.e., {\Gamma} is parametrized by the round circle {S^1=\{ (x,y)\in\mathbb{R}^2: x^2+y^2=1 \}}). Unfortunately, the techniques employed by Douglas and Radó do not work for arbitrary dimension {m} and codimension {n}.

Example 2 The following example gives an idea on the difficulties that one might found while trying to solve Plateau’s problem (in the formulation given above).Let us consider the case {m=n=2}, {\Gamma\subset\mathbb{R}^4\simeq \mathbb{C}^2} defined as

\displaystyle \Gamma:=\{(\zeta^2,\zeta^3)\in\mathbb{C}^2: |\zeta|=1\}

The singular immersed disk

\displaystyle D:=\{(z, w)\in\mathbb{C}^2: z^3=w^2, |z|\leq 1\}

satisfies {\partial D=\Gamma} and the so-called calibration method can be applied to prove that

\displaystyle \mathcal{H}^2(D)<\mathcal{H}^2(\Sigma)

for all smooth oriented {2}-dimensional submanifold {\Sigma\subset\mathbb{R}^4} with {\partial\Sigma = \Gamma}. (Here, {\mathcal{H}^2} stands for the {2}-dimensional Hausdorff measure on {\mathbb{R}^4}.)

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.

This point of view was adopted by several authors: for example, De Giorgi studied the notion of sets of finite perimeter when {n=1}, and Federer and Fleming introduced the notion of currents.

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 ({m}-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 {2} at least:

Theorem 1 (Almgren) Let {T} be an integral rectifiable {m}-dimensional current in a {(m+n)}-dimensional {C^5} Riemannian manifold {M^{m+n}}.If {T} is area-minimizing (i.e., a weak solution to Plateau’s problem), then there exists a closed subset {\textrm{sing}(T)} of {T} such that:

  • {\textrm{sing}(T)} has codimension {\geq 2}: the Hausdorff dimension of {\textrm{sing}(T)} is {\leq (m-2)}, and
  • {\textrm{sing}(T)} is the singular set of {T}: the subset {T\cap (M-(\textrm{supp}(\partial T)\cup\textrm{sing}(T))} is induced by a smooth oriented {m}-dimensional submanifold of {M}.

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 {1/3}) 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 {n=1}; 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 {n\geq 2}; 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., {M^{m+n} = \mathbb{R}^{m+n}}. In fact, this is not a great loss in generality because the statements and proofs in the Euclidean setting {M^{m+n} = \mathbb{R}^{m+n}} can be adapted to arbitrary Riemannian manifolds {M^{m+n}} with almost no extra effort.

2. Federer-Fleming theory of currents

Definition 2 An integral rectifiable {m}-dimensional current in {\mathbb{R}^{m+n}} is a triple {T=(R,\tau,\theta)} where:

  • {R} is a countably {\mathcal{H}^m}-rectifiable set, i.e., {R=R_0\cup\bigcup\limits_{i=1}^{\infty} C_i} where {R_0} has zero {m}-dimensional Hausdorff measure ({\mathcal{H}^m(R_0)=0}) and for all {i\in\mathbb{N}} one has that {C_i\subset M_i} for some {C^1} {m}-dimensional oriented submanifold {M_i\subset\mathbb{R}^{m+n}};
  • {\tau: R\rightarrow\Lambda_m:=\wedge^{m}\mathbb{R}^{m+n}} is an orientation of {R}: {\tau} is a measurable map such that for each {i\in\mathbb{N}} and for {\mathcal{H}^m}-almost every {x\in C_i}, one has that

    \displaystyle \tau(x) = v_1\wedge\dots\wedge v_m\in\Lambda_m

    for an oriented orthonormal basis {\{v_1,\dots, v_m\}\subset T_x M_i}. In other terms, {\tau(x)} is the approximate tangent space of {R} at {x};

  • {\theta:R\rightarrow\mathbb{Z}} is the multiplicity: {\theta} is a measurable {\mathcal{H}^m}-integrable function such that {\theta(x)\in\mathbb{Z}} intuitively describes how many copies of oriented pieces of {M_i} the current {R} has near a {\mathcal{H}^m}-typical point {x\in C_i\subset M_i} (for each {i\in\mathbb{N}}).

An integral rectifiable {m}-dimensional current {T=(R,\tau,\theta)} induces a continuous linear functional on the space {\mathcal{D}^m} of compactly supported and smooth differential {m}-forms {\omega} on {\mathbb{R}^{m+n}} via the formula:

\displaystyle T(\omega)=\int_{R}\theta\langle\omega,\tau\rangle d\mathcal{H}^m.

Remark 3 In general, a {m}-dimensional current is an element of the dual of {\mathcal{D}^m} (i.e., a continuous linear functional on {\mathcal{D}^m}).

Example 3 The notion of integral rectifiable {m}-dimensional current generalizes the definition of an oriented smooth compact {m}-dimensional submanifold {\Sigma^m\subset\mathbb{R}^{m+n}}. Indeed, given such a {\Sigma^m}, we have that the triple {(\Sigma,\tau_{\Sigma},1)} is an integral rectifiable {m}-dimensional current (where {\tau_{\Sigma}} is the orientation of {\Sigma} and {1} is the constant function.)

The notions of boundary, mass and support of an integral rectifiable {m}-dimensional current {T} are defined as follows.

Definition 3 The boundary {\partial T} is the {(m-1)}-dimensional current satisfying Stokes formula:

\displaystyle \partial T(\omega) = T(d\omega) \quad \forall\,\omega\in\mathcal{D}^{m-1}

where {d\omega} is the exterior derivative of {\omega}.The mass of {T} is {\textbf{M}(T) = \int_R |\theta| d\mathcal{H}^m}.

The support {\textrm{supp}(T)} of {T=(R,\tau,\theta)} is the support of the measure {\|T\|:=|\theta|\mathcal{H}^m|_{R}}, i.e., {\textrm{supp}(T)=\{x\in\mathbb{R}^{m+n}: \|T\|(B(x,r))>0 \textrm{ for all } r>0\}}.

Example 4 For the current {T=(\Sigma,\tau_{\Sigma},1)} associated to an oriented smooth compact manifold {\Sigma} with boundary {\partial\Sigma}, the boundary {\partial T} is

\displaystyle \partial T = (\partial\Sigma, \tau_{\partial\Sigma}, 1),

the mass {\textbf{M}(T)} is {\textrm{vol}_m(\Sigma)=\mathcal{H}^m(\Sigma)}, and the support {\textrm{supp}(T)} is {\Sigma}.

These definitions motivate the following generalization of Plateau’s problem (formulated in the introduction of this post):

Generalized Plateau’s problem. Let {\Gamma} be an integral rectifiable {(m-1)}-dimensional current compactly supported in {\mathbb{R}^{m+n}} such that {\partial\Gamma = 0}. Find an integral rectifiable {m}-dimensional current {T} such that {\partial T=\Gamma} and

\displaystyle \textbf{M}(T)\leq \textbf{M}(S)

for all integral rectifiable {m}-dimensional currents {S} with {\partial S = \Sigma = \partial T}.

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 {T} with {\partial T = \Gamma}. In particular, it makes sense to consider

\displaystyle m:=\inf\{\textbf{M}(T): T \textrm{ integral rectifiable current with } \partial T = \Gamma\}

Next, the space {(\mathcal{D}^m)'} of currents has a weak-{\ast} topology associated to the evaluation maps

\displaystyle T\mapsto T(\omega)

for each {\omega\in\mathcal{D}^m}. As it is expected from weak-{\ast} topologies (cf. Banach-Alaoglu theorem), we have a compactness property allowing us to extract a convergent subsequence {T_{\ell}\rightarrow T} from any area-minimizing sequence {T_n} in the sense that {T_n} are integral rectifiable currents with {\partial T_n=\Gamma} and {\textbf{M}(T_n)\rightarrow m}. In particular, {\partial T=\Gamma}.

Furthermore, it is possible to prove that the mass depends on a lower semicontinuous way on the current. Since {T_{\ell}\rightarrow T} and {\textbf{M}(T_{\ell})\rightarrow m}, this means that {T} satisfies

\displaystyle \textbf{M}(T)\leq m

This almost puts us in position to solve the generalized Plateau problem. Indeed, the weak-{\ast} limit {T} constructed above solves the generalized Plateau problem except for the fact that it is not obvious at all that the current {T} is integral rectifiable!

Fortunately, it turns out that {T} 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 {(T_k)_{k\in\mathbb{N}}} be a sequence of integral rectifiable {m}-dimensional currents converging to {T}. Suppose that

\displaystyle \sup\limits_{k\in\mathbb{N}} (\textbf{M}(T_k) + \textbf{M}(\partial T_k)) <\infty

Then, {T} is also an integral rectifiable {m}-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 {T} 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

\displaystyle reg(T)=\{x\in\textrm{supp}(T): \textrm{ for some } r>0, \,\textrm{supp}(T)\cap B(x,r) \textrm{ is a smooth } m-\textrm{submanifold}\}

be the set of (interior) regular points of {T}, and let

\displaystyle sing(T):= \textrm{supp}(T)-(reg(T)\cup\textrm{supp}(\partial T))

be the singular set of {T}. In this language, one wants to understand the size of {sing(T)}.

Remark 4 By requiring {\textrm{supp}(T)\cap B(x,r)} to be a {m}-dimensional submanifold in the definition of {x\in reg(T)}, we are implicitly skipping the interesting question of boundary regularity of {T}! In fact, the study of the boundary regularity of area-minimizing currents is a delicate problem (which is very sensible to the regularity of {\Gamma} in the statement of the generalized Plateau problem): one disposes of results in the codimension {1} case, but there is no satisfactory boundary regularity theory in arbitrary codimensions.

We divide the analysis of the size of {sing(T)} into two parts. In the next section, we will review De Giorgi’s regularity theory in codimension {1}: in particular, we will see that {sing(T)} has codimension {7} for any area-minimizing integral rectifiable current {T\subset\mathbb{R}^{m+1}}. 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 {1}

In this section, we follow De Giorgi’s original approach by studying the regularity of sets {E\subset\mathbb{R}^{m+1}} with finite perimeter. In terms of currents, if we consider the {(m+1)}-dimensional current

\displaystyle T_E(f dx_1\wedge\dots\wedge dx_{m+1}) = \int_E f dx_1\wedge\dots\wedge dx_{m+1}

associated to {E} via integration of {(m+1)}-differential forms and we require boundary current {\partial T_E} has finite mass, then it is possible to prove that {\partial T_E} has the form

\displaystyle \partial T_E = (\partial^*E, \tau_E, 1)

Remark 5 The fact that the integral rectifiable current {\partial T_E} has multiplicity {1} is a very peculiar feature of the codimension {n=1} case!

In this setting, {\partial^*E} is called the essential boundary of {E} and the perimeter of {E} is {\mathcal{H}^m(\partial^* E)}.

For later use, we denote by {\nu_E} the approximate unit normal to {\partial^*E}, i.e., for {\mathcal{H}^m}-a.e. {x}, we take {\nu_E(x)} to be the unique normal vector to the {m}-plane associated to {\tau_E(x)} which is positively oriented (i.e., {\tau_E(x)\wedge\nu_E(x) = e_1\wedge\dots\wedge e_{m+1}} where {e_1,\dots,e_{m+1}} stands for the canonical basis of {\mathbb{R}^{m+1}}).

3.1. De Giorgi’s almost everywhere regularity theorem

In order to study the size of the singular set {sing(\partial T_E)} of a locally perimeter minimizing set {E}, 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:

\displaystyle \textbf{E}(\partial T_E, B(x,r)) := \frac{1}{2 r^m}\int_{B(x,r)\cap\partial^*E} |\nu_E(y) - \overline{\nu_E}(x,r)|^2 d\mathcal{H}^m(y)

where

\displaystyle \overline{\nu_E}(x,r):=\frac{1}{\left|\int_{B(x,r)\cap\partial^*E}\nu_E(y) d\mathcal{H}^m(y)\right|}\int_{B(x,r)\cap\partial^*E}\nu_E(y) d\mathcal{H}^m(y)

By definition, on {B(x,r)}, the excess {\textbf{E}(\partial T_E, B(x,r))} is the variance of {\nu_{E}(y)} from its mean {\overline{\nu_E}(x,r)}.

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 {m\in\mathbb{N}}, there exists a dimensional threshold {\varepsilon=\varepsilon(m)>0} with the following property.Let {E\subset\mathbb{R}^{m+1}} be a set locally minimizing perimeter in an open set {\Omega\subset\mathbb{R}^{m+1}} (i.e., {\mathcal{H}^m(\partial^* E\cap B(x,r))\leq \mathcal{H}^m(\partial^* F \cap B(x,r))} for every {F} such that the symmetric difference {E\Delta F} is compactly contained in a ball {B(x,r)} with closure {\overline{B(x,r)}\subset\Omega}).

Let {x\in\textrm{supp}(\partial T_E)} and suppose that one has a {\varepsilon}-small excess

\displaystyle \textbf{E}(\partial T_E, B(x,r))<\varepsilon

for some ball {B(x,r)} compactly contained in {\Omega}. Then, in an appropriate system of coordinates, {\partial T_E\cap B(x,r/2)} is the graph of a smooth (and even real-analytic) function.

This theorem is sometimes called a {\varepsilon}-regularity theorem because it provides smoothness (of {\partial^* E} near {x}) whenever the excess is below a certain threshold {\varepsilon}. For this reason, we will refer to it as De Giorgi’s {\varepsilon}-regularity theorem in what follows.

Since {\mathcal{H}^m}-almost every {x\in\partial^*E} is a Lebesgue point of the approximate unit normal {\nu_E(x)}, an immediate consequence of De Giorgi’s {\varepsilon}-regularity theorem is the following restriction on the size of the singular set:

Corollary 7 (De Giorgi) If {E\subset\mathbb{R}^{m+1}} locally minimizes the perimeter in an open set {\Omega\subset \mathbb{R}^{m+1}}, then {\partial^*E} is almost everywhere regular in {\Omega}, i.e.,

\displaystyle \mathcal{H}^m(\Omega\cap sing(\partial T_E))=0

The proof of De Giorgi’s {\varepsilon}-regularity theorem is based on the so-called excess decay lemma:

Lemma 8 (Excess decay lemma) For each {m\in\mathbb{N}}, there exists a (small) dimensional constant {\alpha=\alpha(m)\in (0,1)} with the following property.Let {E} be a locally perimeter minimizing set (in some open set {\Omega\subset\mathbb{R}^{m+1}}). Then,

\displaystyle \textbf{E}(\partial T_E, B(x,\alpha r))\leq \frac{1}{2} \textbf{E}(\partial T_E, B(x,r))

whenever {x\in\textrm{supp}(\partial T_E)} and {\textbf{E}(\partial T_E, B(x,r))<\varepsilon=\varepsilon(m)}.

In fact, the excess decay lemma is the starting point of an iteration scheme of the open condition {\textbf{E}(\partial T_E,B(x,r))<\varepsilon} in {x\in\textrm{supp}(\partial T_E)} showing that the approximate unit normal {\overline{\nu_E}(x,r)} at scale {r} is Hölder continuous on {x} uniformly on the scale {r}.

From this uniform Hölder continuity property near {x} with {\textbf{E}(\partial T_E,B(x,r))<\varepsilon}, one can show that, for some {\gamma>0} (and, actually, for all {0<\gamma<1}), the set {\partial^*E} is the graph (in an adequate system of coordinates) of a function {\phi\in C^{1+\gamma}} in a neighborhood of {x}.

Since {E} minimizes the perimeter (locally in {\Omega}), the function {\phi\in C^{1+\gamma}} is a weak solution to the minimal surfaces equation

\displaystyle \textrm{div}\left(\frac{\nabla\phi}{\sqrt{1+|\nabla\phi|^2}}\right) = 0

In particular, we can invoke the regularity theory of quasilinear elliptic PDEs to conclude that the weak solution {\phi\in C^{1+\gamma}} of the previous equation is necessarily smooth (real-analytic).

In summary, we just saw the sketch of proof of DeGiorgi’s {\varepsilon}-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 ({\textbf{E}(\partial T_E,B(x,r))<\varepsilon}), but the excess does not decay as expected. After performing some scalings and rotations (to “blow-up” {\partial^*E} near {x}), one obtains a family {E_h}, {h>0}, of finite perimeter sets contained in the unit ball {B_1^{m+1}} of {\mathbb{R}^{m+1}} such that {0\in\textrm{supp}(\partial T_{E_h})} and the corresponding approximate unit normals deviate very little from an arbitrarily fixed direction (independent of {h}).

This means that {\partial^* E_h} can be approximated by the graph of a function {\phi_h} on the unit ball {B_1^m} of {\mathbb{R}^m} with a very small Lipschitz constant. By using this information to linearize the area functional, one gets that the perimeter of {E_h} has an expansion of the form

\displaystyle \textrm{perimeter of } E_h \sim \textrm{vol}(B_1^m) + \frac{1}{2}\int_{B_1^m} |\nabla\phi_h|^2

Since {E_h} are obtained from {E} by scalings and rotations, {E_h} also minimizes the perimeter. By combining this fact with the previous expansion of the area functional, one can prove that {\phi_h\rightarrow\phi} as {h\rightarrow\infty} and {\phi} is a harmonic function (as {\phi} minimizes the Dirichlet energy {\frac{1}{2}\int_{B_1^m}|\nabla\phi|^2}).

In this way, one gets a contradiction in the limit because any harmonic function {\phi} satisfies the decay estimate

\displaystyle \int_{B_{\alpha}^m}|\nabla\phi|^2 \leq \alpha^m\int_{B_1^m}|\nabla\phi|^2

(where {B_r^m = B(0,r)\subset\mathbb{R}^m}), a property that is incompatible with our assumption that {E} (and a fortiori {E_h}) 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 {r>0} and {x\in\mathbb{R}^{m+n}}, let us consider the scaling {\iota_{x,r}(y) = (y-x)/r}. Given a current {T}, let {T_{x,r}:=(\iota_{x,r})_* T} be defined by {(\iota_{x,r})_* T(\omega) = T((\iota_{x,r})_*\omega)}. The family {T_{x,r}\cap B(0,1)} of currents corresponds to zooming in {T\cap B(x,r)} at {x}.

Theorem 9 If {T} is an area-minimizing current and {x\in\textrm{supp}(T)-\textrm{supp}(\partial T)} is an “interior point”, then any weak-{\ast} limit {S} of the currents {T_{x,r}} (as {r\rightarrow 0}) is a cone without boundary, i.e.,

\displaystyle S_{0,r}=S \,\,\,\, \forall\, r>0 \quad \textrm{and} \quad \partial S=0

which is locally area-minimizing in {\mathbb{R}^{m+n}}, i.e.,

\displaystyle \|S\|(\Omega)\leq \|S'\|(\Omega)

whenever {\textrm{supp}(S-S')} is compactly supported in a bounded open set {\Omega\subset \mathbb{R}^{m+n}}.

Definition 10 Any cone {S} as in the theorem above is called a tangent cone to {T} at {x}.

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 {\varepsilon}-regularity theorem is the following characterization of the regular set of an area-minimizing currents in codimension one. Let {T} be a area-minimizing {m}-dimensional current in {\mathbb{R}^{m+1}}. Then:

{x\in\textrm{supp}(T)} is regular if and only if some tangent cone of {T} at {x} 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 {\varepsilon}-regularity theorem).

Remark 7 As we will see later, this characterization of {reg(T)} 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 {x_h\in sing(T_h)} are singular points of (locally in {\Omega}) area-minimizing currents {T_h}, and {x_h\rightarrow x(\in\Omega)} and {T_h\rightarrow T} as {h\rightarrow\infty}, then {x\in sing(T)}.

This intimate relationship between flat cones and regular points (in the codimension {1} 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 {E} be a set of finite perimeter. Assume that {E} locally minimizes perimeter in some open set {\Omega\subset \mathbb{R}^{m+1}}.

  • If {m\leq 6}, then {sing(\partial^*E\cap\Omega)=\emptyset};
  • If {m=7}, then {sing(\partial^*E\cap\Omega)} is discrete;
  • In general, {\mathcal{H}^{m-7+\delta}(sing(\partial^*E)\cap\Omega)=0} for all {\delta>0}, so that the Hausdorff dimension of the singular set is {\leq (m-7)}.

The proof of this theorem goes as follows. By the principle of persistence of singularities, the tangent cone {S} at a singular point {x\in sing(\partial^*E\cap\Omega)} (obtained the scalings {T_{x,r}}) is singular at the origin. Furthermore, we saw that {S} is a minimal cone (i.e., {S} is a cone without boundary which locally minimizes area). In other words, a tangent cone {S} at {x\in sing(\partial^*E\cap\Omega)} is always a singular minimal cone.

By the series of works of De Giorgi, Fleming, Almgren and Simons, there are no singular minimal cones when {m\leq 6}, and there is a singular minimal cone when {m=7}, namely, Simons’ cone

\displaystyle S=\{(z,w)\in\mathbb{R}^4\times\mathbb{R}^4=\mathbb{R}^8: |z|^2=|w|^2\}

From this, we deduce that the singular set {sing(\partial^*E\cap\Omega)} is empty when {m\leq 6} (and it might be non-empty when {m=7}), 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 {m=7} 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 {m=7}. If {sing(\partial^*E\cap\Omega)} is not discrete for some {E\subset\mathbb{R}^8}, we would have a sequence {(x_h)_{h\in\mathbb{N}}\subset sing(\partial^*E\cap\Omega)} converging to {x\in sing(\partial^*E\cap\Omega)} (by the principle of persistence of singularities).

By making a blowup of {\partial T_E} (on {x}) at scales {r_h:=|x_h-x|>0}, we obtain a tangent cone {S=\lim\limits_{h\rightarrow\infty} (\partial T_E)_{x,r_h}} which is singular at the origin {0} and at some point {p=\lim\limits_{h\rightarrow\infty}(x_h-x)/r_h} in the unit sphere of {\mathbb{R}^8} (again by the principle of persistence of singularities).

Since {S} is a cone, the half-line {L} between {0} and {p} is contained in {S}. By doing a new blowup at the middle point of the segment between {0} and {p}, we obtain a new tangent cone {S'} of the form {S'=S''\times \mathbb{R}} (where the factor {\mathbb{R}} comes from the contribution in the limit of {L} to the blowup) such that {S''} would be a singular minimal cone in {\mathbb{R}^7}, a contradiction with the fact that there are no singular minimal cones in {\mathbb{R}^{m+1}} when {m\leq 6}.

At this point, we dispose of sharp regularity results for solutions of the generalized Plateau problem in codimension {1} (area-minimizing {m}-dimensional currents in {\mathbb{R}^{m+1}}), 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 {m}-dimensional currents in {\mathbb{R}^{m+n}} when {n\geq 2} 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 {\iota_{x,r}} 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 {2}-disk

\displaystyle D:=\{(z, w)\in\mathbb{C}^2: z^3=w^2, |z|\leq 1\}

with boundary {\partial D=\Gamma:=\{(\zeta^2,\zeta^3)\in\mathbb{C}^2: |\zeta|=1\}}.

4.1. Examples of center manifolds and multivalued functions

Note that {D} is locally area-minimizing: indeed, as we already mentioned in Example 2, this is a direct consequence of the calibration method.

More precisely, given an integral rectifiable current {T=(R,\tau,\theta)} (with orientation {\tau}), we say that {T} is calibrated by a smooth closed differential {m}-form {\omega} on an open subset {\Omega\subset \mathbb{R}^{m+n}} with {|\omega(x)|\leq 1} for all {x\in\Omega} whenever

\displaystyle \langle\tau(x),\omega(x)\rangle = 1

for {\|T\|}-almost every {x\in\Omega}.

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 {S\subset \mathbb{C}^d} of complex dimension {k} induces a current that is calibrated by

\displaystyle \omega=\frac{1}{k!}\sum\limits_{l=1}^d dx_l\wedge dy_l

In particular, these facts imply that {D} is locally area-minimizing.

Observe that this shows that Almgren’s theorem 1 is an optimal solution to Plateau’s problem in codimension {>1}: in fact, {D} is a integral rectifiable locally area-minimizing {2}-dimensional current in {\mathbb{C}^2=\mathbb{R}^4} with a singular set {sing(D)=\{(0,0)\}} of Hausdorff dimension {0=2-2}.

By considering the scalings

\displaystyle \begin{array}{rcl} (\iota_{(0,0),r})_{\ast} D=\{(z,w)\in\mathbb{C}^2: (rz,rw)\in D\}=\{(z,w)\in\mathbb{C}^2: rz^3=w^2, |z|\leq 1/r\}, \end{array}

we see that the tangent cone to {D} at {O=(0,0)} (i.e., the weak-{\ast} limit of {(\iota_{(0,0),r})_{\ast} D} as {r\rightarrow 0}) is the integral rectifiable current

\displaystyle S=(\mathbb{R}^2\times\{0\}, e_1\wedge e_2, 2)

associated to the horizontal plane {\mathbb{R}^2\times\{0\}\subset \mathbb{R}^2\times\mathbb{R}^2=\mathbb{R}^4} with multiplicity two. Geometrically, this means that, after scaling (“zooming in near the origin”), the two branches of {D} (corresponding to the two determinations of {w}, i.e., square-root of {z^3}, near {(0,0)}) merge together into the plane {\mathbb{R}^2\times \{0\}}.

It is worth to notice that the tangent cone {\mathbb{R}^2\times\{0\}} of {D} at {(0,0)} is flat, but, nevertheless, {(0,0)} is a singular point of {D} due to its branch nature. In other words, contrary to the case of Plateau’s problem in codimension {1}, 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 {D} (near the origin) is that it behaves differently along the {z} and {w} coordinates. This suggest that we should replace the homogenous scalings {\iota_{(0,0), r}} by non-homogenous scalings taking into account the distinct behaviors of the {z} and {w} variables along {D}. Moreover, by analogy with the case of Plateau’s problem in codimension {1}, 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 {D} near the origin is {\phi_{\lambda}(z,w) = (\lambda^2 z, \lambda^3 w)}. In fact, the main point here is that this scaling fixes {D} (that is, {\phi_{\lambda}(D)=D}) because the functions {w_1(z)} and {w_2(z)} corresponding to the determinations of the square-root of {z^3} (i.e., {w_1(z)^2=w_2(z)^2=z^3}) 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

\displaystyle \mathcal{W}=\{(z,w)\in\mathbb{C}^2: (w-z^2)^2=z^5, |z|\leq1\}

By the calibration method, we have that {\mathcal{W}} is locally area-minimizing.

Also, it is not hard to check that {\mathcal{W}} is singular at the origin. If one tries to play with non-homogenous scalings {(z,w)\mapsto (\lambda^{r} z, \lambda^{s})w}, {r,s\in \mathbb{N}}, then it is possible to verify that the sole scaling producing an interesting (non-trivial and “harmonic”) blowup is

\displaystyle \psi_{\lambda}(z,w) = (\lambda z, \lambda^2 w)

A quick calculation reveals that {(\psi_{\lambda})_{\ast}\mathcal{W}} converges (as {\lambda\rightarrow\infty}) to the current induced by the smooth complex curve

\displaystyle C=\{(z,w)\in\mathbb{C}^2: w=z^2\}

In particular, this non-homogenous scaling of {\mathcal{W}} 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 {\mathcal{W}=\{(w-z^2)^2=z^5\}} is that the induced current has a “regular part” given by the smooth curve {C=\{w-z^2=0\}} and a “singular branching” part coming from the determination of the square-root of {z^5}. This means that the parametrization of the current {\mathcal{W}} with the flat coordinates {(z,w)} is not adequate: instead, it is a better idea to parametrize {\mathcal{W}} in terms of the smooth complex curve {C}, i.e., we have a complex parameter {\zeta(=z)} for the smooth complex curve {C} (“regular part of {\mathcal{W}}”), and we have two functions {u_1(\zeta)} and {u_2(\zeta)} with {u_1(\zeta)^2=u_2(z)^2 = z^5} parametrizing the “singular branching part of {\mathcal{W}}”.

This decomposition led Almgren to introduce the notions of center manifolds (“regular parts”) and multivalued harmonic functions (“singular branching part”). In the case of {\mathcal{W}}, the curve {C} is the center manifold, and the multivalued function {\zeta\mapsto \{u_1(\zeta), u_2(\zeta)\}} 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 {C}) is defined as follows. We consider {Q\in\mathbb{N}} (“number of branches”), and we let

\displaystyle \mathcal{A}_{Q}(\mathbb{R}^{n})=(\mathbb{R}^n)^Q/\sim

be the space of unordered {Q}-tuples of points in {\mathbb{R}^n}. Here, {\sim} is the equivalent relation {(x_1,\dots,x_Q)\sim (x_{\sigma(1)},\dots, x_{\sigma(Q)})} for all {(x_1,\dots, x_Q)\in(\mathbb{R}^n)^Q} and {\sigma\in Sym_Q} is a permutation of {\{1,\dots, Q\}}.

In this setting, a {Q}valued function is simply a function {f:C\rightarrow\mathcal{A}_Q(\mathbb{R}^n)}.

Note that {\mathcal{A}_Q(\mathbb{R}^n)} is a metric space. For example, this space possesses the (Wasserstein) metric

\displaystyle W_2((P_1,\dots, P_Q), (S_1,\dots, S_Q)) := \min\limits_{\sigma\in S_Q} \sqrt{\sum\limits_{k=1}^Q |P_i-S_{\sigma(i)}|^2}

This metric space structure on {\mathcal{A}_{Q}(\mathbb{R}^n)} allows us to talk about Lipschitz {Q}-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 {Q}-valued harmonic functions. First, one tries to isometrically embed {\mathcal{A}_Q(\mathbb{R}^n)} into some Euclidean space {\mathbb{R}^p}. Then, one defines Sobolev spaces, Dirichlet energy and harmonicity (i.e., minimizers of the Dirichlet energy) of {Q}-valued functions with the aid of the nice structures of {\mathbb{R}^p}: for example, the Sobolev space {W^{1,p}(C,\mathcal{A}_Q(\mathbb{R}^n))} is defined as the space of functions {f\in W^{1,p}(C,\mathbb{R}^p)} (in the usual sense) such that {f(x)\in\mathcal{A}_Q(\mathbb{R}^n)} for almost every {x\in C}.

As it turns out, these definitions work well when {\mathcal{A}_Q(\mathbb{R}^n)} is isometrically embedded in {\mathbb{R}^p} in such a way that it is bi-Lipschitz equivalent to a Lipschitz retract of {\mathbb{R}^p}. This led Almgren to prove that, in fact, this is always the case for some {p=p(n,Q)}.

In their proof of Almgren’s theorem, DeLellis-Spadaro proposed an alternative intrinsic approach for the definition of {Q}-valued harmonic functions: the idea is to rely only on the metric structure {W_2} of {\mathcal{A}_{Q}(\mathbb{R}^n)} (without making any mention to ambient spaces like {\mathbb{R}^p}). More precisely, we say that {f:C\rightarrow \mathcal{A}_Q(\mathbb{R}^n)} belongs to the Sobolev space {W^{1,p}(C,\mathcal{A}_Q(\mathbb{R}^n))} if there are {\phi_j\in L^2(C)}, {j=1,\dots, m} such that

  • for each {T\in\mathcal{A}_Q(\mathbb{R}^n)}, the function {x\mapsto W_2(f(x), T)} belongs to the (usual) Sobolev space {W^{1,2}(C)};
  • for each {T\in\mathcal{A}_Q(\mathbb{R}^n)} and {j\in\{1,\dots, m\}}, {|\partial_j W_2(f,T)(x)|\leq \phi_j} for almost every {x\in C}.

In other words, we measure the (Sobolev) regularity of {f:C\rightarrow\mathcal{A}_Q(\mathbb{R}^n)} by testing the Sobolev regularity of the real-valued functions on {C} obtained by measuring the {W_2}-distance of the values {f(x)} of {f} to arbitrary points {T\in\mathcal{A}_{Q}(\mathbb{R}^n)}.

Remark 11 One can show that there are (minimal) functions {|\partial_j f|} such that {|\partial_j f|\leq \phi_j} for any {L^2}-functions {\phi_j} satisfying the inequalities above.

In this context, the Dirichlet energy of {f\in W^{1,2}(C,\mathcal{A}_Q(\mathbb{R}^n))} is

\displaystyle \textrm{Dir}(f):=\int_{C} |Df|^2

where {|Df|:=\sum\limits_{j=1}^m |\partial_j f|}. Moreover, we say that a {Q}-valued function {f\in W^{1,2}(C,\mathcal{A}_Q(\mathbb{R}^n))} is harmonic if {f} minimizes the Dirichlet energy (with Dirichlet or Neumann boundary condition).

In any event, the notion of harmonic {Q}-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 {\Omega\subset \mathbb{R}^m} be a bounded open subset with Lipschitz boundary (e.g., a piece of the center manifold {C}), and {g\in W^{1,2}(\Omega, \mathcal{A}_Q(\mathbb{R}^n))}.Then, there exists {f:\Omega\rightarrow\mathcal{A}_{Q}(\mathbb{R}^n)} minimizing the Dirichlet energy in the class of {h\in W^{1,2}(\Omega, \mathcal{A}_Q(\mathbb{R}^n))} with {h|_{\partial\Omega}=g}. Furthermore:

  • any such {f} is locally {\kappa}-Hölder continuous for a certain {\kappa=\kappa(m,Q)>0}, and {|Df|} is locally {L^p} for some {p=p(m,n,Q)>2};
  • there exists a closed (“singular”) subset {sing(f)\subset \Omega} with Hausdorff dimension {\leq (m-2)} such that the graph

    \displaystyle \{(x,y): x\in\Omega-sing(f), y\in f(x)\}

    of {f} outside {sing(f)} is a smooth embedded {m}-dimensional submanifold of {\mathbb{R}^{m+n}}.

At this point, we are ready to give a global (rough) description of the proof of Almgren’s theorem 1 (assuming the statement of Theorem 12).

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 {Q}-valued harmonic function as the limit of these Lipschitz multivalued functions;
  • finally, one applies the regularity estimate of Theorem 12 to this {Q}-valued harmonic function to conclude the proof of Theorem 1 (i.e., to deduce that the singular set of an area-minimizing integral rectifiable {m}-dimensional current has Hausdorff dimension {\leq (m-2)}).
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