Higher integrability and approximation of minimal currents

HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMALCURRENTS

CAMILLO DE LELLIS AND EMANUELE NUNZIO SPADARO

Abstract. We give a new, simpler proof of the main approximation theorem for areaminimizing current contained in Almgren’s Big regularity paper, [2]. Our proof relies ona new estimate concerning the higher integrability of the quantity called here the excessdensity, which in turn we link to an analogous property of harmonic multiple valuedfunctions.

0. Introduction

In what follows, we consider integer rectifiable m-currents T supported in some opencylinder Cr(y) = Br(y)! Rn " Rm ! Rn and satisfying the following assumption:

!#T = Q !Br(y)" and "T = 0, (H)

where ! : Rm ! Rn # Rm is the orthogonal projection and m, n,Q are fixed positiveintegers. For such currents, we denote by eT the non-negative excess measure and byEx(T, Cr(y)) the cylindrical excess, respectively defined by

eT (A) := M!T (A! Rn)

"$Q |A| for every Borel A " Br(y),

Ex(T, Cr(y)) :=eT (Bs(x))

|Bs(x)| =eT (Bs(x))

#msm.

The following theorem, proved by De Giorgi [5] in the case n = Q = 1, is due in itsgenerality to Almgren, who spends almost the entire third chapter of his Big regularitypaper [2] to accomplish it.

Theorem 0.1. There exist constants C, $, %0 > 0 with the following property. For everymass-minimizing, integer rectifiable m-current T in the cylinder C4 which satisfies (H) andE = Ex(T, C4) < %0, there exist a Q-valued function f % Lip(B1,AQ(Rn)) and a closed setK " B1 such that

Lip(f) & CE!, (0.1a)

graph(f |K) = T (K ! Rn) and |B1 \K| & CE1+!, (0.1b)####M

!T C1

"$Q #m $

ˆ

B1

|Df |2

2

#### & C E1+!. (0.1c)

The most interesting aspects of Theorem 0.1 are the use of multiple-valued functions(necessary when n > 1, as for the case of branched complex varieties) and the gain of

1

2 C. DE LELLIS AND E. N. SPADARO

a small power E! in the three estimates (0.1). Observe that the usual approximationtheorems, which cover the case Q = 1 and stationary currents, are stated with $ = 0.

In this note, we provide a di!erent, much simpler proof of Almgren’s theorem. In doingso we establish a higher integrability estimate for the Lebesgue density dT of the excessmeasure eT , called the excess density,

dT (x) := lim sups!0

eT (Bs(x))

#m sm.

Theorem 0.2. There exist constants p > 1 and C, %0 > 0 such that, for every mass-minimizing, integer rectifiable m-current T satisfying (H) and E = Ex(T, C4) < %0, itholds

ˆ

{d"1}#B2

dp & C Ep. (0.2)

This estimate is not stated in [2], but it can be deduced from some of the argumentstherein. These arguments, which involve quite elaborate constructions and use severalintricate covering algorithms, are the most involved part of Almgren’s proof. In the caseQ = 1, we know a posteriori that T coincides with the graph of a C1," function over B2

(see [5], for instance). However, for Q ' 2 this conclusion does not hold and Theorem 0.2has, therefore, an independent interest. The main contribution of the paper is to give amuch shorter and conceptually clearer derivation of (0.2).

However, we introduce several new ideas even in the derivation of Theorem 0.1 fromTheorem 0.2. In particular:

• we introduce a powerful maximal function truncation technique to approximategeneral integer rectifiable currents with multiple valued functions;

• we give a simple compactness argument to conclude directly a first harmonic ap-proximation of T ;

• we link Theorem 0.2 to a similar higher integrability property for the gradient ofDir-minimizing multiple valued functions, observed here for the first time;

• we give a new proof of the existence of Almgren’s “almost projections” !#.

Remark 0.3. The careful reader will notice two important di!erences between the mostgeneral approximation theorem of Almgren’s book and Theorem 0.1.

First of all, though the smallness hypothesis Ex(T, C4) < %0 is the same, the estimatescorresponding to (0.1) are stated in [2] in terms of the “varifold excess”, a quantity smallerthan Ex. An additional argument, which we report in the appendix, shows that Ex andthe varifold excess are indeed comparable. This is obtained from a strenghtened version ofTheorem 0.1, included in the appendix, see Theorem A.1. Our proof is, to our knowledgenew and might have an independent interest.

Second, the most general result of Almgren is stated for currents in Riemannian mani-folds. However, we believe that such generalization follows from standard modifications ofour arguments and we plan to address this issue elsewhere.

HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS 3

0.1. Graphical approximation of currents. Given a normal m-current T , following[4] we can view the slice map x (# )T,!, x* as a BV function taking values in the spaceof 0-dimensional currents (endowed with the flat metric). Indeed, by a key estimate ofJerrard and Soner (see [4] and [10]), the total variation of the slice map is controlled bythe mass of T and "T . In the same vein, following [6], Q-valued functions can be viewedas Sobolev maps into the space of 0-dimensional currents.

Combining these points of view with standard truncation arguments, we develop a pow-erful and simple Lipschitz approximation technique, which gives a systematic tool to findgraphical approximations of integer rectifiable currents. For this purpose we introduce themaximal function of the excess measure of a current T satisfying (H):

MT (x) := supBs(x)$Br(y)

eT (Bs(x))

#m sm= sup

Bs(x)$Br(y)Ex(T, Cs(x)).

Our main approximation result is the following and relies on an improvement of the usualJerrard–Soner estimate.

Proposition 0.4 (Lipschitz approximation). There exist constants c, C > 0 with thefollowing property. Let T be an integer rectifiable m-current in C4s(x) satisfying (H)and let & % (0, c) be given. Set K :=

$MT < &

%+ B3s(x). Then, there exists u %

Lip(B3s(x),AQ(Rn)) such that graph(u|K) = T (K ! Rn), Lip(u) & C &12 and

|B3s(x) \K| & C

&eT

!{MT > &/2}

". (0.3)

In what follows, we will often choose & = E2" (= Ex(T, C4s(x))2"), for some ' %(0, (2m)%1). The map u given by Proposition 0.4 will then be called the E"-Lipschitz (orbriefly the Lipschitz ) approximation of T in C3s(x). We therefore conclude the followingestimates:

Lip(u) & C E", |B3s(x) \K| & C E%2 " eT!{MT > E2 "/2}

",

ˆ

B3s(x)\K|Du|2 & eT

!{MT > E2 "/2}

".

(0.4)

In particular, the function f in Theorem 0.1 is given by the E"-Lipschitz approximationof T in C1, for a suitable choice of '.

0.2. Harmonic approximation. The second step in the proof of Theorem 0.2 is a com-pactness argument which shows that, when T is mass-minimizing, the approximation f isclose to a Dir-minimizing function w, with an o(E) error.

Theorem 0.5 (o(E)-improvement). Let ' % (0, (2m)%1). For every & > 0, there exists%1 > 0 with the following property. Let T be a rectifiable, area-minimizing m-current inC4s(x) satisfying (H). If E = Ex(T, C4s(x)) & %1 and f is the E"-Lipschitz approximationof T in C3s(x), then

ˆ

B2s(x)\K|Df |2 & & eT (B4s(x)), (0.5)


and there exists a Dir-minimizing w % W 1,2(B2s(x),AQ(Rn)) such thatˆ

B2s(x)

G(f, w)2 +

ˆ

B2s(x)

!|Df |$ |Dw|

"2 & & eT (B4s(x)). (0.6)

This theorem is the multi-valued analog of De Giorgi’s harmonic approximation, whichis ultimately the heart of all the regularity theories for minimal surfaces. Our compactnessargument is, to our knowledge, new (even for n = 1) and particularly robust. Indeed, weexpect it to be useful in more general situations.

0.3. Higher integrability estimates and Theorem 0.2. The first key point in ourproof of Theorem 0.2 is that most of the energy of a Dir-minimizer lies where the gradientis relatively small. The following quantitative version of this statement can be proved viaa classical reverse Holder inequality (see [12] for a di!erent proof and some improvements).Curiously, though Chapter 3 of Almgren’s book has statements about the energy of Dir-minimizing functions in various regions, Theorem 0.6 is stated nowhere and there is nohint to reverse Holder inequalities.

Theorem 0.6 (Higher integrability of Dir-minimizers). Let "& "" " "" Rm be opendomains. Then, there exist p > 2 and C > 0 such that

,Du,Lp(!!) & C ,Du,L2(!) for every Dir-minimizing u % W 1,2(",AQ(Rn)). (0.7)

Theorems 0.5 and 0.6 imply the following key estimate, which leads to Theorem 0.2 viaan elementary “covering and stopping radius” argument.

Proposition 0.7. For every ( > 0, there is %2 > 0 with the following property. Let T be aninteger rectifiable, area-minimizing current in C4s(x) satisfying (H). If E = Ex(T, C4s(x)) &%2, then

eT (A) & ( Esm for every Borel A " Bs(x) with |A| & %2|B4s(x)|. (0.8)

0.4. Almgren’s estimate and Theorem 0.1. Using now Theorem 0.2, we can provethe following estimate, which is explicitely stated by Almgren and is the last ingredient,in conjunction with Proposition 0.4, to derive Theorem 0.1.

Theorem 0.8. There exist constants ), C > 0 such that, for every mass-minimizing,integer rectifiable m-current T in C4 satisfying (H) and E = Ex(T, C4) < %0, it holds

eT (A) & C E!E$ + |A|$

"for every Borel A " B4/3. (0.9)

The proof of Theorem 0.8 is the only part where we follow essentially Almgren’s strategy.The main point is to estimate the size of the set over which the graph of the Lipschitzapproximation f di!ers from T . As in many standard references, in the case Q = 1 this isachieved comparing the mass of T with the mass of graph (f - *E!), where * is a smoothconvolution kernel and # > 0 a suitably chosen constant.

However, for Q > 1, the space AQ(Rn) is not linear and we cannot regularize f byconvolution. To bypass this problem, we follow Almgren and view AQ as a subset of alarge Euclidean space (via a biLipschitz embedding "). We can then convolve the map ".f


and project the convolution back on the set "(AQ). But to do this e#ciently in energeticalterms, we need an “almost” projection, denoted by !#

µ, which is very little expensive interms of energy in the µ-neighborhood of "(AQ(Rn)) (µ is a parameter which will be tunedaccordingly). At this point Theorem 0.2 enters in a crucial way in estimating the size ofthe set where the regularization of " . f is far from "(AQ(Rn)).

We give here a clear and direct proof of the existence of !'µ, which avoids some ofthe technical complications of [2]. The !#

µ are slightly di!erent from Almgren’s almostprojections, but similar in spirit. Moreover, our argument is di!erent and uses e#ciently theKirzsbraun’s Theorem, yielding more explicit estimates in terms of the crucial parametersinvolved.

0.5. Plan of the paper. The paper is organized as follows. Section 1 summarizes thenecessary tools from the theory of multiple valued functions. In particular, in this sectionwe prove a concentration-compactness lemma which is an essential step of Theorem 0.5.Section 2 contains a summary of the Ambrosio-Kirchheim theory of currents with thenecessary improvement of the Jerrard–Soner estimate and gives the proof of Proposition0.4 together with some results on the graphs of Q-valued functions. In Section 3 weproceed with the proof of Theorem 0.5 and in Section 4 with that of the higher integrabilityestimates in Theorem 0.6, Proposition 0.7 and Theorem 0.2. Section 5 is devoted to theproof of Theorem 0.8 and the final derivation of the main Theorem 0.1. In Section 6 weprove the existence of the projections !#

µ. The Appendix contains three sections. Two ofthem are devoted to give a strengthened version of Theorem 0.1, from which we derive acomparison between Ex and the varifold excess. The third one gives a simple proof of thestructure properties of the graphs of Lipschitz multiple valued functions when viewed asinteger rectifiable currents.

1. Multiple valued functions

1.1. Q-valued functions. We follow here the notation and terminology of [6], which di!erslightly from those in Almgren’s big regularity paper [2]. Q-valued functions are maps froma domain " " Rm taking values in the metric space AQ(Rn) of unordered Q-tuple of vectorsin Rn. For the rigorous definitions of AQ, of the metric G, of Sobolev Q-valued maps andof the Dirichlet energy, we refer to [6]. Here, we recall the following theorem (cp. to [6,Theorem 2.1 and Corollary 2.2]).

Theorem 1.1 (The maps " and !). There exist N = N(Q, n) and an injective function" : AQ(Rn) # RN with the following three properties:

(i) Lip(") & 1;(ii) Lip("%1|Q) & C(n, Q), where Q = "(AQ);

(iii) |Df | = |D(" . f)| a.e. for every f % W 1,2(",AQ).

Moreover, there exists a Lipschitz projection ! : RN # Q which is the identity on Q.

Many properties of classical Sobolev functions are inherited by Q-functions. In particu-lar, we have the following results (cp. to [6, Section 2.2]).


Lemma 1.2 (Lipschitz approximation of Sobolev maps). Let " " Rm be a Lipschitzdomain and f % W 1,2(",AQ). Then, for every % > 0, there exists f% % Lip(",AQ) suchthat

ˆ

!

G(f, f%)2 +

ˆ

!

!|Df |$ |Df%|

"2 & %.

If f |&! % W 1,2("",AQ), then f% can be chosen to satisfy alsoˆ

&!

G(f, f%)2 +

ˆ

&!

!|Df |$ |Df%|

"2 & %.

Lemma 1.3 (Interpolation lemma). There exists a constant C = C(m, n,Q) with thefollowing property. Assume f % W 1,2(Br,AQ(Rn)) and g % W 1,2("Br,AQ(Rn)) are givenmaps such that f |&Br % W 1,2("Br,AQ(Rn)). Then, for every % %]0, r[ there is a functionh % W 1,2(Br,AQ(Rn)) such that h|&Br = g and

Dir(h, Br) & Dir(f, Br) + % Dir(g, "Br) + % Dir(f, "Br) + C %%1

ˆ

&Br

G(f, g)2.

Moreover, if f and g are Lipschitz, then h is as well Lipschitz with the estimate

Lip(h) & C

&Lip(f) + Lip(g) + %%1 sup

&Br

G(f, g)

'. (1.1)

1.2. Concentration compactness. The aim of this section is to show the following.

Lemma 1.4 (Concentration Compactness). Let (gl)l(N be a sequence in W 1,2(",AQ) withsupl Dir(gl, ") < +/. Then, there are maps +j % W 1,2(",AQj), with Q =

(Jj=1 Qj and

J ' 1, and points ylj % Rn, with |yl

j $ yli|# +/ for i 0= j, such that, up to a subsequence

(note relabeled), the Q-valued maps #l =(J

j=1!,ylj. +j" satisfy

liml!+)

,G(gl, #l),L2(!) = 0 . (1.2)

Moreover, if "& is an open subset of " and Jl a sequence of Borel sets with |Jl|# 0, then

lim infl

)ˆ

!!\Jl

|Dgl|2 $ˆ

!!|D#l|2

*' 0, (1.3)

and lim inf l

´

(|Dgl|2 $ |D#l|2) = 0 holds, if and only if lim inf l

´

(|Dgl|$ |D#l|)2 = 0.

Before coming to the proof, we recall two lemmas of [6]. Here, s(T ) and d(T ) are,respectively, the separation and the diameter of a point T =

(i !Pi":

s(T ) := min$|Pi $ Pj| : Pi 0= Pj

%and d(T ) := max

i,j|Pi $ Pj|.

Lemma 1.5. (Retractions [6, Lemma 3.7]). Let T % AQ, r < s(T )/4. Then, there existsa retraction - : AQ # Br(T ) such that

(i) G(-(S1), -(S2)) < G(S1, S2) if S1 /% Br(T ),(ii) -(S) = S for every S % Br(T ).


Lemma 1.6. (Splitting [6, Lemma 3.8]). For every 0 < % < 1, we set .(%, Q) = (%/3)3Q.

Then, for every T % AQ, there exists a point S % AQ such that

.(%, Q) d(T ) < s(S) and G(S, T ) < % s(S).

Proof of Lemma 1.4. First of all, by [6, Proposition 2.12], we can find gl % AQ(Rn) suchthat

ˆ

G(gl, gl)2 & c

ˆ

|Dgl|2 & C,

where c and C are constants independent of l. We prove (1.2) by induction on Q anddistinguish two cases.

Case 1: lim inf l d(gl) < /. After passing to a subsequence, we can then find yl % Rn suchthat the functions ,yl

. gl are equi-bounded in the W 1,2-distance. Hence, by the Sobolevembedding [6, Proposition 2.11], there exists a Q-valued + such that ,yl

. gl converges to +in L2. Note that, when Q = 1, we are always in this case.

Case 2: liml d(gl) = +/. By Lemma 1.6 there are points Sl % AQ such that

s(Sl) ' .1/8 d(gl) and G(Sl, gl) & s(Sl)/8.

Set rl = s(Sl)/4 and let /l be the retraction into Brl(Sl) provided by Lemma 1.5. Thus,

Sl =(J

i=1 ki !P il ", with mini*=j |P i

l $P jl | = s(Sl). In principle, the numbers I and ki depend

on l but, up to a subsequence, we can assume that they do not depend on l.Clearly, the functions hl = /l . gl satisfy Dir(hl, ") & Dir(gl, ") and can be decomposed

as the superposition of ki-valued functions zil , with ki < Q,

hl =J+

i=1

#zi

l

$, with ,G(zi

l , ki

#P i

l

$),) & rl.

The existence of #l such that ,G(hl, #l),L2 # 0 follows, hence, by induction and, with-out loss of generality, we also can assume that liml |yl

i $ ylj| = +/ for i 0= j. Showing

,G(hl, gl),L2 # 0, therefore, completes the proof of (1.2).To this aim, recall first that {gl 0= hl} = {G (gl, Sl) > rl} 1 {G (gl, gl) > rl/2}. Thus,

| {gl 0= hl} | & | {G (gl, gl) > rl/2} | &C

r2l

ˆ

{G(gl,gl)>rl2 }

G (gl, gl) &C

(d(gl))2.

Since d(gl) # +/, we conclude |{gl 0= hl}| # 0. Next, since /l(gl) = gl and Lip(/l) = 1,we have G(hl, gl) & G(gl, gl). Therefore, by Sobolev embedding, for m ' 3 we infer

ˆ

B2

G(hl, gl)2 =

ˆ

{gl *=hl}G(hl, gl)

2 & 2

ˆ

{hl *=gl}G(hl, gl)

2 + 2

ˆ

{hl *=gl}G(gl, gl)

2

& 4

ˆ

{hl *=gl}G(gl, gl)

2 & ,G (gl, gl),2L2" |{hl 0= gl}|1%

22"

& C

d(gl)4

m#2

)ˆ

B2

|Dgl|2*m+2

m#2

.


Recalling again that d(gl) diverges, this shows ,G(hl, gl),L2 # 0. The obvious modificationwhen m = 2 is left to the reader.

Now we come to the proof of (1.3). Arguing as in case 2, we find hl =(

i !zil" such that

,G(hl, gl),L2 # 0, ,G(,%yil. zi

l , +i),L2 # 0 and |Dhl| & |Dgl|. Therefore, we conclude that

D(" . ,%yli. zi

l )'0 D(" . +i), (1.4)

and hence

Dir(+i, "&) =

ˆ

!!|D(" . +i)|2 & lim inf

l

ˆ

!!\Jl

|D(" . ,%yli. zi

l )|2 = lim infl

ˆ

!!\Jl

|Dzil |2. (1.5)

Summing over i, we obtain (1.3). As for the final claim of the lemma, let # =(

i !+i" andassume Dir(gl, ") # Dir(#, "). Set Jl := {gl 0= hl} and recall that |Jl| # 0. Thus, by(1.3), we conclude that

´

Jl|Dgl|2 # 0 and hence, that

!|Dgl|$ |Dhl|

"# 0 strongly in L2.

On the other hand, we also infer

lim supl

+

i

ˆ

|D(" . ,%yli. zi

l )|2 = lim supl

ˆ

|Dhl|2 &ˆ

!

|D#|2 .

In conjunction with (1.5), this estimate leads to liml

´

|D(" . ,%yli. zi

l )|2 =´

|D(" . +i)|2,which, in turn, by (1.4), implies D(" . ,%yl

i. zi

l ) # D(" . +i) strongly in L2. Therefore,

|Dhl|# |D#| in L2, thus concluding the proof. !

2. The Lipschitz approximation

2.1. The modified Jerrard–Soner estimate. For the sake of brevity, we do not intro-duce the machinery of metric space valued BV functions, developed by Ambrosio in [3],which nevertheless remains the most elegant framework for this theory – cp. to [4]. Weadopt the definitions and the standard notation due to Federer, see [8] and [11]. An inte-ger rectifiable 0-current S in Rn with finite mass is simply a finite sum of Dirac’s deltas:S =

(hi=1 )i $xi , where h % N, )i % {$1, 1} for every i and the xi’s are (not necessarily

distinct) points in Rn. The space of such measures, denoted by I0(Rn), is a Banach spacewhen endowed with the flat norm

F(S) := sup$)S,1* : 1 % C1(Rn), ,1,) , ,D1,) & 1

%,

where )S, 1* =(

i )i 1(xi). Note that F($x, $y) = |x$ y| if |x$ y| & 1.Let T be an integer rectifiable m-dimensional normal current on C4. The slicing map

x (# )T,!, x* takes values in I0(Rm+n) and is characterized by (see Section 28 of [11])ˆ

B4

,)T,!, x* , 2(x, ·)

-dx = )T,2 dx* for every 2 % C)

c (C4).

Note that, in particular, supp ()T,!, x*) 1 !%1({x}). Moreover, (H) implies that, if wewrite )T,!, x* =

(i )i$(x,yi), then

(i )i = Q.

Our estimates concerns the push-forwards of the slices )T,!, x* into the vertical direction,

Tx := q'

!)T,!, x*

"% I0(Rn), (2.1)


where q : Rm+n # Rn is the orthogonal projection on the last n components. Tx ischaracterized through the identity

ˆ

B4

)Tx, 1*3(x) dx = )T,3(x) 1(y) dx* for every 3 % C)c (B4), 1 % C)

c (Rn).

Proposition 2.1 (Modified BV estimate). Let T be an integer rectifiable current in C4

satisfying (H). For every 1 % C)c (Rn), set $((x) := )Tx, 1*. If ,1,) , ,D1,) & 1, then

$( % BV (B4) and satisfies

!|D$(|(A)

"2 & 2 eT (A)M (T (A! Rn)) for every Borel A " B4. (2.2)

Note that (2.2) is a refined version of the usual Jerrard–Soner estimate, where the righthand side would rather be (T (A! Rn))2 (cp. to [4]). Note also that assumption (H) canbe dropped if in (2.2) eT is replaced by its total variation.

Proof. It is enough to prove (2.2) for every open set A 1 B4. To this aim, recall that

|D$(|(A) = sup

&ˆ

A

$((x) div3(x) dx : 3 % C)c (A, Rm), ,3,) & 1

'. (2.3)

For any vector field 3 as in (2.3), (div 3(x)) dx = d', where

' =+

j

3j dxj and dxj = ($1)j%1dx1 2 · · · 2 dxj%1 2 dxj+1 2 · · · 2 dxm.

Moreover, by the characterization of the slice map, we haveˆ

A

$((x) div3(x) dx =

ˆ

B4

)Tx, 1(y)* div3(x) dx = )T,1(y) div 3(x) dx*

= )T,1 d'* = )T, d(1 ')* $ )T, d1 2 '* = $)T, d1 2 '* , (2.4)

where in the last equality we used the hypothesis "T C4 = 3.Observe that the m-form d1 2 ' has no dx component, since

d1 2 ' =m+

j=1

n+

i=1

($1)j%1 "1

dyi(y) 3j(x) dyi 2 dxj.

Let 4e be the m-vector orienting Rm and write 4T = (4T · 4e)4e + 4S (see Section 25 of [11] forthe scalar product on m-vectors). We then conclude that )T, d1 2 '* = )4S · ,T, , d1 2 '*andˆ

A+Rn

|4S|2 d ,T, =

ˆ

A+Rn

.1$

!4T · 4e

"2/

d ,T, & 2

ˆ

A+Rn

.1$

!4T · 4e

"/d ,T, = 2 eT (A).


Since |d1 2 '| & ,D1,) ,3,) & 1, the Cauchy–Schwartz inequality yields

ˆ

A

$((x) div 3(x) dx & | )T, d1 2 '* | = |)4S · ,T, , d1 2 '*| & |d1 2 '|ˆ

A+Rn

|4S| d ,T,

&)ˆ

A+Rn

|4S|2 d ,T,* 1

2 0M(T (A! Rn))

&4

20eT (A)

0M(T (A! Rn)).

Taking the supremum over all such 3’s, we conclude (2.2). !

2.2. The Lipschitz approximation technique. For a vector measure 5 in B4r, |5| de-notes its total variation and M(5) its local maximal function:

M(5)(x) := sup0<s<4 r%|x|

|5|(Bs(x))

#m sm.

We recall the following proposition (see for instance Section 6.6.2 of [7], up to the necessaryelementary modifications), a fundamental ingredient in the proof of Proposition 0.4.

Proposition 2.2. There is a dimensional constant C with the following property. If 5 isa vector measure in B4r, / %]0,/[ and J) := {x % B3r : M(5) & /}, then

|J)| &C

/|5|(B4r). (2.5)

If in addition 5 = Df for some f % BV (B4r), then

|f(x)$ f(y)| & C / |x$ y| for a.e. x, y % J). (2.6)

Proof of Proposition 0.4. Since the statement is invariant under translations and dilations,without loss of generality we assume x = 0 and s = 1. Consider the slices Tx % I0(Rn)of T (as defined in (2.1)). Recall that M(T A ! Rn) =

´

A M(Tx) for every open set A(cp. to [11, Lemma 28.5]). Therefore,

M(Tx) & limr!0

M(T Cr(x))

#m rm& MT (x) + Q for almost every x.

Without loss of generality, we can assume c < 1. Hence, for almost every point in K,being & < 1, we have that M(Tx) < Q + 1. On the other hand, M(Tx) ' Q for every x,because !'T = Q !B4". Thus, Tx is the sum of Q positive Dirac’s delta for every x % K,that is, Tx =

(i $gi(x) for some measurable functions gi. We set g :=

(i !gi", so that

g : K # AQ(Rn).


For every 1 % C)c (Rn), by Proposition 2.1 we deduce that

M(|D$(|)(x)2 = sup0<r"4%|x|

)|D$(|(Br(x))

|Br|

*2

& sup0<r"4%|x|

2 eT (Br(x))M(T, Cr(x))

|Br|2

& sup0<r"4%|x|

2eT (Br(x))!eT (Br(x)) + Q |Br|

"

|Br|2

& 2 MT (x)2 + 2 Q MT (x) & C MT (x).

Hence, by Proposition 2.2, this implies the existence of a constant C > 0 such that

|$((x)$ $((y)| =

#####+

i

1(gi(x))$+

i

1(gi(y))

##### & C &12 |x$ y| for a.e. x, y % K.

Taking the supremum over all the 1 % C)c (Rn) with ,1,) , ,D1,) & 1, we deduce that

F(g(x)$ g(y)) & C &1/2 |x$ y|. (2.7)

It is well-known that there is a constant C such that G(T1, T2) & C F(T1 $ T2), for everyT1, T2 % AQ(Rn) " I0(Rn), if F(T1 $ T2) is small enough. Therefore, from (2.7), since& < c and s = 1, for c small enough, we infer that g can be viewed as a Lipschitz map to(AQ(Rn),G). Recalling [6, Theorem 1.7], we can extend g to a map u : B3 # (AQ(Rn),G)with constant C &1/2. Clearly, u(x) = Tx for almost every point x % K, which impliesgraph(u|K) = T (K ! Rn). Finally, (0.3) follows directly from Proposition 2.2. !

2.3. Graphs of Q-valued functions. We conclude the section with a result on the graphsof Q-valued functions. Given a Lipschitz f : " # AQ, we set f(x) =

(i !(x, fi(x))" and

consider Df =(

i

#Dfi

$its di!erential (see [6, Section 1.3]). We set, moreover,

##Jfi

## (x) =1

det!Dfi(x) ·Dfi

T (x)",

and, denoting by 4e the standard m-vector e1 2 . . . 2 em in Rm,

4Tfi(x) =Dfi(x)#4e##Jfi

## (x)=

(e1 + "1fi(x)) 2 · · · 2 (em + "mfi(x))##Jfi

## (x)% %m(Rm+n).

The current graph(f) induced by the graph of f is, hence, defined by the following position:

)graph(f), #* =

ˆ

!

+

i

2# (x, fi(x)) , 4Ti(x)

3 ##Jfi

## (x) dHm(x) 5 # % Dm(Rm+n).

As one expects, we have the formula

M (graph(f)) =

ˆ

!

+

i

##Jfi

## dHk =

ˆ

!

+

i

1det

!Dfi ·DfT

i

"dHk. (2.8)

Moreover "graph(f) is supported in ""!Rn and is given by the current graph(f |&!). Allthese facts are proved in Appendix C (see also [2, Section 1.5(6)]).

The following is a Taylor expansion for the mass of the graph of a Q-valued function.


Proposition 2.3. There is a constant C > 1 such that, for every g % Lip(",AQ(Rn)) withLip(g) & 1 and for every Borel set A " ", it holds

1$ C%1Lip(g)2

2

ˆ

A

|Dg|2 & egraph(g)(A) & 1 + C Lip(g)2

2

ˆ

A

|Dg|2. (2.9)

Proof. Note that det(Dfi · DfTi )2 = 1 + |Dfi|2 +

(|"|,2(M

"i )2, where ' is a multi-index

and M"i the corresponding minor of order |'| of Dfi. Since

41 + x2 & 1 + x2

2 and M"fi&

C |Df ||"| & C |Df |2 Lip(f)|"|%2 & C |Df |2 when |'| ' 2, we conclude

M (graph(f |A)) =+

i

ˆ

A

)1 + |Dfi|2 +

+

|"|,2

(M"fi)2

* 12

& Q |A|+ˆ

A

!12 |Df |2 + C |Df |4

"& Q |A|+ 1

2 (1 + C Lip(f)2)´

A |Df |2.

On the other hand, exploiting the lower bound 1 + x2

2 $x4

4 &4

1 + x2,

M (graph(f |A)) '+

i

ˆ

A

01 + |Dfi|2 '

+

i

ˆ

A

!1 + 1

2 |Dfi|2 $ 14 |Dfi|4

"

'+

i

ˆ

A

)1 + 1

2 |Dfi|2 $ 14 Lip(f)2|Dfi|2

*

= Q |A|+ 12

!1$ 1

4Lip(f)2" ´

A |Df |2 .

This concludes the proof. !

3. The o(E)-improved approximation

In this section we prove Theorem 0.5. Both arguments for (0.5) and (0.6) are by con-tradiction and builds upon the construction of a suitable comparison current.

3.1. Proof of (0.5). Without loss of generality, assume x = 0 and s = 1. Arguing bycontradiction, there exist a constant c1, a sequence of currents (Tl)l(N and correspondingLipschitz approximations (fl)l(N such that

El := Ex(Tl, C4) # 0 and

ˆ

B2\Kl

|Dfl|2 ' c1 El.

Set Hl := {MTl& E2 "

l /2} " B3. Since Tl and graph(fl) coincide over Kl, the Taylorexpansion (2.9) gives

´

Kl\Hl|Dfl|2 & C eTl

(Kl \Hl). Together with (0.4), this leads to

c1 El &ˆ

Bs\Hl

|Dfl|2 & C eTl(B4 \Hl), 5 s % [2, 3],

which in turn, for 2 c2 = c1/C, implies

eTl(Hl +Bs) & eTl

(Bs)$ 2 c2 El. (3.1)


Since Lip(fl) & C E"l # 0, the Taylor expansion and (3.1) give, for l big enough,

ˆ

Hl#Bs

|Dfl|2

2& eTl

(Hl +Bs) & eTl(Bs)$ c2 El, 5 s % [2, 3]. (3.2)

Our aim is to show that (3.2) contradicts the minimality of Tl. To this extent, weconstruct a competitor current in di!erent steps.

Step 1: splitting. Consider the maps gl := fl/4

El. Since supl Dir(gl, B3) < / and|B3 \Hl|# 0, we can find maps +j and #l =

(Jj=1!,yl

j. +j" as in Lemma 1.4 such that

(a1) .l :=´

B3G(gl, #l)2 # 0;

(b1) lim inf l(Dir(gl, " +Hl)$Dir(#l, ")) ' 0 for every " " B3.

Let # :=(

j !+j" and note that |D#l| = |D#|.

Step 2: choice of a suitable radius. From the estimates in (0.4), one gets

M!Tl $ graph(fl), C3

"= M

!Tl, (B3 \Kl)! Rn

"+ M

!graph(fl), (B3 \Kl)! Rn

"

& Q |B3 \Kl|+ El + Q |B3 \Kl|+ C |B3 \Kl|Lip(fl)

& El + C E1%2"l & C E1%2"

l . (3.3)

With a slight abuse of notation, we write (Tl $ graph(fl)) "Cr for )Tl $ graph(fl), 3, r*,where 3(z, y) = |z| and introduce the real valued function 1l given by

1l(r) := E2"%1l M

!(Tl $ graph(fl)) "Cr

"+ Dir(gl, "Br) + Dir(#, "Br) + .%1

l

ˆ

&Br

G(gl, #l)2.

From (a1), (b1) and (3.3), lim inf l

´ 3

2 1l(r) dr < /. By Fatou’s Lemma, there is r % (2, 3)and a subsequence, not relabeled, such that liml 1l(r) < /. Hence, it follows that:

(a2)´

&BrG(gl, #l)2 # 0,

(b2) Dir(#l, "Br) + Dir(gl, "Br) & M for some M < /,(c2) M

!(Tl $ graph(fl)) "Br

"& C E1%2"

l .

Step 3: Lipschitz approximation of #l. We now apply Lemma 1.2 to the +j’s and findLipschitz maps +j with the following requirements:

(i) Dir(+j, Br) & Dir(+j, Br) + c2/(2 Q),(ii) Dir(+j, "Br) & Dir(+j, "Br) + 1/Q,(iii)´

&BrG(+j, #)2 & c2

2/(26C Q (M + 1)), where C is the constant in the interpolation

Lemma 1.3.

The function 6l :=(

!,yli. +i" is, then, a Lipschitz approximation of #l which, for (i)-(iii),

(b1), (b2) and (3.2), satisfies, for l big enough,

(a3) Dir(6l, Br) & Dir(#, Br) + c2/2 & 2 eTl(Br)$ c2,

(b3) Dir(6l, "Br) & Dir(#, "Br) + 1 & M + 1,(c3)

´

&BrG(6l, #l)2 & c2

2/(26C (M + 1)).


Step 4: patching graph(6l) and Tl. Next, apply the interpolation Lemma 1.3 to 6l andgl with % = c2/(24(M + 1)). We then find maps 7l such that 7l|&Br = gl|&Br and, from (a2),(a3)-(c3), for l large enough,

Dir (7l, Br) & Dir (6l, Br) + % Dir (6l, "Br) + % Dir(gl, "Br) + C %%1

ˆ

&Br

G (6l, gl)2

& 2 E%1l eTl

(Br)$ c2 +c2

8+

c2

8+

c2

4& 2 E%1

l eTl(Br)$

c2

2. (3.4)

Moreover, from the last estimate in Lemma 1.3, if follows that Lip(7l) & CE"%1/2l , since

Lip(gl) & C E"%1/2l , Lip(6l) &

+

j

Lip(+j) & C and ,G(6l, gl),) & C + C E"%1/2l .

Set zl :=4

El 7l and consider the current Zl := graph(7l). Since zl|&Br = fl|&Br , "Zl =graph(fl) "Br. Therefore, from (c2), M("(Tl Br $ Zl)) & CE1%2"

l . From the isoperi-metric inequality (see [11, Theorem 30.1]), there exists an integral current Rl such that"Rl = "(Tl Cr $ Zl) and M(Rl) & CE(1%2")m/(m%1).

Set finally Wl = Tl (C4 \ Cr) + Zl + Rl. By construction, it holds obviously "Wl = "Tl.Moreover, since ' < 1/(2m), for l large enough, Wl contradicts to the minimality of Tl:

M(Wl)$M(Tl) & Q |Br|+!1 + C E2 "

l

" ˆ

Br

|Dzl|2

2+ C E

(1#2 ")mm#1

l $Q|Br|$ eTl(Br)

(3.4)

&!1 + C E2 "

l

" )eTl

(Br)$c2 El

4

*+ C E

(1#2 ")mm#1

l $ eTl(Br)

& $c2El + CE1+2"l + C E

(1#2 ")mm#1

l < 0.

3.2. Proof of (0.6). The proof is again by contradiction. Let (Tl)l be a sequence withvanishing El := Ex(Tl, C4) and contradicting (0.6), and perform again Steps 1 and 3.Clearly, since (0.6) does not hold, up to extraction of a subsequence, we can assume that

(i) either liml

´

B2|Dgl|2 >

´

|D#|2,(ii) or, for some j, +j is not Dir-minimizing in B2.

Indeed, in case one between (i) and (ii) does not hold, it su#ces to set w = #l, because,when each +j is harmonic, infx(B2 G(,yl

i. +i(x), ,yl

j. +j(x)) # / and, by the Maximum

principle [6, Proposition 3.5], #l is harmonic for l large enough as well.In case (i), since, for large l,

ˆ

Br

|D#l|2 &ˆ

Br

|Dgl|2 $ 2 c2 & E%1l eT (Br)$ c2,

for some positive constant c2, we can arguing exactly as in the proof of (0.5).In case (ii), we find a competitor for +j and, hence, new functions #l such that #l|&Br =

#l|&Br and

liml

ˆ

Br

|D#l|2 & liml

ˆ

Br

|D#l|2 & liml

ˆ

Br

|Dgl|2 $ 2 c2 & E%1l eT (Br)$ c2.


We then can argue as above with #l in place of #l, thus concluding the proof.

4. Higher integrability estimates

We come now to the proof of the various higher integrability estimates, culminating withthe proof of one of the two main result in the paper, Theorem 0.2.

4.1. Higher integrability of Dir-minimizers. In this section we prove Theorem 0.6.The theorem is a corollary of Proposition 4.1 below and a Gehring’s type lemma due toGiaquinta and Modica (see [9, Proposition 5.1]).

Proposition 4.1. Let 2 (m%1)m < s < 2. Then, there exist C = C(m, n,Q, s) such that, for

every u : " # AQ Dir-minimizing,

)$ˆ

Br(x)

|Du|2* 1

2

& C

)$ˆ

B2r(x)

|Du|s* 1

s

, 5 x % ", 5 r < min$1, dist(x, "")/2

%.

Proof. Let u : " # AQ(Rn) be a Dir-minimizing map and let 3 = " . u : " # Q " RN .Since the estimate is invariant under translations and rescalings, it is enough to prove itfor x = 0 and r = 1. We assume, therefore " = B2. Let 3 % RN be the average of 3 onB2. By Fubini’s theorem, there exists * % [1, 2] such that

ˆ

&B#

(|3$ 3|s + |D3|s) & C

ˆ

B2

(|3$ 3|s + |D3|s) & C,D3,sLs(B2).

Consider 3|&B# . Since 12 > 1

s $1

2 (m%1) , we can use the embedding W 1,s("B*) 8# H1/2("B*)

(see, for example, [1]). Hence, we infer that443|&B# $ 3

44H

12 (&B#)

& C ,D3,Ls(B2) , (4.1)

where , · ,H1/2 = , · ,L2 + | · |H1/2 and | · |H1/2 is the usual H1/2-seminorm. Let 3 be theharmonic extension of 3|&B# in B*. It is well known (one could, for example, use the resultin [1] on the half-space together with a partition of unity) that

ˆ

B#

|D3|2 & C(m) |3|2H

12 (&B#)

. (4.2)

Therefore, using (4.1) and (4.2), we conclude ,D3,L2(B#) & C ,D3,Ls(B2). Now, since! . 3|&B# = u|&B# and ! . 3 takes values in Q, by the the minimality of u and the Lipschitzproperties of ", "%1 and !, we conclude

)ˆ

B1

|Du|2* 1

2

& C

5ˆ

B#

|D3|26 1

2

& C

)ˆ

B2

|D3|s* 1

s

& C

)ˆ

B2

|Du|s* 1

s

.

!


4.2. Weak Almgren’s estimate. Here we prove Proposition 0.7. Without loss of gen-erality, we can assume s = 1 and x = 0. Let f be the E"-Lipschitz approximation in C3,with ' % (0, 1/(2m)). Fix & = (/4 and choose %2(() & %1(&). Arguing as in Step 4 ofsubsection 3.1, we find a radius r % (2, 3) and a current R such that

"R = (T $ graph(f)) "Br and M(R) & CE(1%")m/(m%1).

Hence, by the minimality of T and using the Taylor expansion in Proposition 2.3, we have

M(T Cr) & M(graph(f) Cr + R) & M(graph(f) Cr) + C Ex(T, C4)(1#2") m

m#1

& Q |Br|+ˆ

Br

|Df |2

2+ C Ex(T, C4)

1++ , (4.3)

where 5 is a fixed constant. On the other hand, using again the Taylor expansion for thepart of the current which coincides with the graph of f , we deduce as well that

M(T Cr) ' M!T ((Br \K)! Rn)

"+ M

!T ((Br +K)! Rn)

"

' M!T ((Br \K)! Rn)

"+ Q |Br +K|+

ˆ

Br#K

|Df |2

2$ C Ex(T, C4)

1++ .

(4.4)

Subtracting (4.4) from (4.3), by the choice of %2, we deduce from (0.5),

eT (Br \K) &ˆ

Br\K

|Df |2

2+ CE1++ & (E

2+ CE1++ . (4.5)

Let now A " B1 be such that |A| & %2 |B4|. Combining (4.5) with the Taylor expansionand Theorem 0.6, we finally get, for some constants C and q > 1 (independent of E) andfor %2(() su#ciently small,

eT (A) & eT (A \K) +

ˆ

A

|Df |2

2+ C E1++ & eT (Br \K) +

ˆ

A

|Dw|2

2+

( E

4+ C E1++

& 3 ( E

4+ C|A|1%1/qE + C E1++ & ( E.

4.3. Proof of Theorem 0.2. The theorem is a consequence of the following estimate:there exists constants 9 ' 2m and . > 0 such that, for every c % [1, (9 E)%1] and s % [2, 4]with s + 2/ m

4c & 4,

ˆ

{, c E"d"1}#Bs

d & 9%-

ˆ

{ c E$ "d"1}#B

s+ 2m$c

d . (4.6)

Iterate (4.6) to obtainˆ

{,2 k+1 E"dT"1}#B2

dT & 9%k -

ˆ

{, E"dT"1}#B4

dT & 9%k - 4m E, (4.7)


for every k & L := 6(log,(:/E)$1)/27 (note that, since 9 ' 2m, it holds 2(

k 9%2k/m & 2).Therefore, setting

Ak = {92k%1 E & dT < 92k+1 E} for k = 1, . . . , L,

A0 = {dT < 9 E} and AL+1 = {92L+1 E & dT & 1},for p < 1 + ./2, we conclude the theorem:ˆ

B2

dpT =L+1+

k=0

ˆ

Ak#B2

dpT &L+1+

k=0

9(2k+1) (p%1) Ep%1

ˆ

Ak#B2

dT(4.7)

& CL+1+

k=0

9k (2p%-) Ep & C Ep.

We now come to the proof of (4.6). Let NB be the constant in Besicovich’s coveringtheorem and choose P % N so large that NB < 2P%1. Set 9 = max{2m, 1/%2(2%2 m%P )} and. = $ log,(NB/2P%1), where %2 is the constant in Proposition 0.7.

Let c and s be any real numbers as above. First of all, we prove that, for a.e. x %{9 c E & dT & 1} +Bs, there exists rx such that

E(T, C4rx(x)) & c E and E(T, C*(x)) ' c E 5* %]0, 4 rx[. (4.8)

Indeed, since dT (x) = limr!0 E(T, Cr(x)) ' 9 c E ' 2mc E and

E(T, C*(x)) =eT (B*(x))

#m *m& 4m E

*m& c E for * ' 4

m4

c,

it su#ces to choose 4rx = min{* & 4/ m4

c : E(T, C*(x)) & cE}. Note that rx & 1/ m4

c.Consider now the current T restricted to C4rx(x). We note that, for the choice of 9,

setting A = {9 c E & dT},

Ex(T, C4rx(x)) & c E & E

9 E& %2

!2%2m%P

",

|A| & c E |B4rx(x)|c E 9

& %2

!2%2m%P

"|B4rx(x)|.

Hence, we can apply Proposition 0.7 to T C4rx(x) to getˆ

Brx (x)#{, c E"dT"1}dT &

ˆ

A

dT & eT (A) & 2%2 m%P eT (B4rx(x))

& 2%2 m%P (4 rx)m #m Ex(T, C4rx(x))

(4.8)

& 2%P eT (Brx(x)). (4.9)

Thus,

eT (Brx(x)) =

ˆ

Brx (x)#{dT >1}dT +

ˆ

Brx (x)#{ c E$ "dT"1}

dT +

ˆ

Brx (x)#{dT < c E$ }dT

&ˆ

A

dT +

ˆ

Brx (x)#{ c E$ "dT"1}

dT +c E

9#m rm

x

(4.8), (4.9)

&!2%P + 9%1

"eT (Brx(x)) +

ˆ

Brx (x)#{ c E$ "dT"1}

dT . (4.10)


Therefore, recalling that 9 ' 2m ' 4, from (4.9) and (4.10) we infer thatˆ

Brx (x)#{, c E"dT"1}dT &

2%P

1$ 2%P $ 9%1

ˆ

Brx (x)#{ c E$ "dT"1}

dT & 2%P+1

ˆ

Brx (x)#{ c E$ "dT"1}

dT .

Finally, by Besicovich’s covering theorem, we choose NB families of disjoint balls Brx(x)whose union covers {9 c E & dT & 1} + Bs and, recalling that rx & 2/ m

4c for every x, we

concludeˆ

{, c E"dT"1}#Bs

dT & NB 2%P+1

ˆ

{ c E$ "dT"1}#B

s+ 2m$c

dT ,

which, for the above defined ., implies (4.6).

5. Almgren’s estimate and Theorem 0.1

5.1. Proof of Theorem 0.8. Let ' % (0, (2m)%1) and fix the E"-Lipschitz approximationf . As pointed out in the introduction, the strategy here is to consider a suitable convolutionof the approximation f in order to find a competitor with energy over B3/2 \K which is asuperlinear power of the excess. In order to do this, we need to regularize " . f and to useAlmgren’s map !# whose properties are stated and proved in Proposition 6.1.

One of the main point in the proof is the following consequence of Theorem 0.2: since|Df |2 & C dT and dT & E2 " & 1 in K, there exists q = 2 p > 2 such that

)ˆ

K

|Df |q* 1

q

& C E12 . (5.1)

Proposition 5.1. Let T be as in Theorem 0.1 and let f be its E"-Lipschitz approximation.Then, there exist constants $, C > 0 and a subset B % [1, 2] with |B| > 1/2 such that, forevery s % B, there exists a Q-valued function g % Lip(Bs,AQ) which satisfies g|&Bs = f |&Bs,Lip(g) & C E" and

ˆ

Bs

|Dg|2 &ˆ

Bs#K

|Df |2 + C E1+!. (5.2)

Proof. We give an explicit construction of g& := " . g starting from f & := " . f and theprojection !#

µ given in Proposition 6.1 with a constant µ > 0 to be fixed later: then,composing with "%1, we recover g. In order to simplify the notation, we simply write !#

in place of !#µ.

To this aim, let µ > 0 and % > 0 be parameters and 1 < r1 < r2 < r3 < 2 be radii tobe fixed later. Let 3 % C)

c (B1) be a standard mollifier in RN and, for the sake of brevity,let lin(h1, h2) denote the linear interpolation in Br \ Bs between two functions h1|&Br andh2|&Bs . The function g& is defined as follows:

g& :=

78889

888:

4E lin

.f !-E, !#

.f !-E

//in Br3 \Br2 ,

4E lin

.!#

.f !-E

/, !#

.f !-E- 3%

//in Br2 \Br1 ,

4E !#

.f !-E- 3%

/in Br1 .

(5.3)


Clearly g&|&Br3= f &|&Br3

. We pass now to estimate its energy.

Step 1. Energy in Br3 \Br2. By the estimate on the linear interpolation, it follows directlythat

ˆ

Br3\Br2

|Dg&|2 & C

ˆ

Br3\Br2

|Df &|2 + C

ˆ

Br3\Br2

|D(!# . f &)|2+

+C E

(r3 $ r2)2

ˆ

Br3\Br2

####f &4E$ !#

)f &4E

*####2

& C

ˆ

Br3\Br2

|Df &|2 +C E µ2#nQ+1

r3 $ r2, (5.4)

where we used |!#(P )$ P | & C µ2#nQfor all P % Q.

Step 2. Energy in Br2 \ Br1. Here, using the same interpolation inequality and the L2

estimate on convolution, we getˆ

Br2\Br1

|Dg&|2 & C

ˆ

Br2\Br1

|Df &|2 +C

(r2 $ r1)2

ˆ

Br2\Br1

|f & $ 3% - f &|2

&C

ˆ

Br2\Br1

|Df &|2 +C %2

(r2 $ r1)2

ˆ

B1

|Df &|2 = C

ˆ

Br2\Br1

|Df &|2 +C %2 E

(r2 $ r1)2. (5.5)

Step 3. Energy in Br1. For this estimate we use the fine bounds on the projection !#. (see

Proposition 6.1). To this aim, consider the set Z :=;

x % Br1 : dist.

f !-E- 3%,Q

/> µnQ

<.

Then, one can estimateˆ

Br1

|Dg&|2 &.1 + C µ2#nQ

/ ˆ

Br1\Z|D (f & - 3%)|2 + C

ˆ

Z

|D (f & - 3%)|2 =: I1 + I2. (5.6)

We consider I1 and I2 separately. For the first we have

I1 &.1 + C µ2#nQ

/ ˆ

Br1

(|Df &| - 3%)2

&.1 + C µ2#nQ

/ ˆ

Br1

!(|Df &|;K) - 3%

"2+

.1 + C µ2#nQ

/ ˆ

Br1

!(|Df &|;B1\K) - 3%

"2+

+ 2.1 + C µ2#nQ

/ 5ˆ

Br1

!(|Df &|;K) - 3%)

"2

6 125ˆ

Br1

!(|Df &|;B1\K) - 3%

"2

6 12

.

(5.7)

Next we notice that the following two estimates hold for the convolutions:ˆ

Br1

!(|Df &|;K) - 3%

"2 &ˆ

Br1+%

(|Df &|;K)2 &ˆ

Br1#K

|Df &|2 +

ˆ

Br1+%\Br1

|Df &|2 (5.8)


and, using Lip(f &) & C E" and |B1 \K| & C E1%2",ˆ

Br1

!(|Df &|;Br1\K) - 3%)

"2 & C E2"44;Br1\K - 3%

442

L2

& C E2"44;Br1\K

442

L1 ,3%,2L2 &

C E2%2 "

%N. (5.9)

Hence, putting (5.8) and (5.9) in (5.7), we get

I1 &.1 + C µ2#nQ

/ ˆ

Br1#K

|Df &|2 + C

ˆ

Br1+%\Br1

|Df &|2 +C E2%2 "

%N+ C E

12

)C E2%2 "

%N

* 12

&ˆ

Br1#K

|Df &|2 + C µ2#nQE + C

ˆ

Br1+%\Br1

|Df &|2 +C E2%2 "

%N+

C E32%"

%N/2. (5.10)

For what concerns I2, first we argue as for I1, splitting in K and B1 \K, to deduce that

I2 & C

ˆ

Z

((|Df &|;K) - 3%)2 +

C E2%2 "

%N+

C E32%"

%N/2. (5.11)

Then, regarding the first addendum in (5.11), we note that

|Z|µ2nQ &ˆ

Br1

####f &4E- 3% $

f &4E

####2

& C %2. (5.12)

Hence, using the higher integrability of |Df | in K, that is (5.1), we obtain

ˆ

Z

((|Df &|;K) - 3%)2 & |Z|

q#2q

5ˆ

Br1

((|Df &|;K) - 3%)q

6 2q

& C E

)%

µnQ

* 2 (q#2)q

. (5.13)

Hence, putting all the estimates together, (5.6), (5.10), (5.11) and (5.13) giveˆ

Br1

|Dg&|2 &ˆ

Br1#K

|Df &|2 + C

ˆ

Br1+%\Br1

|Df &|2+

+ C E

5µ2#nQ

+E1%2"

%N+

E12%"

%N/2+

)%

µnQ

* 2 (q#2)q

6. (5.14)

Now we are ready to estimate the total energy of g& and conclude the proof of theproposition. We start fixing r2 $ r1 = r3 $ r2 = :. With this choice, summing (5.4), (5.5)and (5.14),ˆ

Br3

|Dg&|2 &ˆ

Br3#K

|Df &|2 + C

ˆ

Br1+3&\Br1

|Df &|2+

+ C E

5µ2#nQ+1

:+

%2

:2+ µ2#nQ

+E

12%"

%N/2+

)%

µnQ

* 2 (q#2)q

6.


We set % = Ea, µ = Eb and : = Ec choosing

a =1$ 2 '

2 N, b =

1$ 2 '

4 N n Qand c =

1$ 2 '

2nQ+2 N n Q.

Now, for a choice of a constant C > 0 su#ciently large, there is a set B " [1, 2] with|B| > 1/2 such that, for every r1 % B, it holds

ˆ

Br1+3&\Br1

|Df &|2 & C :

ˆ

Br1

|Df &|2 & C E1+ 1#"

2nQ+2 N n Q .

Then, for a suitable $ = $(', n, N,Q) and for s = r3, we conclude (5.2).For what concerns the Lipschitz constant of g&, we notice that it is bounded by

789

8:

Lip(g&) & C Lip(f & - 3%) & C Lip(f &) & C E" in Br1 ,

Lip(g&) & C Lip(f &) + C .f !%f !'.%.L%/ & C(1 + %

/) Lip(f &) & C E" in Br2 \Br1 ,

Lip(g&) & C Lip(f &) + C E1/2 µ2#nQ

/ & C E" + C E1/2 & C E" in Br3 \Br2 .

!

Now we are ready for the conclusion of the proof of Theorem 0.8. Consider the setB " [1, 2] given in Proposition 5.1 and, as done in subsection 4.2, choose r % B and ainteger rectifiable current R such that

"R =!T $ graph(f)

""Br and M(R) & CE(1%2")m/(m%1).

Since g|&Bs = f |&Bs , we use graph(g) + R as competitor for the current T . In this way weobtain, for a suitable ),

M (T Cs) & Q |Bs|+ˆ

Bs

|Dg|2

2+ C E1+"

(5.2)

& Q |Bs|+ˆ

Bs#K

|Df |2

2+ C E1+$. (5.15)

On the other hand, again using Taylor’s expansion (2.9),

M (T Cs) = M (T (Bs \K)! Rn) + M (graph(f |Bs#K))

' M (T (Bs \K)! Rn) + Q |K +Bs|+ˆ

K#Bs

|Df |2

2$ C E1+$. (5.16)

Hence, from (5.15) and (5.16), we get eT (Bs \K) & C E1+$.This is enough to conclude the proof. Indeed, for A " B1, using the higher integrability

of |Df | in K, possibly changing ), we get

eT (A) & eT (A +K) + eT (A \K) &ˆ

A#K

|Df |2

2+ C E1+$

& C |A +K|q#2

q

)ˆ

A#K

|Df |q* 2

p

+ C E1+$ & C E.|A|

q#2q + E$

/.


5.2. Proof of Theorem 0.1. Choose ' < min{(2m)%1, (2(1 + )))%1)}, where ) is theconstant in Theorem 0.8 and let f be the E"-Lipschitz approximation of T C4/3.

Clearly (0.1a) follows directly from (0.4) for $ < '. Set A = {MT > E2 "/2} " B4/3.Applying (0.9) to A, since by (2.5) |A| & CE1%2", we get (0.1b), for some positive $,

|B1 \K| & C E%2 " eT (A) & C E1+$%2 " + CE1+$%2(1+$)" & C E1+!.

On the other hand, (0.1c) is consequence of (0.9) and (2.9). Indeed, if we set & = graph(f):####M

!T C1

"$Q #m $

ˆ

B1

|Df |2

2

#### & eT (B1 \K) + e"(B1 \K) +

####e"(B1)$ˆ

B1

|Df |2

2

####(0.9), (2.9)

& C E1+$ + C |B1 \K|+ C Lip(f)2

ˆ

B1

|Df |2

& C!E1+$ + E1+2 "

"= C E1+!.

6. Almgren’s projections !#µ

In this section we complete the proof of Theorem 0.1 showing the existence of the almostprojections !#

µ. Compared to the original maps introduced by Almgren, our !#’s have theadvantage of depending on a single parameter. Our proof is di!erent from Almgren’sand relies heavily on the classical Theorem of Kirszbraun on the Lipschitz extensions ofRd–valued maps. A feature of our proof is that it gives more explicit estimates.

Proposition 6.1. For every µ > 0, there exists !#µ : RN(Q,n) # Q = "(AQ(Rn)) such that:

(i) the following estimate holds for every u % W 1,2(", RN),ˆ

|D(!#µ . u)|2 &

.1 + C µ2#nQ

/ ˆ

{dist(u,Q)"µnQ}|Du|2 + C

ˆ

{dist(u,Q)>µnQ}|Du|2, (6.1)

with C = C(Q, n);(ii) for all P % Q, it holds |!#

µ(P )$ P | & C µ2#nQ.

From now on, in order to simplify the notation we drop the subscript µ. We divide theproof into two parts: in the first one we give a detailed description of the set Q; then, wedescribe rather explicitly the map !#

µ.

6.1. Linear simplicial structure of Q. In this subsection we prove that the set Q canbe decomposed in a families of sets {Fi}nQ

i=0, here called i-dimensional faces of Q, with thefollowing properties:

(p1) Q = 8nQi=0 8F(Fi F ;

(p2) F := 8Fi is made of finitely many disjoint sets;(p3) each face F % Fi is a convex open i-dimensional cone, where open means that for

every x % F there exists an i-dimensional disk D with x % D " F ;(p4) for each F % Fi, F \ F " 8j<i 8G(Fj G.


In particular, the family of the 0-dimensional faces F0 contains an unique element, theorigin {0}; the family of 1-dimensional faces F1 consists of finitely many lines of the formlv = {: v : : %]0, +/[} with v % SN%1; F2 consists of finitely many 2-dimensional conesdelimited by two half lines lv1 , lv2 % F1; and so on.

To prove this statement, first of all we recall the construction of ". After selecting asuitable finite collection of non zero vectors {el}h

l=1, we define the linear map L : Rn Q # RN

given by

L(P1, . . . , PQ) :=!P1 · e1, . . . , PQ · e1= >? @

w1

, P1 · e2, . . . , PQ · e2= >? @w2

, . . . , P1 · eh, . . . , PQ · eh= >? @wh

".

Then, we consider the map O : RN # RN which maps (w1 . . . , wh) into the vector(v1, . . . , vh) where each vi is obtained from wi ordering its components in increasing order.Note that the composition O.L : (Rn)Q # RN is now invariant under the action of the sym-metric group PQ. " is simply the induced map on AQ = RnQ/PQ and Q = "(AQ) = O(V )where V := L(Rn Q).

Consider the following equivalence relation 9 on V :

(w1, . . . , wh) 9 (z1, . . . , zh) if

Awi

j = wik : zi

j = zik

wij > wi

k : zij > zi

k

5 i, j, k , (6.2)

where wi = (wi1, . . . , w

iQ) and zi = (zi

1, . . . , ziQ) (that is two points are equivalent if the map

O rearranges their components with the same permutation). We let E denote the set ofcorresponding equivalence classes in V and C := {L%1(E) : E % E}. The following fact isan obvious consequence of definition (6.2):

L(P ) 9 L(S) implies L(P0(1), . . . , P0(Q)) 9 L(S0(1), . . . , S0(Q)) 5 ! % PQ .

Thus, !(C) % C for every C % C and every ! % PQ. Since " is injective and is induced byO.L, it follows that, for every pair E1, E2 % E , either O(E1) = O(E2) or O(E1)+O(E2) = 3.Therefore, the family F := {O(E) : E % E} is a partition of Q.

Clearly, each E % E is a convex cone. Let i be its dimension. Then, there exists ai-dimensional disk D " E. Denote by x its center and let y be any other point of E. Then,by (6.2), the point z = (1 + %) y$ % x belongs as well to E for any % > 0 su#ciently small.The convex envelope of D8{z}, which is contained in E, contains in turn an i-dimensionaldisk centered in y. This establishes that E is an open convex cone. Since O|E is a linearinjective map, F = O(E) is an open convex cone of dimension i. Therefore, F satisfies(p1)-(p3).

Next notice that, having fixed w % E, a point z belongs to E \ E if and only if

• wij ' wi

k implies zij ' zi

k for every i, j and k;• there exists r, s and t such that wr

s > wrt and zr

s = zrt .

Thus, if d is the dimension of E, E \ E " 8j<d 8G(Ej G, where Ed is the family of d-dimensional classes. Therefore,

O(E \ E) " 8j<d 8H(Fj H, (6.3)


from which (recalling F = O(E)) we infer that

O(E \ E) + F = O(E \ E) +O(E) = 3. (6.4)

Now, since O(E \ E) " O(E) " O(E) = F , from (6.4) we deduce O(E \ E) " F \ F .On the other hand, it is simple to show that F " O(E). Hence, F \ F " O(E) \ F =O(E) \ O(E) " O(E \ E). This shows that F \ F = O(E \ E), which together with (6.3)proves (p4).

6.2. Construction of !#µ. The construction is divided into three steps:

(1) first we specify !#µ in Q;

(2) then we find an extension on a µnQ-neighborhood of Q, QµnQ ;(3) finally we extend the !#

µ to all RN .

6.2.1. Construction on Q. The construction of !#µ on Q is made through a recursive pro-

cedure whose main building block is the following lemma.

Lemma 6.2. Let b > 2 and D % N. There exists a constant C such that the followingholds for every , %]0, 1[. Let V d " RD be a d-dimensional cone and let v : "Bb+V d # RD

satisfy Lip(v) & 1 + , and |v(x) $ x| & , . Then, there exists an extension w of v,w : Bb + V d # RD, such that

Lip(w) & 1 + C ,, |w(x)$ x| & C4

, and w(x) = 0 5x % B1 + Vd.

Proof. First extend v to B1 +Vd by setting it identically 0 there. Note that such a functionis still Lipschitz continuous with constant 1+C , . Indeed, for x % "Bb+V d and y % B1+V d,we have that

|v(x)$ v(y)| = |v(x)| & |x|+ , = b + , & (1 + C ,)(b$ ,) & (1 + C ,) |x$ y|.Let w be an extension of v to Bb + V d with the same Lipschitz constant, whose existenceis guaranteed by the classical Kirzbraun’s Theorem, see [8, Theorem 2.10.43]. We claimthat w satisfies |w(x)$ x| & 1 + C

4, , thus concluding the lemma.

To this aim, consider x % Bb \ B1 and set y = b x/|x| % "Bb. Consider, moreover,the line r passing through 0 and w(y), let ! be the orthogonal projection onto r and setz = !(w(x)). Note that, if |x| & C , , then obviously |w(x) $ x| & |x| + |w(x)| & C , .Thus, without loss of generality, we can assume that |x| ' C , for some constant , . Inthis case, the conclusion is clearly a consequence of the following estimates:

|z $ w(x)| & C4

, , (6.5)

|x$ z| & C ,. (6.6)

To prove (6.5), note that Lip(! . w) & 1 + C, and, hence,

|z $ w(y)| & (1 + C,)|x$ y| & b$ |x|+ C , (6.7)

|z| = |! . w(x)$ ! . w(0)| & (1 + C,)|x| & |x|+ C ,.

Then, by the triangle inequality,

|z| ' |w(y)|$ |w(y)$ z| ' b (1$ ,)$ b + |x|$ C, ' |x|$ C,. (6.8)


Since |x| ' C , , the left hand side of (6.8) can be supposed nonnegative and we obtain(6.5),

|z $ w(x)|2 = |w(x)|2 $ |z|2 & (1 + C,)2|x|2 $ (|x|$ C,)2 & C,.

For what concerns (6.6), note that####x$

|x|b

w(y)

#### &|x|b|y $ w(y)| & |x|C , & C ,. (6.9)

On the other hand, since by (6.7) |z$w(y)| & b$|x|+C, & b$, & |w(y)| and w(y)·z ' 0,we have also

####z $|x|b

w(y)

#### =

####|z|$|x|b|w(y)|

#### & ||z|$ |x||+####|x|$

|x|b|w(y)|

#### & C,,

which together with (6.9) gives (6.6). !

Now we pass to the construction of the map !#µ. To fix notation, let Sk denote the k-

skeleton of Q, that is the union of all the k-faces, Sk = 8F(FkF . For every k = 1 . . . , nQ$1

and F % Fk, let Fa,b denote the set

Fa,b =$x % Q : dist(x, F ) & a , dist(x, Sk%1) ' b

%,

where a, b > 0 are given constants. In the case of maximal dimension F % FnQ, for everya > 0, Fa denotes the set

Fa =$x % F : dist(x, SnQ%1) ' a

%.

Next we choose constants 1 = cnQ%1 < cnQ%2 < . . . < c0 such that, for every 1 & k &nQ $ 1, each family {F2ck,ck#1

}F(Fkis made by pairwise disjoint sets. Note that this is

possible: indeed, since the number of faces is finite, given ck one can always find a ck%1

such that the F2ck,ck#1’s are pairwise disjoint for F % Fk.

Moreover, it is immediate to verify that

nQ%1B

k=1

B

F(Fk

F2ck,ck#18

B

F(FnQ

FcnQ#1 8B2c0 = Q.

To see this, let Ak = 8F(FkF2ck,ck#1

and AnQ = 8F(FnQFcnQ#1 : if x /% 8nQk=1Ak, then

dist(x, Sk%1) & ck%1 for every k = 1, . . . , nQ, that means in particular that x belongs toB2c0 .

Now we are ready to define the map !#µ inductively on the Ak’s. On AnQ we consider

the map fnQ = Id . Then, we define the map fnQ%1 on AnQ8AnQ%1 starting from fnQ and,in general, we define inductively the map fk on 8nQ

l=kAl knowing fk+1.Each map fk+1 : 8nQ

l=k+1Al # Q has the following two properties:

(ak+1) Lip(fk+1) & 1 + C µ2#nQ+k+1and |fk+1(x)$ x| & C µ2#nQ+k+1

;


(bk+1) for every k-dimensional face G % Fk, setting coordinates in G2ckck#1in such a way

that G +G2ck,ck#1" Rk ! {0} " RN , fk+1 factorizes as

fk+1(y, z) = (y, hk+1(z)) % Rk ! RN%k 5 (y, z) % G2ckck#1+

nQB

l=k+1

Al.

The constants involved depend on k but not on the parameter µ.Note that, fnQ satisfies (anQ) and (bnQ) trivially, because it is the identity map. Given

fk+1 we next show how to construct fk. For every k-dimensional face G % Fk, settingcoordinates as in (bk+1), we note that the set Vy := G2ckck#1

+!{y}! RN%k

"+ B2ck

(y, 0)is the intersection of a cone with the ball B2ck

(y, 0). Moreover, hk+1(z) is defined onVy+(B2ck

(y, 0)\Bck(y, 0)). Hence, according to Lemma 6.2, we can consider an extension wk

of hk+1|{|z|=2ck} on Vy+B2ck(again not depending on y) satisfying Lip(wk) & 1+C µ2#nq+k

,

|z $ wk(z)| & C µ2#nq+kand wk(z) ; 0 in a neighborhood of 0 in Vy.

Therefore, the function fk defined by

fk(x) =

A(y, wk(z)) for x = (y, z) % G2ck,ck#1

" Ak,

fk+1(x) for x %CnQ

l=k+1 Al \ Ak,(6.10)

satisfies the following properties:

(ak) First of all the estimate

|fk(x)$ x| & C µ2#nQ+k(6.11)

comes from Lemma 6.2. Again from Lemma 6.2, we conclude Lip(fk) & 1 +C µ2#nQ+k+1

on every G2ck,ck#1. Now, every pair of points x, y contained, respectively,

into two di!erent G2ck,ck#1and H2ck,ck#1

are distant apart at least one. Therefore,

|fk(x)$ fk(y)|(6.11)

& |x$ y|+ C µ2#nQ+k &.1 + Cµ2#nQ+k

/|x$ y| .

This gives the global estimate Lip(fk) & 1 + C µ2#nQ+k.

(bk) For every (k $ 1)-dimensional face H % Fk%1, setting coordinates in H2ck#1,ck#2in

such a way that H +H2ck#1,ck#2" Rk%1 ! {0} " RN%k+1, fk factorizes as

fk(y&, z&) = (y&, hk(z

&)) % Rk%1 ! RN%k+1 5 (y&, z&) % H2ck#1,ck#28

nQB

l=k

Al.

Indeed, when H " "G, with G % Fk+1 and z& = (z&1, z) where (y, z) is the coordinatesystem selected in (bk+1) for G, then

hk(z&) = (z&1, wk(z)) .

After nQ steps, we get a function f0 = !#0 : Q# Q which satisfies

Lip(!#0) & 1 + C µ2#nQ

and |!#0(x)$ x| & C µ2#nQ

.


Moreover, since the extensions wk coincide with the projection in balls BCµ2#nQ+k#1 aroundthe origin, hence, in particular on balls Bµ, it is easy to see that, for every face F % Fk,the map !#

0 coincides with the projection on F for x % Fµ,2ck#1, that is

!#0(x) = !F (x) 5 x % Fµ,2ck#1

. (6.12)

6.2.2. Extension to QµnQ. Now we need extend the map !#0 : Q # Q to a neighborhood

of Q preserving the same Lipschitz constant.We start noticing that, since the number of all the faces is finite, when µ is small

enough, there exists a constant C = C(N) with the following property. If x, y are twopoints contained, respectively, in Fµi+1 \ 8j<i 8G(Fj Gµj+1 and Hµi+1 \ 8j<i 8G(Fj Gµj+1 ,where F 0= H % Fi, then

dist(x, y) ' C µi. (6.13)

The extension !#1 is defined inductively. We start this time from a neighborhood of the

0-skeleton of Q. i.e. the ball Bµ(0). The extension g0 has the constant value 0 on Bµ(0)(note that this is compatible with the !#

0 by (6.12)).Now we come to the inductive step. Suppose we have an extension gk of !#

0, defined onthe union of the µl+1-neighborhoods of the l-skeletons Sl, for l running from 0 to k, thatis, on the set

%k := Q 8Bµ 8kB

l=1

B

F(Fl

Fµl+1 .

Assume that Lip(gk) & 1 + C µ2#nQ. Then, we define the extension of gk to %k+1 in the

following way. For every face F % Fk+1, we set

gk+1 :=

Agk in (Sk)µk+1 + Fµk+2 ,

!F in {x % RN : |!F (x)| ' 2 ck} + Fµk+2 .(6.14)

Consider now a connected component C of %k+1 \ %k. As defined above, gk+1 maps aportion of C into the closure K of a single face of Q. Since K is a convex closed set, wecan use Kriszbraun’s Theorem to extend gk+1 to C keeping the same Lipschitz constant ofgk, which is 1 + C µ2#nQ

.Next, notice that if x belongs the intersection of the boundaries of two connected com-

ponents C1 and C2, then it belongs to %k. Thus, the map gk+1 is continuous. We nextbound the global Lipschitz constant of gk+1. Indeed consider points x % Fµk+2 \ %k andy % F &

µk+2 \ %k, with F, F & % Fk+1. Since by (6.13) |x$ y| ' C µk, we easily see that

|gk+1(x)$ gk+1(y)| & 2 µk+1 + |gk(!F (x))$ gk(!F !(y))|

& 2 µk+1 + (1 + C µ2#nQ)|!F (x)$ !F !(y)|

& 2 µk+1 + (1 + C µ2#nQ)!|x$ y|+ 2 µk+1

"& (1 + C µ2#nQ

) |x$ y|.

Therefore, we can conclude again that Lip(gk+1) & 1 + C µ2#nQ, finishing the inductive

step.


After making the step above nQ times we arrive to a map gnQ which extends !#0 and is

defined in a µnQ-neighborhood of Q. We denote this map by !#1.

6.2.3. Extension to RN . Finally, we extend !#1 to RN with a fixed Lipschitz constant.

This step is immediate recalling the Lipschitz extension theorem for Q-valued functions.Indeed, taken "%1 .!#

1 : SµnQ # AQ, we find a Lipschitz extension h : RN # AQ of it withLip(h) & C. Clearly, the map !#

µ := " . h fulfills all the requirements of Proposition 6.1.

Appendix A. A variant of Theorem 0.1

Theorem A.1. There are constants C, ', %1 > 0 such that the following holds. Assume Tsatifes the assumptions of Theorem 0.1 with E4 := Ex(T, C4) < %1 and set Er := Ex(T, Cr).Then there exist a radius s %]1, 2[, a set K " Bs and a map f : Bs # AQ(Rn) such that:

Lip(f) & CE"s , (A.1a)

graph(f |K) = T (K ! Rn) and |Bs \K| & CE1+"s , (A.1b)

####M!T Cs

"$Q #msm $

ˆ

Bs

|Df |2

2

#### & C E1+"s . (A.1c)

The theorem will be derived from the following lemma, which in turn follows fromTheorem 0.1 through a standard covering argument.

Lemma A.2. There are constants C, ., %2 > 0 such that the following holds. Assume T isan area-minimizing, integer recitifiable current in C*, satisfying (H) and E := Ex(T, C*) <%2. Set r = *(1$ 4 E-). Then there exist a set K " Br and a map f : Br # AQ(Rn) suchthat:

Lip(f) & CE-, (A.2a)

graph(f |K) = T (K ! Rn) and |Br \K| & CE1+-rm, (A.2b)####M

!T Cr

"$Q #mrm $

ˆ

Br

|Df |2

2

#### & C E1+-rm. (A.2c)

Proof. Without loss of generality we prove the lemma for * = 1. Fix . > 0 and %2 > 0and assume T as in the statement. We choose a family of balls Bi = BE'(7i) satisfyingthe following conditions:

(i) the numer N of such balls is bounded by CE%m-;(ii) B4E'(7i) " B1 and {BE'/2(7i)} covers Br = B1%4E' ;(iii) each Bi intersects at most M balls Bj.

The constants C and M are dimensional and do not depend on E, . and %2. Moreover,observe that

Ex(T, C4E'(7i)) & 4%mE%m-Ex(T, C1) & C E1%m-.

Fix now %2 such that %1%m-2 & %0, with %0 the constant in Theorem 0.1. Applying (the

obvious scaled version of) Theorem 0.1, for each Bi we obtain a set Ki " Bi and a map


fi : Bi # AQ(Rn) such that

Lip(fi) & CE(1%m-)!, (A.3)

graph(fi|Ki) = T (Ki ! Rn) and |Bi \Ki| & CE(1%m-)(1+!)Em-, (A.4)####M

!T CE'(7i)

"$Q #mEm- $

ˆ

Bi

|Df |2

2

#### & C E(1%m-)(1+!)Em-. (A.5)

Set next I(i) := {j : Bj +Bi 0= 3} and Ji := Ki +D

j(I(i) Kj. By (iii) and (A.4), we have

|Bi \ Ji| & CE(1%m-)(1+!)+m-. (A.6)

Define K :=C

Ji. Since fi|Ji#Jj = fj|Jj#Ji , there is a function f : K # AQ(Rn) such thatf |Ji = fi. Choose . so small that (1$m.)(1 + $) ' 1 + .. Then, (A.2b) holds because of(i) and (A.6).

We claim next that f satisfies the Lipschitz bound (A.2a). First take x, y % K suchthat |x$ y| & E-/2. Then, by (ii), x % BE'/2(7i) for some i and hence x, y % Bi. By thedefinition of K, x % Jj " Kj for some j. On the other hand, Bj +Bi 0= 3 and thus, by thedefinition of Jj, we necessarily have x % Ki. For the same reason we conclude y % Ki. Itfollows from (A.3) and the choice of . & (1$m.) $ that

|f(x)$ f(y)| = |fi(x)$ fi(y)| & CE-|x$ y|.

Next, assume that x, y % K and |x $ y| ' E-/2. On the segment ) = [x, y], fix N &8E%-|x $ y| points +i with +0 = x, +N = y and |+i+1 $ +i| & E-/4. We can choose +i sothat, for each i % {1, N $ 1}, Bi := BE'/8(+i) " Br. Obviously, if . and %2 are chosen

small enough, (A.2b) implies that Bi + K 0= 3 and we can select zi % Bi + K 0= 3. Butthen |zi+1 $ zi| & E-/2 and hence |f(zi+1) $ f(zi)| & CE2-. Setting zN = +N = y andz0 = +0 = x, we conclude the estimate

|f(x)$ f(y)| &N+

i=0

|f(i + 1)$ f(i)| & CNE2- & CE-|x$ y| .

Thus, f can be extended to Br with the Lipschitz bound (A.2a). Finally, a simple argumentusing (A.2a), (A.2b), (A.5) and (i) gives (A.2c) and concludes the proof. !

Proof of Theorem A.1. Let . be the constant of Lemma A.2 and choose ' & ./(2 + .).Set r0 := 2 and E0 := Ex(T, Cr0), r1 := 2(1$4E-

0 ) and E1 := Ex(T, Cr1). Obviously, if %1 issu#ciently small, we can apply Lemma A.2 to T in Cr0 . We also assume of having chosen

%1 so small that 2(1 $ 4E-0 ) > 1. Now, if E1 ' E1+-/2

0 , then f satisfies the conclusion ofthe theorem. Otherwise we set r2 = r1(1 $ 4E-

1 ) and E2 := Ex(T, Cr2). We continue thisprocess and stop only if

(a) either rN < 1;

(b) or EN ' E1+-/2N%1 .


First of all, notice that, if %1 is chosen su#ciently small, (a) cannot occur. Indeed, we have

Ei & E(1+-/2)i

0 & %1+i-/21 and thus

logri

2=

+log(1$ 4E-

i ) ' $8+

E-i ' $8

+%-+i-2/21 ' $8 %-

1

%-2/21

1$ %-2/21

. (A.7)

Clearly, for %1 su#ciently small, the right and side of (A.7) is larger than log(2/3), whichgives ri ' 4/3.

Thus, the process can stop only if (b) occurs and in this case we can apply Lemma A.2to T in CrN#1 and conclude the theorem for the radius s = rN . If the process does notstop, we conclude that Ex(T, CrN ) # 0. If s := limN rN , we then conclude that s > 1 andthat Ex(T, Cs) = 0. But then, because of (H), this implies that there are Q points qi % Rn

(not necessarily distinct) such that T Cs =(

i !Bs ! {qi}". Thus, if we set K = Bs andf ;

(i !qi", the conclusion of the theorem holds trivially. !

Appendix B. The varifold excess

As pointed out in Remark 0.3, though the approximation theorems of Almgren have(essentially) the same hypotheses of Theorem 0.1, the main estimates are stated in termsof the “varifold excess” of T in the cylinder C4. More precisely, consider the representationof the rectifiable current T as 4T ,T,. As it is well-known, 4T (x) is a simple vector of theform v1 2 . . . 2 vm with )vi, vj* = $ij. Let ,x be the m-plane spanned by v1, . . . , vm andlet !x : Rm+n # ,x be the orthogonal projection onto ,x. Finally, for any linear mapL : Rm+n # Rm, denote by ,L, the operator norm of L. Then, the varifold excess isdefined by

VEx(T, Cr(x0)) =

ˆ

Cr(x0)

,!x $ !,2 d,T,(x) , (B.1)

whereas

Ex(T, Cr(x0)) =

ˆ

Cr(x0)

|4T (x)$ 4em|2 d,T,(x) . (B.2)

The two quantities di!er. If on the one hand VEx & CEx for trivial reasons (indeed,,!x $ !, & C,4T (x)$ 4em, for every x), VEx might, for general currents, be much smallerthan Ex. However, Almgren’s statements can be easily recovered from Theorem 0.1 thanksto the following proposition.

Proposition B.1. There are constants %3, C > 0 with the following properties. Assume Tis as in Theorem A.1 and consider the radius s given by its conclusion. If Ex(T, C2) & %3,then Ex(T, Cr) & CVEx(T, Cr).

Proof. Note that there are constants c0, C1 such that |4T (x)$ 4em| & C1 and |4T (x)$ 4em| &C1,!x $ !, if |4T (x) $ 4em| < c0. Let now D := {x % Cr : |4T (x) $ 4em| > c0}. We can thenwrite

Ex(T, Cr) & C1VEx(T, Cr) + 2M(T D).


On the other hand, from the bounds (A.1), it follows immediately that M(T D) &CEx(T, Cr)1+". If %3 is chosen su#ciently small, we conclude

2%1Ex(T, Cr) & Ex(T, Cr)$ CEx(T, Cr)1+" & C1VEx(T, Cr) .

!

Appendix C. Push-forward of currents under Q-functions

In this appendix we give the rigorous definition of the integer rectifiable current associ-ated to the graph of a Q-valued function and prove that the boundary of such currents isgiven by the graph of the trace of the function.

Given a Q-valued function f : Rm # AQ(Rn), we set f =(

i !(x, fi(x))", f : Rm #AQ(Rm+n). Let R % Dk(Rm) be a rectifiable current associated to a k-rectifiable set Mwith multiplicity /. In the notation of [11], R = ,(M, /, 7), where 7 is a simple Borelk-vector field orienting M . If f is a proper Lipschitz Q-valued function, we can define thepush-forward of T under f as follows.

Definition C.1. Given R = ,(M, /, 7) % Dk(Rm) and f % Lip(Rm,AQ(Rn)) as above, wedenote by Tf,R the current in Rm+n defined by

)Tf,R, #* =

ˆ

M

/+

i

,# . fi, D

M fi#7-dHk 5 # % Dk(Rm+n), (C.1)

where(

i

#DM fi(x)

$is the di!erential of f restricted to M .

Remark C.2. Note that, by Rademacher’s theorem [6, Theorem 1.13] the derivative of aLipschitz Q-function is defined a.e. on smooth C1 and, hence, also on rectifiable sets.

As a simple consequence of the Lipschitz decomposition in [6, Proposition 1.6], thereexist {Ej}j(N closed subsets of ", positive integers kj,l, Lj % N and Lipschitz functionsfj,l : Ej # Rn, for l = 1, . . . , Lj, such that

Hk(M \ 8jEj) = 0 and f |Ej =

Lj+

l=1

kj,l !fj,l" . (C.2)

From the definition, Tf,R =(

j,l kj,lfj,l#(R Ej) is a sum of rectifiable currents defined bythe push-forward under single-valued Lipschitz functions. Therefore, it follows that Tf,R isrectifiable and coincides with ,

!f(M), /f , 4Tf

", where

/f (x, fj,l(x)) = kj,l/(x) and 4Tf (x, fj,l(x)) =DM fj,l#7(x)

|DM fj,l#7(x)|5 x % Ej.

By the standard area formula, using the above decomposition of Tf,R, we get an explicitexpression for the mass of Tf,R:

M (Tf,R) =

ˆ

M

|/|+

i

1det

!DM fi · (DM fi)T

"dHk. (C.3)


C.1. Boundaries of Lipschitz Q-valued graphs. With a slight abuse of notation, whenR = !"" % Dm(Rm) is given by the integration over a Lipschitz domain " " Rm of thestandard m-vector 4e = e1 2 · · · 2 em, we write simply Tf,! for Tf,R. We do the same forTf,&!, understanding that "" is oriented as the boundary of !"". We give here a proof ofthe following theorem.

Theorem C.3. For every " Lipschitz domain and f % Lip(",AQ), " Tf,! = Tf,&!.

This theorem is of course contained also in Almgren’s monograph [2]. However, ourproof is di!erent and considerably shorter. The main building block is the following smallvariant of [6, Homotopy Lemma 1.8].

Lemma C.4. There exists a constant cQ with the following property. For every closed cubeC " Rm centered at x0 and u % Lip(C,AQ), there exists h % Lip(C,AQ) with the followingproperties:

(i) h|&C = u|&C, Lip(h) & cQ Lip(u) and ,G(u, h),L% & cQ Lip(u) diam(C);(ii) u =

(Jj=1 !uj", h =

(Jj=1 !hj", for some J ' 1 and Lipschitz (multi-valued) maps

uj, hj; each Thj ,C is a cone over Tuj ,&C:

Thj ,C = !(x0, aj)"!!Tuj ,&C , for some aj % Rn.

Proof. The proof is essentially contained in [6, Lemma 1.8]. Indeed, (i) follows in a straight-forward way from the conclusions there. Concerning (ii), we follow the inductive argumentin the proof of [6, Lemma 1.8]. By the obvious invariance of the problem under translationand dilation, it is enough to prove the following. If we consider the cone-like extensionof a vector-valued map u, h(x) =

(i !,x,ui (x/,x,)", where ,x, = supi |xi| is the uni-

form norm, then Th,C1 = !0" !!Tu,&C1 , with C1 = [$1, 1]m. This follows easily from thedecomposition Tu,&C1 =

(j,l kj,luj,l#(R Ej) described in the previous subsection. Indeed,

settingFj = {tx : x % Ej, 0 & t & 1},

clearly h decomposes in Fj as u in Ej and hj,l#(R Fj) = !0"!! uj,l#(R Ej). !Proof of Theorem C.3. Without loss of generality, we can can reduce to the case where" is the unit cube [0, 1]m. First of all, assume that " is biLipschitz equivalent to asingle cube and let 2 : " # [0, 1]m be the corresponding homeomorphism. Set g =f . 2%1. Define 2 : "! Rn # [0, 1]m ! Rn. Following [11, Remark 27.2 (3)] and using thecharacterization Tf,! = ,(f("), /f , 4Tf ), it is simple to verify that 2#Tf,! = Tg,[0,1]m and

analogously 2#Tf,&! = Tg,&[0,1]m . So, since the boundary and the push-forward commute,the case of " biLipschiz equivalent to [0, 1]m is reduced to the the case " = [0, 1]m.

Next, using a grid-type decomposition, any ", can be decomposed into finitely manydisjoint "i " ", all homeomorphic to a cube via a biLipschitz map, with the property that8"i = ". The conclusion for " follows then from the corresponding conclusion for each "i

and the obvious cancellations for the overlapping portions of their boundaries.

Assuming therefore " = [0, 1]m, the proof is by induction on m. For m = 1, by theLipschitz selection principle (cp. to [6, Proposition 1.2]) there exist single-valued Lipschitz


functions fi such that f =(

i !fi". Hence, it is immediate to verify that

"Tf,! =+

i

"Tfi,! =+

i

!$fi(1) $ $fi(0)

"= Tf |(!

.

For the inductive argument, consider the dyadic decompositions of scale 2%l of ",

" =B

k({0,...,2l%1}m

Qk,l, with Qk,l = 2%l (k + [0, 1]m) .

In each Qk,l, let hk,l be the cone-like extension given by Lemma C.4 and

Tl :=+

k

Thk,l,Qk,l= Thl

,

with hl the Q-function which coincides with hk,l in Qk,l. Note that the hl’s are equi-Lipschitzand converge uniformly to f by Lemma C.4 (i).

By inductive hypothesis, since each face F of "Qk,l is a (m$1)-dimensional cube, "Tf,F =Tf,&F . Taking into account the orientation of "F for each face, it follows immediately that

"Tf,&Qk,l= 0. (C.4)

Moreover, by Lemma C.4, each Thk,l,Qk,lis a sum of cones. Therefore, using (C.4) and

"(!0" !!T ) = T $ !0" !! "T (see [11, Section 26]), "(Tl Qk,l) = "Thk,l,Qk,l= Tf,&Qk,l

.Considering the di!erent orientations of the boundary faces of adjacent cubes, it followsthat all the contributions cancel except those at the boundary of ", thus giving "Tl = Tf,&!.

The integer m-rectifiable currents Tl, hence, have all the same boundary, which is integerrectifiable and has bounded mass. Moreover, the mass of Tl can be easily bounded usingthe formula (C.3) and the fact that sup Lip(hl) < /. By the compactness theorem forintegral currents (see [11, Theorem 27.3]), there exists an integral current S which is theweak limit for a subsequence of the Tl (not relabeled). Clearly, "S = liml!) "Tl = Tf,&!.We claim that Tf,! = S, thus concluding the proof.

To show the claim, notice that, since hl # f in L), then supp (S) 1 graph(f). So,we need only to show that the multiplicity of the currents S and Tf,! coincide almosteverywhere. Consider a point x % Ej, for some Ej in (C.2). From the Lipschitz continuityof f and hl, in a neighborhood U of x, hl and S can be decomposed in the same way as f ,

hl|U =

Lj+

p=1

!hl,p" and S (U ! Rn) =

Lj+

p=1

Sp,

where the hl,p’s are kj,p-valued and the Sp are integer rectifiable m-currents with disjointsupports. By definition, the density of Tf,! in (x, fj,p(x)) is kj,p. On the other hand, since

!'Sp = liml

!'Thl,p,U = kj,l !U" and supp (Sp) + ({x}! Rn) = (x, fj,p(x)),

it follows that the density of Sp (and hence of S) in (x, fj,p(x)) equals kj,p. Since |"\8jEj| =0, this implies S = Tf,!. !


References

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Zurich universityE-mail address: [email protected] and [email protected]

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