Partial regularity for quasi minimizers of perimeter

perimeter Luigi Ambrosio Dipartimentodi Matemati a viaAbbiategrasso, 205 27100Pavia, Italia Emanuele Paolini

S uola Normale Superiore

piazzadeiCavalieri, 7

56126 Pisa, Italia

3rd November 1999

Abstra t

Let ER n

be a quasi minimizer of perimeter, that is, a set su h that P(E;B

 (x)) 

(1+!())P(F;B(x))forall variationsF withF4EbB(x)andforagivenfun tion!

with lim

!0

!()=0. Weprovethat, up toa losed setwithdimensionat mostn 8,

for all <1thesetEis an(n 1)-dimensionalC 0;

manifold. Thisresultisobtained

ombiningtheDeGiorgiandReifenbergregularitytheoriesforareaminimizers. Moreover

weprovethat,inthe asen=2,E isabi-lips hitz urve.

Introdu tion

Theaimofthispaperistostudytheregularityofquasiminimizersofperimeter. We onsider

thefollowingmultipli ativequasiminimality ondition

P(E;B



(x))(1+!())P(F;B

 (x))

forall variationsF oftheset E su h thatF4E bB



(x)and foragivenfun tion ! whi h is

innitesimalas!0.

Itiswellknown(see[24,6℄)that withtheassumption!()  2

(forsome ;>0) the

boundaryofaquasiminimizer anbesplitintotheunionofaC 1;

relativelyopenhypersurfa e

and a losedsingularset withHausdor dimensionat mostn 8(emptyifn7). Therst

partial regularity result of this kind was given by De Giorgi [10℄ who proved that for lo al

minimizers(quasiminimizerswith!0)thesingularsethaszero(n 1)-dimensionalHausdor

measure. AfterwardsMassari[16,17℄ extended thesameresultto sets withgeneralizedmean

urvature in L p

(R n

)with p>n. Wenoti e that setswith mean urvature in L p

withpn

are quasiminimizerswith !()o(1) 2

,= p n

2p

, soin the asep=n weonly knowthat

!()isinnitesimalas!0.

In1992DeGiorgiproposedanexampleofaquasiminimizerintheplanehavingasingular

pointat the origin. Gonzales, Massari and Tamanini proved in [15℄ that the exampleof De

Giorgi, whose boundaryis theunion of two bilogarithmi spirals, is indeed a set with mean

urvature in L 2

(R 2

). This example shows that in generalwe annot expe t the boundary of

a quasi minimizer to be lo ally a Lips hitz graph; however, De Giorgi onje tured that the

boundaryofaquasiminimizerislo ally parameterizable(outof asingularset, ifn8)bya

bilips hitzmapdened onanopenballofR n 1

.

Therst attempt to provethe De Giorgi onje ture hasbeenmade bythe se ond author

in [19℄, proving that if n  7 the boundary of a set with mean urvature in L n

is lo ally

parameterizablefor any <1withamap  su h that both and  1

are C 0;

; this implies

thatEisasurfa e,atleastinthetopologi alsense. Inthispaper,adoptinganewte hnique,

weprovethat thesamepropertyholds forquasiminimizers,foranynumberof dimensionsn,

outofa losedsingularsetwithdimensionatmostn 8(Theorem4.10). ThefullDeGiorgi

onje ture,namelytheC 0;1

regularityof and  1

,isstillanopenproblem; weproveitonly

in the ase n = 2, by showing that any hord ar parameterization, is, thanks to the quasi

theoriesfor areaminimizers: rstweuseavariantofDe Giorgide aytheorem (forthemean

atness insteadoftheex ess),developedin theregularitytheoryforvarifolds, toshowthat if

EissuÆ iently atinaballB



(x)thenitremains atonsmallers ales(if!()  2

there

is an improvement, leading to C 1;

regularity); we also use a density lowerbound for quasi

minimizers to transform the mean estimate on the atness into a pointwise one. Then, the

Reifenbergtopologi aldisk ondition anbeappliedtoshowthatE\B

 (x)isparameterizable by a C 0; map , with ; 1 2 C 0;

, for some < . Finally, using a standarddimension

redu tionargumentweshowthat thesetof pointswhi harenotsuÆ iently atonanys ale

hasdimensionat mostn 8.

We point out that we have onned our dis ussion to lo al minimizers in R n

, but our

arguments,ofalo alnature,applytolo alminimizersinanopensetR n

aswell,withonly

minormodi ations.

Related resulthavebeenobtained bySemmes [22℄ forthe lass of hord-ar surfa es (see

also [21℄). SuÆ ient onditions for the existen e of bi-lips hitz parameterizationshavebeen

foundbyToro[25,26℄.

1 Quasi minimizers

First,wespe ifyourmainnotations. Inthefollowingndenotesaxedinteger,weassumen2

and set m =n 1. If ER n

, jEj is theLebesgue measure of E, '

E

(x) isthe hara teristi

fun tion ofE,!

n

istheLebesguemeasureoftheunitballB

1 (0)and E:=fx2R n :8>0 0<jE\B  (x)j<! n  n g:

Finally,E4F isthesymmetri dieren e(EnF)[(F nE).

We refer to [13℄ for the denitions on erning the notions of Ca ioppoli sets, the

peri-meterP(E;),themeasuresD'

E

andjD'

E

j,theinnernormalve tor

E

(x)andtheredu ed

boundary  E;were allthatD' E = E jD' E jandthat P(E;B)=jD' E j(B)=H m (  E\B)

foranyBorelsetBR N

ifE hasniteperimeterin R n

.

Inthisse tionweintrodu ethe lassofquasiminimizersofperimetertowhi hourpartial

regularitytheoremapplies; following[2℄and[6℄,weadopt amultipli ativedenition.

Denition 1.1(Quasi minimizers) Let !:(0;+1) ! (0;+1℄ be an in reasing, fun tion

with lim

!0

!()=0.

Wewilldenote byM

!

the familyof measurablesets E su hthat

P(E;B



(x))(1+!())P(F;B

 (x))

wheneverx2E,!()<+1andE4F bB

 (x).

The main example of quasi minimizers is given by sets with pres ribed mean urvature,

as we will see in Se tion 2. Thefollowingproposition,whose proof dire tly follows from the

denition,willbeveryoftenusedinblow-uparguments.

Proposition1.2 (res aling properties ofquasi-minimizers) Let E 2 M

! ,  > 0 and x2R n . Then 1  (E x)2M ! 0 with! 0 (t):=!(t).

Inthefollowingthreepropositionsweestabilishupperandlowerboundsforareaand

peri-meter ofquasiminimalsets.

Proposition1.3 (density upper bound) If E 2 M

! , then given x 2 R n and  > 0 we have P(E;B  (x))n! n 2 m (1+!(2)) m :

We may suppose that P(E;B



(x)) 6= 0 and !(2) < +1, sin e otherwise the statement is

trivial. Lety2E\B  (x)and<;wehave P(E;B  (x))  P(E;B 2 (y))(1+!(2))P(EnB 2 (y);B 2 (y))  (1+!(2))(n! n (2) m +P(E;B 2 (y)nB 2 (y)))

andletting !the on lusion follows.

2

Theproof ofthefollowing ompa tnesstheorem anbea hievedbya lassi al omparison

argument(seee.g. [15,Th. 1.1℄,[3,Th. 4.2.5℄).

Proposition1.4 ( ompa tnessofquasi-minimizers) Let(E

k

)beasequen eof!

k

-minimal

sets and suppose that !

k

! ! pointwise, with !() ! 0 as ! 0. Then there existsa

sub-sequen e(E kj )whi h onverges inL 1 lo toasetE 2M ! . Moreover, if E k onvergein L 1 lo to E then D' Ek  *D' E ; jD' Ek j  *jD' E j; ask!1. Proposition1.5 LetE2M !

,x2E. Then,for all >0,

1 (2+!()) n  jE\B  (x)j ! n  n 1 1 (2+!()) n : Proof: Deneg():=jE\B 

(x)j. Thisisanin reasingfun tion su h thatforalmostall>0

maxfP(E\B  (x);B  (x));P(EnB  (x);B  (x))gg 0 ();

moreovertheisoperimetri inequalitygives

n! 1 n n g() m n P(E\B  (x);R n )=P(E;B  (x))+P(E\B  (x);B  (x)): (1) ComparingE withEnB 

(x)forall>weobtain

P(E;B  (x))(1+!())P(EnB  (x);B  (x))

so,letting!,ifis a ontinuitypointof!weget

P(E;B  (x) )(1+!())P(EnB  (x);B  (x) )=(1+!())P(EnB  (x);B  (x)): (2)

Putting(1)and(2)together, foralmost all,weobtain

n! 1 n n g() m n (2+!())g 0 () that is (g() 1 n ) 0  ! 1 n n 2+!()

and integrating weget therst inequality. The se ond inequality isobtainedif we repla eE

byits omplementaryset.

2

Proposition1.6 (density lowerbound) There exists a onstant 

! > 0su h that if E 2 M ! ,x2E and< ! then P(E;B  (x)) ! m 2  m :

Suppose by ontradi tionthat there exist asequen e of sets E k 2 M ! , asequen e of points x k 2E k

andasequen eofradii

k !0su hthat  m k P(E k ;B  k (x k ))< ! m 2 :

Bytheres alingpropertiesofquasi-minimizersthesetsF

k = 1  k (E k x k )belongtoM ! k with ! k (t):=!( k t). Wenotethat! k

!0sobyLemma1.4,uptoasubsequen e,wemaysuppose

thatthesequen e(F

k

) onvergestoasetF2M

0

. Fromthelowersemi- ontinuityofperimeter

weobtain ! m 2 liminf k !1  m k P(E k ;B  k (x k )) = liminf k !1 P(F k ;B 1 (0))P(F;B 1 (0)): (3)

Now, by Proposition 1.5 there exist onstants >0 and C < !

n su h that  n  jF k \ B  (0)jC n and,sin ejF\B  (0)j=lim k !1 jF k \B  (0)jweobtain02F. ThereforeasF

isaminimalset,weknowthat(seeforexample[13,(5.16)℄)P(F;B

1 (0))! m andwehavea ontradi tionwith(3). 2 Corollary 1.7 If E2M ! then H m (En  E)=0: In parti ular jD' E j =H m

E and in Proposition 1.4 we have lo al onvergen e of E

k to

E in thesense of Kuratowski 1

.

Proof:

ByProposition1.6weknowthatthemeasureH m

 

E=P(E;
)haspositivem-dimensional

densityonthesetEandinparti ularonEn 

E. Sin e,forallBorelsetsC withH m

(C)<

+1,H m

C has0m-dimensionaldensityinH m

almostallpointsofR n

nC(seeforinstan e

[23,Th. 3.5℄), we on ludethatH m (En  E)=0. IfE k

onvergetoEandsatisfytheassumptionsofProposition1.4,itis learthatanypoint

x 2 E an be approximated by points in E

k , by the onvergen e of D' Ek to D' E . The

density lowerbound and the onvergen e of jD'

Ek

j to jD'

E

j ensurethat any limit point of

E

k

belongstoE.

2

2 Boundaries with pres ribed mean urvature

Inthisse tionwepresentthemainexamplesofquasi-minimalsets. GivenH 2L 1

(R n

)wesay

thatasetER n

has(generalized)mean urvatureH ifEisalo alminimizerofthefun tional

F H (E)=P(E;R n )+ Z E H(x)d x

that is, ifforanyFR n withF4E bR n we haveF H (E)F H

(F). Wewould liketo show

that ifH 2L p

(R n

)with pnand E is aset withmean urvatureH then, forasuitable!,

E2M

! .

Firstof allit isnotdiÆ ult toshowthat thereexists anin reasing fun tion :[0;+1)!

[0;+1)with(t)!0ast!0su hthatgivenanysetFR n withE4F bB  (x)thefollowing istrue P(E;B  (x))P(F;B  (x))+()jE4Fj m n : (4) 1

Wesaythatasequen eofsets(X

k

) onvergestoXinthesenseofKuratowskiif

x k 2X k ; x k !x ) x2X x2X ) 9x k 2X k ; x k !x:

P(E;B  (x))P(F;B  (x))+ Z E4F jH(x)jdx andifwedene ():= sup x2R n jjHjj L n (B(x)) ;

from Holderinequalityweget

Z E4F jHjjjHjj L n (B  (x)) jE4Fj m n sothat(4)holds.

Itis notdiÆ ultto see (see[15℄) thatProposition1.5, Proposition1.4 andthen also

Pro-position 1.6 hold for the the sets satisfying (4) so that, if E4F b B

 (x), x 2 E and  is suÆ ientlysmall, jE4Fj m n ! m n n  m  (n)P(E;B  (x))

forsome onstant (n)>0,when e

P(E;B  (x)) 1 1 (n)() P(F;B  (x)) that isE2M ! with!():= (n)() 1 (n)()

forsmalland!():=+1elsewhere.

A tuallyit ouldhavebeenpossibletouseinequality(4)inthedenitionofquasiminimizers,

but thedenition,asitisgiven,seemstobemoregeneral.

3 Flatness

In this se tion we introdu e two quantities whi h measure the atness of E; the rst one,

analogous to the L 2

norm, is useful in the regularity theory for varifolds; the se ond one,

analogoustothesupnorm,hasbeenusedbyReifenbergin hisregularitytheory.

Denition 3.1 LetE beaCa ioppoliset. Then wedenethe mean atness and Reifenberg

atness ofE inB(x;)toberespe tively A E (x;) :=  m 2 min A2G(n) Z E\B  (x) d 2 (y;A)dH m (y);  E (x;) :=  1 min A2G(n);A3x max y2E\B  (x) d(y;A) !

whereG(n) isthe family ofaÆne hyperplanes of R n

andd(y;A)isthe distan e ofthe pointy

from A.

Thefollowingproposition omesdire tlyfrom thedenition.

Proposition3.2 (res aling properties) If E isaCa ioppoliset and>0wehave

A E (x;) = A E (x;)  E (x;) =  E (x;): If,moreover, B  (y)B  (x) wehave A E (y;)   m 2 A E (x;)  E (y;)   1  E (x;):

1 1 ! ! then  E (x;)C 1 [A E (x;2)℄ 1 m+2 : Proof:

LetAbeanhyperplanesu hthat

A E (x;2)=(2) m 2 Z E\B2(x) d 2 (y;A)d H m (y) andletA 0

bethehyperplanethroughxparalleltoA. Givenanyz2B

 (x)dene:= 1 3 d(z;A) and := 1 2 d(x;A)sothatd(z;A 0

)3+2. Notethatandbe aused(x;A)2,

sobothB  (z)andB  (x)are ontainedinB 2

(x) andwehavethat d(y;A)2ify2B

 (z)

whiled(y;A) ify2B

(x). Thenwehave(re allingalsoProposition1.6)

A E (x;2)  (2) m 2 Z E\B  (z) d 2 (y;A)dH m (y)  (2) m 2 4 2 ! m 2  m A E (x;2)  (2) m 2 Z E\B(x) d 2 (y;A)dH m (y)  (2) m 2  2 ! m 2  m that is   2  1 2! m A E (x;2)  1 m+2   2  2 ! m A E (x;2)  1 m+2 :

So,foreveryz2B

 (x)wehave d(z;A 0 )3+22  3 4 1 m+2 +2    2 ! m A E (x;2)  1 m+2 ;

that isthe laimwithC

1 =2(3
4 1 m+2 +2)(2=! m ) 1 m+2 . 2 Lemma3.4 Let(E k )M ! onvergingin L 1 lo

toasetE andletx2R n . Then A E (x;)limsup k !1 A E k (x;) 8>0: Proof:

Wewillrstprovetheinequalityforall>0su hthatH m

(E\B



(x))=0(thisistruefor

almostevery). InviewofTheorem 1.4,weknowthat

H m E k  *H m E:

LetAbeanhyperplanesu hthat

A E (x;)= m 2 Z E\B  (x) d 2 (y;A)dH m (y); dene '(y) := d 2

(y;A) and onsider the measures 

k := 'H m E k and  := 'H m E. Sin e(B  (x))=0wehave k (B  (x))!(B 

(x))(see[13,AppendixA℄,[3,Th. 1.2.7℄)and

then limsup k !1 A Ek (x;)  limsup k !1  m 2  k (B  (x) ) =  m 2 (B  (x))=A E (x;):

Byouterapproximationthesameinequalityholdsforany>0.

Thefollowingde aytheoremfor themean atnessplaysa entral rolein thispaper;boththe

statementandtheproofarereminis entoftheregularitytheoryofvarifoldsdevelopedbyAllard

[1℄andBrakke[7℄(seealso[23℄).

Theorem4.1 (de ay) Forall 2(0;1) andall M>0thereexists"

1

(dependingonly on,

M and!)su hthat,for every E2M

! ,the onditions x2E; A E (x;)" 1 ; MA E (x;) p !() imply A E (x;)C 3  2 A E (x;) whereC 3

isa onstant dependingonly onn.

Proof:

If ! is a linear fun tion a proof is given in [3℄; in the general ase the proof is similar, see

alsothede aytheoremforthemean atnessofthejump setoftheMumford{Shahfun tional

provedin[4℄,wherenolinearityassumptionon! ismade. The onstantC

3

isrelatedtosome

pointwiseestimatesonharmoni fun tions ofmvariables.

2

Lemma4.2 (iteration) There exists a positive onstant "

2 =" 2 (!) su h that if E 2 M ! , x2E,A E (x;r)" 2 and p !(r)" 2

thenfor all 2(0;r)

A E (x;)C m+2 3 max ( sup 2[;r℄ p !()  ; A E (x;r) r )

and, inparti ular, lim

!0 A E (x;)=0. Proof: Take = C 1 3 , M = m 3 , " 2 =" 1 (;M;!)(where C 3 and " 1

are the onstants given by

Theorem4.1).

Wewill prove,byindu tion,that forallintegersk0

A E (x; k r) k max ( sup 2[ k r;r℄ r  p !() ;A E (x;r) ) : (5)

The ase k = 0 is trivial, so assuming that (5) is true for k we prove that (5) holds also

for k+1. If MA E (x; k r)  p !( k

r) we an apply Theorem 4.1 (in fa t, A

E (x; k r)  max n p !(r) ;A E (x;r) o " 2 )toobtain A E (x; k +1 r)A E (x; k r) k +1 max ( sup 2[ k +1 r;r℄ r  p !() ;A E (x;r) ) :

Otherwise,byLemma3.2weget

A E (x; k +1 r)   m 2 A E (x; k r) m 2 p !( k r) M =  q !( k r) k +1 r sup 2[ k +1 r;r℄ p !()  :

To on ludetheproof,noti e that givenany2(0;r)there exists anintegerk su hthat

 k +1 r k r, sothat A E (x;)   m 2 A E (x; k r)  C m+2 3  k rmaxf sup 2[;r℄ p !()  ;A E (x;r)g  C m+3 3 maxf sup 2[;r℄ p !()  ;A E (x;r)g:

lim !0  sup 2[;r℄ p !()  =0:

Infa t,given anysequen e

k !0,let k 2[ k ;r℄besu h that p !( k )  k =sup 2[;r℄ p !()  . If  k !0then k p !(k) k  p !( k

)!0,otherwisethere exists >0su h that

k

 forallk

andwe on ludenoti ingthat

k p !(k) k  k p !(r)  !0. 2 IfXR n

and>0wewilldenoteby(X)



theopen-neighborhoodofX:

(X)



:=fx2R n

:d(x;X)<g:

Denition 4.3 LetS bea losedset ofR n

, x

0

2S,R>0and 0mnan integer. Then

wesaythatS satisesthe (";R ;m)-Reifenberg onditioninx

0

if givenanyballB

 (x)B R (x 0 )

there existsan m-dimensionalplane through x su hthat

S\B  (x)  () " \B  (x)  (S) " :

Thefollowingtheorem anbefoundin[20℄.

Theorem4.4 (Reifenberg) Given0<<1,thereexists"

3

>0su hthatifSisa losedset

inR n satisfyingthe(" 3 ;R ;m)-Reifenberg onditioninx 0

2SforsomeR>0,andsomeinteger

0  m  n, then there exists an open set U, BR

16 (x 0 )UB R (x 0 ) and an homeomorphism :D m ! S \U (D m

is the losed m dimensional unit disk) su h that both  and  1 are -Holder maps. Lemma4.5 Let E 2 M ! , x 2 R n

, > 0, " 2 (0;1=2), A an hyperplane through x and let

 A :R n !R n

be the orthogonal proje tion onto A. If E\B



(x) (A)

"

then atleast oneof

the followingstatements istrue:

1.  A (E\B  (x))A\B (1 ") (x); 2. P(EnB  (x);B  (x) )4"m! m  m : Proof:

InthisproofweassumethatthesetEissu hthatEisexa tlyequaltothetopologi al

bound-aryofE. This anbedonerepla ingE withthesetE  :=fx2R n :limsup !0 jE\B(x)j ! n  n >0g

whi h diersfrom E bya zeroLebesguemeasure set. Taking into a ountProposition 1.5 it

anbeprovedthatthetopologi alboundaryofE  isE  =E. Letx 1 ;:::;x n

beasystemof oordinatessu hthatA=fx

n =0ganddeneE + =fx n  "g\B  (x) and E =fx n  "g\B 

(x) . From theassumptions on E weknowthat E

interse ts neitherE +

nor E . So, being E losed and E +

, E ompa t sets, we on lude

thatE haspositivedistan efromE +

andE . Sooneofthefollowingstatementsmusthold:

1. EE ,E\E + =;; 2. EE + ,E\E =;; 3. E\E + =;,E\E =;; 4. EE + [E .

Supposethat 1istrue,thenwe laimthat therststatementofthepropositionholds. Let

A + =fx n ="g\B  (x) and A =fx n = "g\B  (x) . Given y 2A\B (1 ")) (x) denote by y +

and y respe tivelythe interse tionsof  1

A

(x) with A +

and A . Sin e y 2E while

y +

62E theremust ertainlybeapointin thesegment[x ;x +

℄whi hbelongstoE (andthat

is ontainedinB



(x)),soy2



(E\B(x;r)).

  " 

these ondstatement:

P(F;B  (x) )H m (F\B  (x) )H m ((A) " \B  (x))4"m! m  m :

Ifstatement4holdstheproofisthatof ase3whereinsteadofEwe onsiderits omplementary

set.

2

Denition 4.6(singular set) ForE2M

! dene

(E):=fx2E:limsup

!0 A

E

(x;)>0g:

Notethat (E) isa losedset. Infa t,givenanyx2En(E),sin elim

 A

E

(x;)=0

we anndaradiussu hthatA

E

(x;2)2 m 2

"

2

andforally2B

 (x)wegetA E (y;) 2 m+2 A E (x;2)  " 2

so that by Lemma 4.2 lim

 A E (y;)= 0; hen e B  (x)\(E) = ;. In

viewofTheorem4.4thefollowingtheoremgivesaregularityresultfortheset En(E).

Theorem4.7 (partial regularity) Let E 2 M

! and x 0 2En(E). For all 0<" <" 2 (where" 2

isthe onstant given inLemma4.2)there existsr>0su hthat the setE satises

the (";r;m)-Reifenberg onditioninx

0 . Proof: Choose onstants 1 , 2 , 3

andr su hthat

 3  " 2 ;  3 < ! m 8! m 1 (1+!( ! )) ;  3  1 2 ;  2 <   3 C 1  m+2 ;  1   2 3C 3  m+2  2 ;  1   2 3  m+2 " 2 ; r< ! ; p !(2r)" 2 2 ; p !(2r)  2 C m+2 3 ; A E (x;3r) 1 whereC 1 ,C 3 and" 2

arethe onstantsgiveninProposition3.3andLemma4.2.

ConsideranyballB

 (x)B

r (x

0

)with enter x2E. ByLemma3.2wehaveA

E (x;2r) 3 2
m+2 A E (x 0 ;3r) 3 2
m+2  1

. andbyLemma4.2weobtain

A E (x;2)  C m+2 3 maxf p !(2r);  r A E (x;2r)g  C m+2 3 maxf  2 C m+2 3 ;  3 2  m+2  1 g 2 :

Now,byProposition3.3weget

E (x;)C 1 (A E (x;2)) 1 m+2 < 3

sothereexistsanhyperplane

Athroughxsu hthat

 1 sup y2E\B  (x) d(y;A)< 3

whi h means that E \B



(x) (A)

3

. To on lude the proof we only need to prove that

A\B



(x)(E)

"

. ByProposition1.6andthedenition ofM

! , givenany2(; ! ) ! m 2  m P(E;B  (x))P(EnB  (x);B  (x))(1+!())

andletting !weobtain

P(EnB  (x);B  (x) ) ! m 2(1+!( + ))  m >4 3 ! m 1  m :

So,inviewofLemma4.5,weobtainthattheproje tion

A (E\B  (x)) ontainsA\B (1  3 ) (x).

Therefore given any point y 2 A\B

 (x) we an nd a point y 0 2 A\B (1  3 ) (x) with jy 0 yj 3 andapointz2E\B  (x)with A (z)=y 0 . Sin eE\B  (x) (A)  3  wehave jz y 0 j 3 when e jy zj2 3 ". 2

k ! k all R>0, lim k !1 sup x2B R (0)\(E k ) d(x;(E))=0: Proof:

Supposeby ontradi tion that there exist ">0and asequen e(x

k )with x k 2 (E k ) whi h

onvergestoapointxandsu hthatd(x

k

;(E))>"forallk. Sin ed(x

k ;x)!0thenx62(E) so that lim !0 A E

(x;) = 0. In view of Lemma 3.4 we annd a radius  > 0 su h that

limsup k !1 A E k (x;)A E (x;)andA E (x;)<2 m 2 " 2

. Sowe anndanintegerksu h

that A E k (x;)<2 m 2 " 2 andjx k xj<=2toobtainA E k (x k ;=2)2 m+2 A E k (x;)" 2 .

ByLemma4.2we on ludethatx

k 62(E

k ).

2

Proposition4.9 (redu tion ofsingularities) Letl0. GivenE2M

! su hthatH l ((E))> 0thereexistsE 1 2M 0 su hthat H l ((E 1 ))>0. Proof:

See[23,Appendix A℄.Firstofallre allthat H l 1 ((E))>0,where H l 1 (A):=inff X i ! l  l i : [ i B i (x i )Ag:

By[23,3.6(2)℄ we anndapointx2(E)with positiveH l

1

-density,that isapointx su h

that thereexist">0andasequen e

k !0su h that H l 1 ((E)\B  k (x))>" l k : (6) LetE k := E x  k

. Uptoasubsequen ewemaysupposethatE

k onvergestoasetE 1 2M 0 .

If wesuppose, by ontradi tion, that H l 1 ((E 1 )\B 2

(0)) =0we annd afamily of balls

B j (x j )su h that S j B j (x j )(E 1 )\B 2 (0) and P j ! l  l j <"=2. Sin e(E 1 )\B 1 (0) isa

ompa tset ithaspositivedistan efrom R n n S j B  j (x j

)and, inviewofLemma 4.8,we an

ndksu hthat(E

k )\B 1 (0) S j B  j (x j

). ThismeansthatH l 1 ((E k )\B 1 (0))<"=2,that is H l 1 ((E)\B  k (x))< l k " 2 :

Thisisin ontradi tionwith(6).

2

Theorem4.10 (maintheorem) LetER n

,E 2M

!

. Then there exista losed set(E)

withHausdordimensionnotgreaterthann 8,su hthatEn(E)isaC 0;

m-dimensional

manifold for all<1.

Proof:

TheregularityofthesetEn(E)followsfromTheorem4.7andTheorem4.4. Itiswellknown

thatifE2M

0

thenH l

((E))=0foralll>n 8(seeforinstan e[23℄)so,byProposition4.9,

itisalsotrueforanyE2M

! .

2

5 The two-dimensional ase

In this se tion we prove that quasi minimizers in R 2

are lo ally bilips hitz parameterizable

withanopenintervalofR. Theproofisa hievedusingas ompetitorinthedenition ofquasi

minimalitytheset F givenbythefollowinglemma.

Lemma5.1 LetEbeaCa ioppolisetandlet:[t

1 ;t

2

℄!Ebealips hitz,inje tivefun tion.

Then there exists a Ca ioppoli set F su h that E4F is ontained in the onvex hull K of

([t

1 ;t

2

℄) andsu h that

P(F;K)H 1 (E\K) Z t 2 t1 j 0 (t)jdt+j(t 1 ) (t 2 )j:

First some notation. We will onsider normal urrents with multipli ity in Z

2

. If  is a

parametri lips hitz urve, we denote by ~ the 1-dimensional urrent (with multipli ity 1)

indu ed by . Likewise if E is a Ca ioppoli set, we denote by ~

E the 2-dimensional normal

urrentonEwithmultipli ity1. Notethatevery2-dimensionalnormal urrentwithmultipli ity

in Z

2

anbewritten as ~

E for someCa ioppoliset E and jD'

E j =k

~

Ek. See[23℄ forbasi

notationandmainfa ts aboutthetheoryof urrents.

Wedenote byS thesegment having (t

1

)and (t

2

) asendpoints and set = ([t

1 ;t

2 ℄).

Noti e that both~ and  ~

E are re tiable1-dimensional urrentswith multipli ity1

sup-portedonthere tiableset . Sin ethereisno hoi efortheorientationof tangentlines(we

onsiderorientationinZ

2

),thetwo urrentsareequal.

ConsidertheparameterizationofS givenby :[0;1℄!R 2 denedby (t)=t(t 2 )+(1 t)(t 1

)andlet~=~ ~ . Clearlyspt~ [SK,where Kisthe onvexhullof .

Sin e ~ = 0there exists a 2-dimensional urrent ~ R (with multipli ity in Z 2 ) su h that  ~

R=~. Moreover([0;1℄)K andbeingR onstantonR 2

nK,we analsoassumespt ~ R K. Let ~ F := ~ E ~

R. ClearlyE4FK(infa t ~ F (R 2 nK)= ~ E (R 2 nK))and  FFE[ S. Sin e  ~ F ( nS)= ~ E ( nS) ~ ( nS)+~ ( nS)=0 (mod2) weobtain that k ~ Fk( nS) =0. But k ~ Fk=jD' F j = H 1   F and weobtain H 1 (  F \ ( nS))=0. So   F(En )[S,upto H 1

-negligiblesets, andP(F;K)=H 1 (  F\K) H 1 (E\K) H 1 ( )+H 1

(S)when e the on lusion follows.

2

Theorem5.2 Let ER 2

, E 2M

!

. Then E islo ally parameterizable overa line, with a

bilips hitz map. That is,E isalips hitz 1-dimensionalmanifold.

Proof:

Letz

0

beanypointofE. InviewofTheorem4.10we anndanopenneighbourhoodBofz

0

su hthatthereexistsanhomeomorphism:[0;L℄!E\B. WemaysupposethatBB

r (z 0 ) with!(2r)< 1 16

. Supposealsothat isa hord-ar parameterizationsothat is 1-lips hitz.

Letusprovethat  1 isalsolips hitz. Lett 1 ;t 2

2[0;L℄ and onsider theset F given by Lemma 5.1. If K is the onvexhull of

([t 1 ;t 2 ℄)noti ethatKB 

(x)forsomex2E andsomejt

2 t 1 j. BeingE4F bB  (x),

bytheminimalityofE andbyProposition1.3weget

P(E;B  (x)) P(F;B  (x)) !() 1+!() P(E;B  (x)) 4(1+!(2))!() 1+!()  1 2 

ontheotherhandLemma5.1 impliesthat

P(F;B  (x)) P(E;B  (x))j(t 2 ) (t 1 )j jt 2 t 1 j andwe on lude jt 2 t 1 j2j(t 2 ) (t 1 )j:

whi hmeansthat 1

islips hitz.

2

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