Introduction_to_Geometric_Measure_Theory-Urs_Lang.pdf

IntroductiontoGeometricMeasureTheory

UrsLang

April20,2005

Abstract

Thesearethenotestofourone-hourlecturesIdeliveredat

thespringschool“GeometricMeasureTheory:OldandNew”that

tookplaceinLesDiablerets,Switzerland,fromApril3–8,2005(see

http://igat.epfl.ch/diablerets05/ ).Theﬁrstthreeoftheselec-

tureswereintendedtoprovidethefundamentalsofthe“old”theoryof

rectiﬁablesetsandcurrentsineuclideanspaceasdevelopedbyBesi-

covitch,Federer–Fleming,andothers.Thefourthlecture,independent

ofthepreviousones,discussedsomemetriquespacetechniquesthat

areusefulinconnectionwiththenewmetricapproachtocurrentsby

Ambrosio–Kirchheim.OthershortcoursesweregivenbyG.Alberti,

M.Cs¨ornyei,B.Kirchheim,H.Pajot,andM.Z¨ahle.

Contents

Lecture1:Rectiﬁability 3

Lipschitzmaps............................. 3

Dierentiability............................. 4

Areaformula.............................. 8

Rectiﬁablesets.............................10

Lecture2:Normalcurrents 14

Vectors,covectors,andforms.....................14

Currents.................................15

Normalcurrents............................18

Resultsforn-currentsinR

n ......................21

Lecture3:Integralcurrents 22

Integerrectiﬁablecurrents.......................22

Thecompactnesstheorem.......................23

Minimizingcurrents..........................23

Lecture4:Somemetricspacetechniques 27

Embeddings...............................27

Gromov–Hausdorconvergence....................28

Ultralimits ...............................31

References 34

Lecture1:Rectiﬁability

Lipschitzmaps

LetX,Ybemetricspaces,andlet2[0,1).Amapf:X!Yis-

Lipschitzif

d(f(x),f(x 0 ))d(x,x 0 )forallx,x 0 2X;

fisLipschitzif

Lip(f):=inf{2[0,1):fis-Lipschitz}<1.

Thefollowingbasicextensionresultholds,see[McS]andthefootnote

in[Whit].

1.1Lemma(McShane,Whitney)

SupposeXisametricspaceandAX.

(1)Forn2N,every-Lipschitzmapf:A!R

n admitsa p n-Lipschitz

n .

(2)ForanysetJ,every-Lipschitzmapf:A!l 1 (J)hasa-Lipschitz

extension ¯ f:X!l 1 (J).

Proof:(1)Forn=1,put

¯ f(x):=inf{f(a)+d(a,x):a2A}.

Forn2,f=(f 1 ,...,f n ),extendeachf i separately.

(2)Forf=(f j ) j2J ,extendeachf j separately. 2

In(1),thefactor p ncannotbereplacedbyaconstant<n 1/4 ,cf.[JohLS]

and[Lan].Inparticular,LipschitzmapsintoaHilbertspaceYcannotbe

extendedingeneral.However,ifXisitselfaHilbertspace,onehasagain

anoptimalresult:

1.2Theorem(Kirszbraun,Valentine)

IfX,YareHilbertspaces,AX,andf:A!Yis-Lipschitz,thenfhas

a-Lipschitzextension ¯ f:X!Y.

m intoacompletemetricspaceY;itisusefulinconnection

withthedeﬁnitionofrectiﬁablesets(Def.1.13).WecallametricspaceY

Lipschitzk-connectedifthereisaconstantc1suchthatevery-Lipschitz

mapf:S k !Yadmitsac-Lipschitzextension ¯ f:B k+1 !Y;hereS k and

B k+1 denotetheunitsphereandclosedballinR

k+1 ,endowedwiththein-

ducedmetric.EveryBanachspaceisLipschitzk-connectedforallk0.

ThesphereS n isLipschitzk-connectedfork=0,...,n−1.

extension ¯ f:X!R

See[Kirs],[Val],or[Fed,2.10.43].Ageneralizationtometricspaceswith

curvatureboundswasgivenin[LanS].

ThenextresultcharacterizestheextendabilityofpartiallydeﬁnedLips-

chitzmapsfromR

m )

LetYbeacompletemetricspace,andletm2N.Thenthefollowing

statementsareequivalent:

(1)YisLipschitzk-connectedfork=0,...,m−1.

(2)Thereisaconstantcsuchthatevery-Lipschitzmapf:A!Y,

m ,hasac-Lipschitzextension ¯ f:R

m !Y.

TheideaoftheproofgoesbacktoWhitney[Whit].Compare[Alm1,

Thm.(1.2)]and[JohLS].

Proof:Itisclearthat(2)implies(1).Nowsupposethat(1)holds,and

letf:A!Ybea-Lipschitzmap,AR

m isoftheformx+[0,2 k ] m

forsomek2Zandx2(2 k

Z) m .DenotebyCthefamilyofalldyadiccubes

m \Athataremaximal(withrespecttoinclusion)subjecttothe

condition

diamC2d(A,C).

Theyhavepairwisedisjointinteriors,coverR

m \A,andsatisfy

d(A,C)<2diamC,

forotherwisethenextbiggerdyadiccubeC 0 containingCwouldstillfulﬁll

diamC 0 =2diamC2(d(A,C)−diamC)2d(A,C 0 ).

m thek-skeletonofthiscubicaldecomposition.Extendf

toaLipschitzmapf 0 :A[ 0 !Ybyprecomposingfwithanearestpoint

retractionA[ 0 !A.Then,fork=0,...,m−1,successivelyextendf k

tof k+1 :A[ k+1 !YbymeansoftheLipschitzk-connectednessofY.As

A[ m =R

m , ¯ f:=f m isthedesiredextensionoff.

Dierentiability

Recallthefollowingdeﬁnitions.

1.4Deﬁnition(GˆateauxandFr´echetdierential)

SupposeX,YareBanachspaces,fmapsanopensetUXintoY,and

x2U.

(1)ThemapfisGˆateauxdierentiableatxifthedirectionalderivative

D v f(x)existsforeveryv2Xandifthereisacontinuouslinearmap

L:X!Ysuchthat

L(v)=D v f(x)forallv2X.

ThenListheGˆateauxdierentialoffatx.

1.3Theorem(LipschitzmapsonR

m .AsYiscomplete,assume

w.l.o.g.thatAisclosed.AdyadiccubeinR

Denoteby k R

(2)Themapfis(Fr´echet)dierentiableatxifthereisacontinuouslinear

mapL:X!Ysuchthat

lim

v!0

f(x+v)−f(x)−L(v)

kvk =0.

ThenL=:Df x isthe(Fr´echet)dierentialoffatx.

ThemapfisFr´echetdierentiableatxifisGˆateauxdierentiable

atxandthelimitin

L(u)=lim

t!0 (f(x+tu)−f(x))/t

existsuniformlyforuintheunitsphereofX,i.e.forall>0thereisa

>0suchthat

kf(x+tu)−f(x)−tL(u)k|t|

whenever|t|andu2S(0,1)X.

1.5Lemma(dierentiableLipschitzmaps)

SupposeYisaBanachspace,f:R

m !YisLipschitz,x2R

m ,Disa

m !Yislinear,

andL(u)=D u f(x)forallu2D.ThenfisFr´echetdierentiableatxwith

dierentialDf x =L.

m !YisLipschitzandGˆateauxdierentiableat

x,thenfisFr´echetdierentiableatx.

Proof:Let>0.ChooseaﬁnitesetD 0 Dsuchthatforeveryu2S m−1

thereisau 0 2D 0 with|u−u 0 |.Thenthereisa>0suchthat

kf(x+tu 0 )−f(x)−tL(u 0 )k|t|

whenever|t|andu 0 2D 0 .Givenu2S m−1 ,picku 0 2D 0 with|u−u 0 |;

then

kf(x+tu)−f(x)−tL(u)k

|t|+kf(x+tu)−f(x+tu 0 )k+|t|kL(u−u 0 )k

(1+Lip(f)+kLk)|t|

forall|t|.

1.6Theorem(Rademacher)

EveryLipschitzmapf:R

m !R

n isdierentiableatL m -almostallpoints

inR

m .

Thiswasoriginallyprovedin[Rad].

densesubsetofS m−1 ,D u f(x)existsforeveryu2D,L:R

Inparticular,iff:R

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