Publ. Ma . 45 (2001), 265–333
REAL ANALYSIS, QUANTITATIVE TOPOLOGY, AND
GEOMETRIC COMPLEXITY
S ephen Semmes
Abs ac
In his pape , we gi e an o e iew o some opics in ol ing beha -
io o homeomo phisms and ways in which eal analysis can a ise
in geome ic se ings.
Con en s
1. Fini e polyhed a and combina o ial pa ame e iza ion
p oblems 266
2. Mappings and dis o ion 275
3. The ma hema ics o good beha io much o he ime,
and he BMO ame o mind 282
4. Quan i a i e opology, and calculus on singula spaces 288
5. Uni o m ec ifiabili y 298
5.1. Smoo hness o Lipschi z and bilipschi z mappings 304
5.2. Smoo hness and uni o m ec ifiabili y 309
5.3. A class o a ia ional p oblems 313
Appendices
Appendix A. Fou ie ans o m calcula ions 315
Appendix B. Mappings wi h b anching 318
Re e ences 320
In gene al, he e can be significan complica ions in ol ed wi h home-
omo phisms and hei beha io . Some aspec s o his a e e iewed in
Sec ion 1; see also [Sem9]. On he o he hand, he e a e ways in which
looking a wha happens on a e age, and a ia ions o his, can ha e use-
ul ea u es, as in eal analysis. He e we discuss some opics ela ed o
hese hemes. In Sec ion 2 we conside a basic geome ic ques ion abou
2000 Ma hema ics Subjec Classifica ion. 42B99.
Key wo ds. BMO, bilipschi z mappings, uni o m ec ifiabili y.
The au ho was suppo ed by he U.S. Na ional Science Founda ion.
266 S. Semmes
mappings in he plane and he dis o ion o dis ances. The analy ic no-
ion o “bounded mean oscilla ion” a ises om his, and is desc ibed in
Sec ion 3. Sec ion 4 conce ns ela ions be ween unc ions and in eg als
o hei de i a i es, as on Euclidean spaces. I one is wo king on some-
hing like a su ace wi h a well-beha ed pa ame e iza ion by a Euclidean
space, hen p ope ies o unc ions on he su ace can be educed o anal-
ogous ques ions on he Euclidean space, o which he e a e nume ous
classical esul s; he e we conside si ua ions in which his may no be
a ailable. Finally, Sec ion 5 deals wi h “uni o m ec ifiabili y”, in which
a su ace can ha e nice p ope ies on a e age, bu no a all poin s. The
appendices con ain some supplemen s o he ea lie sec ions.
This su ey o igina ed wi h he John J. Ge gen Memo ial Lec u es
a Duke Uni e si y in Janua y, 1998. The au ho would like o hank
he Ma hema ics Depa men a Duke Uni e si y o he oppo uni y o
gi e hese lec u es.
1. Fini e polyhed a and combina o ial pa ame e iza ion
p oblems
Fix a posi i e in ege d, and le Pbe a d-dimensional polyhed on.
We assume ha Pis a fini e union o d-dimensional simplices, so ha
Phas “pu e” dimension d.
P oblem 1.1. How can one ell i Pis a PL (piecewise-linea ) mani old?
In o he wo ds, when is Plocally PL-equi alen o Rda each poin ?
To be p ecise, Pis locally PL-equi alen o Rda a poin x∈Pi
he e is a neighbo hood o xin Pwhich is homeomo phic o an open se
in Rd h ough a mapping which is piecewise-linea .
This is eally jus a pa icula example o a gene al issue, conce ning
exis ence and complexi y o pa ame e iza ions o a gi en se . P ob-
lem 1.1 has he nice ea u e ha fini e polyhed a and piecewise-linea
mappings be ween hem can, in p inciple, be desc ibed in fini e e ms.
Be o e we y o add ess P oblem 1.1 di ec ly, le us e iew some
p elimina y ma e s. I will be con enien o hink o Pas being like a
simplicial complex, so ha i is made up o simplices which a e always
ei he disjoin o mee in a whole ace o some (lowe ) dimension. Thus
we can speak abou he e ices o P, he edges, he 2-dimensional aces,
and so on, up o he d-dimensional aces.
Since Pis a fini e polyhed on, i s local s uc u e a any poin is
p e y simple. Namely, Plooks like a cone o e a (d−1)-dimensional
polyhed on a e e y poin . To make his p ecise, imagine ha Qis some
Real Analysis, Quan i a i e Topology, e c. 267
fini e polyhed on in some Rn, and le zbeapoin inRnwhich is affinely-
independen o Q, i.e., which lies in he complemen o an (affine) plane
ha con ains Q. (We can always eplace Rnwi h Rn+1, i necessa y, o
ensu e ha he e is such a poin .) Le c(Q) deno e he se which consis s
o all ays in Rnwhich emana e om zand pass h ough an elemen
o Q. We include zi sel in each o hese ays. This defines he “cone
o e Qcen e ed a z”. I does no eally depend on he choice o z,in
he sense ha a diffe en choice o zleads o a se which is equi alen o
he one jus defined h ough an in e ible affine ans o ma ion.
I xis a “ e ex” o P, in he sense desc ibed abo e, hen he e is a
na u al way o choose a (d−1)-dimensional polyhed on Qso ha Pis
he same as he cone o e Qcen e ed a xin a neighbo hood o x. Le
us call Q he link o Pa x. (Ac ually, wi h his desc ip ion Qis only
de e mined up o piecewise-linea equi alence, bu his is adequa e o
ou pu poses.)
Now suppose ha xis no a e ex. One can s ill ealize Pas a
cone o e a (d−1)-dimensional polyhed on nea x, bu one can also do
some hing mo e p ecise. I xis no a e ex, hen he e is a posi i e
in ege kand a k-dimensional ace Fo Psuch ha xlies in he in e io
o F. In his case he e is a (d−k−1)-dimensional polyhed on Qsuch ha
Pis locally equi alen o Rk×c(Q) nea x, wi h xin Pco esponding
o a poin (y,z)inRk×c(Q), whe e zis he cen e o c(Q). This same
polyhed on Qwo ks o all he poin s in he in e io o F, and we call
Q he link o F.
Basic Fac 1.2. Pis e e ywhe e locally equi alen o Rdi and only
i all o he a ious links o P(o all dimensions) a e piecewise-linea ly
equi alen o s anda d sphe es (o he same dimension).
He e he “s anda d sphe e o dimension m” can be aken o be he
bounda y o he s anda d (m+ 1)-dimensional simplex.
Basic Fac 1.2 is s anda d and no ha d o see. The “i ” pa is im-
media e, since one knows exac ly wha he cone o e a s anda d sphe e
looks like, bu o he con e se he e is a bi mo e o check. A use ul
obse a ion is ha i Qis a j-dimensional polyhed on whose cone c(Q)
is piecewise-linea ly equi alen o Rj+1 in a neighbo hood o he cen-
e o c(Q), hen Qmus be piecewise-linea ly equi alen o a s anda d
j-dimensional sphe e. This is p e y easy o e i y, and one can use i e-
pea edly o he links o Po codimension la ge han 1. (A well-known
poin he e is ha one should be ca e ul no o use adial p ojec ions o
in es iga e links a ound e ices, bu sui able pseudo- adial p ojec ions,
268 S. Semmes
o fi wi h he piecewise-linea s uc u e, and no jus he opological
s uc u e.)
A nice ea u e o Basic Fac 1.2 is ha i se s up a na u al induc ion in
he dimensions, since he links o Palways ha e dimension less han P.
This leads o he ollowing ques ion.
P oblem 1.3. I Qis a fini e polyhed on which is a k-dimensional PL
mani old, how can one ell i Qis a PL sphe e o dimension k?
I is easonable o assume he e ha Qis a PL-mani old, because o
he way ha one can use Basic Fac 1.2 and induc ion a gumen s.
P oblem 1.3 is pa o he ma e o he Poinca ´e conjec u e, which
would seek o say ha Qis a PL sphe e as soon as i is homo opy-
equi alen o a sphe e. This has been es ablished in all dimensions ex-
cep 3 and 4. (Compa e wi h [RouS].) In dimension 4 he Poinca ´e con-
jec u e was se led by M. F eedman [F e] in he “ opological” ca ego y
(wi h o dina y homeomo phisms (con inuous mappings wi h con inuous
in e ses) and opological mani olds), bu i emains unknown in he PL
case. The PL case is equi alen o he smoo h e sion in his dimension,
and bo h a e equi alen o he o dina y opological e sion in dimen-
sion 3. (A b ie su ey ela ed o hese s a emen s is gi en in Sec ion 8.3
o [F eQ].) Al hough he Poinca ´e conjec u e is known o hold in he
PL ca ego y in all highe dimensions ( han 4), i does no always wo k in
he smoo h ca ego y, because o exo ic sphe es (as in [Mil1], [Ke M]).
I he PL e sion o he Poinca ´e conjec u e is ue in all dimen-
sions, hen his would gi e one answe o he ques ion o ecognizing
PL mani olds among fini e polyhed a in P oblem 1.1. Specifically, ou
polyhed on Pwould be a PL mani old i and only i i s links a e all
homo opy-equi alen o sphe es (o he co ec dimension).
This migh seem like a p e y good answe , bu he e a e s ong di -
ficul ies conce ning complexi y o ma e s o homo opy. In o de o
ak-dimensional polyhed on Q o be a homo opy sphe e, i has o be
simply connec ed in pa icula , a leas when k≥2. In o he wo ds,
i should be possible o con inuously de o m any loop in Q o a single
poin , o , equi alen ly, o ake any con inuous mapping om a ci cle
in o Qand ex end i o a con inuous mapping om a closed disk in o Q.
This ex ension can en ail eno mous complexi y, in he sense ha he
filling o he disk migh ha e o be o much g ea e complexi y han he
o iginal loop i sel .
This is an issue whose geome ic significance is o en emphasized by
G omo . To desc ibe i mo e p ecisely i is help ul o begin wi h some
ela ed algeb aic p oblems, conce ning fini ely-p esen ed g oups.
Real Analysis, Quan i a i e Topology, e c. 269
Le Gbe a g oup. A fini e p esen a ion o Gis gi en by a fini e lis
g1,g
2,...,g
no gene a o s o G oge he wi h a fini e se 1,
2,...,
m
o “ ela ions”. The la e a e (fini e) wo ds made ou o he gi’s and
hei in e ses. Le us assume o con enience ha he se o ela ions
includes he in e ses o all o i s elemen s, and also he emp y wo d.
The j’s a e equi ed o be i ial, in he sense ha hey ep esen he
iden i y elemen o G. This implies ha a bi a y p oduc s o conjuga es
o he j’s also ep esen he iden i y elemen , and he final equi emen
is ha i wis any wo d in he gi’s and hei in e ses which ep esen s
he iden i y elemen in G, hen i should be possible o ob ain w om
some p oduc o conjuga es o he j’s h ough cancella ions o subwo ds
o he o m g−1
igiand gig−1
i.
Fo ins ance, he g oup Z2can be desc ibed by wo gene a o s a,
band one ela ion, aba−1b−1. As ano he conc e e example, he e is
he (Baumslag-Soli a ) g oup wi h wo gene a o s x,yand one ela-
ion x2yx−1y−1.
Suppose ha a g oup Gand fini e p esen a ion o Ga e gi en and
fixed, and le wbe a wo d in he gene a o s o Gand hei in e ses.
Gi en his in o ma ion, how can one decide whe he w ep esen s he
iden i y elemen in G? This is called “ he wo d p oblem” ( o G). I is
a amous esul ha he e exis fini e p esen a ions o g oups o which
he e is no algo i hm o sol e he wo d p oblem. (See [Man].)
To unde s and wha his eally means, le us fi s no ice ha he se o
i ial wo ds o he gi en p esen a ion is “ ecu si ely enume able”. This
means ha he e is an algo i hm o lis ing all o he i ial wo ds. To
do his, one simply has o ha e he algo i hm sys ema ically gene a e all
possible conjuga es o he ela ions, all possible p oduc s o conjuga es o
ela ions, and all possible wo ds de i ed om hese h ough cancella ions
as abo e. In his way he algo i hm will cons an ly gene a e i ial
wo ds, and e e y i ial wo d will e en ually show up on he lis .
Howe e , his does no gi e a fini e p ocedu e o de e mining ha
a gi en wo d is no i ial. A p io i one canno conclude ha a gi en
wo d is no i ial un il one goes h ough he en i e lis o i ial wo ds.
The eal ouble comes om he cancella ions. In o de o es ablish
he i iali y o a gi en wo d w, one migh ha e o make de i a ions
h ough wo ds which a e eno mously la ge , wi h a lo o collapsing a
he end. I one had a bound o he size o he wo ds needed o a leas
one de i a ion o he i iali y o a gi en wo d w, a bound in e ms o an
effec i ely compu able (o “ ecu si e”) unc ion o he leng h o w, hen
he wo d p oblem would be algo i hmically sol able. One could simply
sea ch h ough all de i a ions o a mos a compu able size.
270 S. Semmes
This would no be e y efficien , bu i would be an algo i hm. As
i is, e en his does no always wo k, and he e a e fini ely-p esen ed
g oups o which he de i a ions o i iali y may need o in ol e wo ds
o non ecu si e size compa ed o he gi en wo d.
One should keep in mind ha o a gi en g oup and a gi en p esen a-
ion he e is always some unc ion (n) on he posi i e in ege s so ha
i ial wo ds o leng h a mos nadmi de i a ions o hei i iali y
h ough wo ds o size no g ea e han (n). This is ue simply because
he e a e only fini ely many wo ds o size a mos n, and so one can
ake (n) o be he maximum size incu ed in some fini e collec ion o
de i a ions. The poin is ha such a unc ion may no be bounded
by a ecu si e unc ion. This means ha could be eally huge, la ge
han any owe o exponen ials, o ins ance.
The same kind o phenomenon occu s geome ically, o deciding
whe he a loop in a gi en polyhed on can be con inuously con ac ed
o a poin . This is because any fini e p esen a ion o a g oup Gcan be
coded in o a fini e polyhed on, in such a way ha he g oup Gis ep e-
sen ed by he undamen al g oup o he polyhed on. This is a well-known
cons uc ion in opology.
No e ha while he undamen al g oup o a space is no mally defined
in e ms o con inuous (based) loops in he space and he con inuous
de o ma ions be ween hem, in he case o fini e polyhed a i is enough
o conside polygonal loops and de o ma ions which a e piecewise-linea
(in addi ion o being con inuous). This is ano he s anda d ac , and i
p o ides a con enien way o hink abou complexi y o loops and hei
de o ma ions.
Al hough a bi a y fini e p esen a ions can be coded in o fini e poly-
hed a, as men ioned abo e, his is no he same as saying ha hey
can be coded in o compac mani olds. I u ns ou ha his does wo k
when he dimension is a leas 4, i.e., o each n≥4 i is ue ha
e e y fini e p esen a ion can be coded in o a compac PL mani old o
dimension n. This ype o coding can be used o con e algo i hmic
unsol abili y esul s o p oblems in g oup heo y in o algo i hmic un-
sol abili y s a emen s in opology. Fo ins ance, he e does no exis
an algo i hm o decide when a gi en fini e p esen a ion o a g oup ac-
ually defines he i ial g oup, and, simila ly, he e does no exis an
algo i hm o deciding when a gi en mani old (o dimension a leas 4)
is simply-connec ed. See [BooHP], [Ma 1], [Ma 2], [Ma 3] o mo e
in o ma ion and esul s.
Le us men ion ha in dimensions 3 and less, i is no ue ha
a bi a y fini ely-p esen ed g oups can be ealized as undamen al g oups
Real Analysis, Quan i a i e Topology, e c. 271
o compac mani olds. Fundamen al g oups o mani olds a e e y special
in dimensions 1 and 2, as is well known. The si ua ion in dimension 3
is mo e complica ed, bu he e a e subs an ial es ic ions on he g oups
ha can a ise as undamen al g oups. As an aspec o his, one can
look a es ic ions ela ed o Poinca ´e duali y. In a diffe en ein, he
undamen al g oup o a 3-dimensional mani old has he p ope y ha all
o i s fini ely-gene a ed subg oups a e fini ely-p esen ed. See [Sco], and
Theo em 8.2 on p. 70 o [Hem1]. See also [Jac]. In ano he di ec ion,
he e a e ela i ely ew abelian g oups which can a ise as subg oups o
undamen al g oups o 3-dimensional mani olds. See [Eps], [E aM],
Theo ems 9.13 and 9.14 on p. 84 o [Hem1], and p. 67–69 o [Jac]. A
any a e, i is a la ge open p oblem o know exac ly wha g oups a ise
as undamen al g oups o 3-dimensional mani olds.
See also [Thu] and Chap e 12 o [Eps+] conce ning hese g oups.
The book [Eps+] ea s a numbe o opics ela ed o compu abili y and
g oups, and no jus in connec ion wi h undamen al g oups o 3-man-
i olds. This includes b oad classes o g oups o which posi i e esul s
and me hods a e a ailable. See [Fa ] as well in his ega d.
Beginning in dimension 5, i is known ha he e is no algo i hm o
deciding when a compac PL mani old is piecewise-linea ly equi alen o
a s anda d (PL) sphe e. This is a esul o S. No iko . See Sec ion 10
o [VolKF], and also he appendix o [Nab]. No e ha in dimensions
less han o equal o 3, such algo i hms do exis . This is classical o
dimensions 1, 2; see [Rub1], [Rub2], [Tho] conce ning dimension 3,
and ela ed p oblems and esul s.
Imagine ha we ha e a connec ed PL mani old Mo some dimen-
sion n≥5 whose equi alence o a s anda d sphe e is ue bu “ha d” o
check. Acco ding o he solu ion o he Poinca ´e conjec u e in hese
dimensions, Mwill be equi alen o an n-sphe e i i is homo opy-
equi alen o Sn. Fo s anda d easons o algeb aic opology, his will
happen exac ly when Mis simply-connec ed and has i ial homology
in dimensions 2 h ough n−1. (Specifically, his uses Theo em 9 and
Co olla y 24 on pages 399 and 405, espec i ely, o [Spa]. I also uses he
exis ence o a deg ee-1 mapping om M o Sn o ge s a ed (i.e., o ha e
a mapping o which he a o emen ioned esul s can be applied), and he
ac ha he homology o Mand Sn anish in dimensions la ge han n,
and a e equal o Zin dimension n. To ob ain he deg ee-1 mapping
om M o Sn, one can s a wi h any poin in Mand a neighbo hood
o ha poin which is homeomo phic o a ball. One hen collapses he
complemen o ha neighbo hood o a poin , which gi es ise o he
272 S. Semmes
desi ed mapping.) The anishing o homology can be de e mined algo-
i hmically, and so i he equi alence o Mwi h an n-sphe e is “ha d”
o algo i hmic e ifica ion, hen he p oblem mus occu al eady wi h
he simple-connec i i y o M.
To de e mine whe he Mis simply-connec ed i is enough o check
ha a fini e numbe o loops in Mcan be con ac ed o a poin , i.e., some
collec ion o gene a o s o he undamen al g oup. I his is “ha d”, hen
i means ha he complexi y o he con ac ions should be eno mous
compa ed o he complexi y o M. Fo i he e we e a bound in e ms
o a ecu si e unc ion, hen one could e e se he p ocess and use his
o ge an algo i hm which could decide whe he Mis PL equi alen o
a sphe e, and his is no possible.
I Mis a ha d example o a PL mani old which is equi alen o an
n-sphe e, hen any mapping om M o he sphe e which ealizes his
equi alence mus necessa ily be o e y high complexi y as well. Be-
cause o he p eceding discussion, his is also ue o mappings which
a e homo opy-equi alences, o e en which me ely induce isomo phisms
on π1, i one includes as pa o he package o da a enough in o ma ion
o jus i y he condi ion ha he induced mapping on π1be an isomo -
phism. (Fo a homo opy equi alence, o ins ance, one could include he
mapping om M o he n-sphe e, a mapping g om he n-sphe e o M
which is a homo opy-in e se o , and mappings which gi e homo opies
be ween ◦gand g◦ o he iden i y on he n-sphe e and M, espec-
i ely.) This is because one could use he mapping o educe he p oblem
o con ac ing a loop in M o a poin o he co esponding p oblem o
he n-sphe e, whe e he ma e o bounds is s aigh o wa d.
Simila conside a ions apply o he p oblem o deciding when a fini e
polyhed on Pis a PL mani old. Indeed, gi en a PL mani old Mwhose
equi alence o a sphe e is in ques ion, one can use i o make a new
polyhed on Pby aking he “suspension” o M. This is defined by
aking wo poin s yand zwhich lie ou side o a plane ha con ains M,
and hen aking he union o all o he (closed) line segmen s ha go
om ei he o yo z o a poin in M. One should also be ca e ul o
choose yand zso ha hese line segmen s ne e mee , excep in he
i ial case o line segmen s om yand z o he same poin xin M,
wi h xbeing he only poin o in e sec ion o he wo segmen s. (One
can imagine yand zas lying on “opposi e sides” o an affine plane ha
con ains M.)
I Mis equi alen o a sphe e, hen his ope a ion o suspension p o-
duces a PL mani old equi alen o he sphe e o 1 la ge dimension, as
one can easily check. I Mis no PL equi alen o a sphe e, hen he
Real Analysis, Quan i a i e Topology, e c. 273
suspension Po Mis no a PL mani old a all. This is because Mis he
link o Pa he e ices yand z, by cons uc ion, so ha one is back o
he si ua ion o Basic Fac 1.2.
Jus as he e a e PL mani olds Mwhose equi alence wi h a sphe e
is ha d, he use o he suspension shows ha he e a e polyhed a P o
which he p ope y o being a PL mani old is ha d o es ablish. Th ough
he ype o a gumen s ou lined abo e, when PL coo dina es exis o a
polyhed on P, hey may ha e o be o eno mous complexi y compa ed
o he complexi y o Pi sel . This wo ks mo e obus ly han jus o PL
coo dina es, i.e., i applies o objec s which a e p ecise enough o deal
wi h simple-connec i i y o he links o P(which a e local, as opposed
o mo e global aspec s o P). Again, his ollows he discussion abo e.
We ha e ocussed on piecewise-linea coo dina es o fini e polyhed a
o he sake o simplici y, bu simila hemes o complexi y come up
much mo e gene ally, and in a numbe o diffe en ways. In pa icula ,
exis ence and complexi y o pa ame e iza ions is o en ela ed in a s ong
manne o he beha io o some hing like π1, some imes in a localized
o m, as wi h he links o a polyhed on. Fo opology o mani olds in
high dimensions, π1and he filling o loops wi h disks come up in he
Whi ney lemma, o ins ance. This conce ns he sepa a ion o c ossings
o submani olds h ough he use o embedded 2-dimensional disks, and i
can be e y use ul o making some geome ic cons uc ions. (A e y nice
b ie e iew o some o hese ma e s is gi en in Sec ion 1.2 o [DonK].)
Localized π1- ype condi ions play a c ucial ole in aming heo ems in
geome ic opology. Some e e ences ela ed o his a e [Bin1], [Bin2],
[Bin3], [Bu ], [Bu C], [Can1], [Can2], [Da e1], [Da e2], [Edw1],
[Moi], [Rus1], [Rus2].
As ano he ype o example, one has he amous “double suspension”
esul s o Edwa ds and Cannon [Can1], [Can3], [Da e2], [Edw2]. He e
one s a s wi h a fini e polyhed on Hwhich is a mani old wi h he same
homology as a sphe e o he same dimension, and one akes he suspen-
sion (desc ibed abo e) o he suspension o H o ge a new polyhed on K.
The esul is ha Kis ac ually homeomo phic o a sphe e. A key poin
is ha His no equi ed o be simply-connec ed. When π1(H)=0,
i is no possible o he homeomo phism om K o a s anda d sphe e
o be piecewise-linea , o e en Lipschi z con inuous (which means ha
he dis ance be ween wo poin s in he image is always bounded by a
cons an imes he dis ance be ween hei p eimages in K). Conce ning
he la e , see [SieS]. No much is known abou he complexi y o he
homeomo phisms in his case. (We shall say a bi mo e abou his in
Sec ion 5.)
280 S. Semmes
diffe en , and one keeps cons an s like hose om he linea decay con-
di ion. This comes ou clea ly in he p oo , and we shall see mo e abou
i la e .
This ype o exponen ial decay occu s in a simple way in he example
abo e, in (2.10). (This also comes up in [Joh].) One can ob ain his
om he p esence o 'log in he angle coo dina e in he image. The
use o he loga i hm he e is no acciden al, bu fi s exac ly wi h he
equi emen s on he mapping. Fo ins ance, i one diffe en ia es log
in o dina y Ca esian coo dina es, hen one ge s a quan i y o size 1/ ,
and his is balanced by he in he fi s pa o he pola coo dina es in
(2.10), o gi e a esul which is bounded.
I may be a bi su p ising, o disappoin ing, ha uni o m app oxima-
ion o he diffe en ial o does no wo k he e. A e all, we did ha e
“uni o m” (o “sup emum”) bounds in he hypo hesis (2.1), and so one
migh hope o ha e he same kind o bounds in he conclusion. This
ype o ailu e o sup emum bounds is qui e common, and in much he
same manne as in he p esen case. We shall e u n o his in Sec ion 3.
How migh one p o e (2.11), o he exponen ial decay bounds o
P(λ)? Le us s a wi h a sligh ly simple si ua ion. Imagine ha we
ha e a ec ifiable cu e γin he plane whose o al leng h is only sligh ly
la ge han he dis ance be ween i s wo endpoin s. I he leng h o γ
we e equal o he dis ance be ween he endpoin s, hen γwould ha e o
be a s aigh line segmen , and no hing mo e. I he leng h is sligh ly
la ge , hen γhas o s ay close o he line segmen ha joins i s end-
poin s. In analogy wi h (2.11), we would like o say ha he angen s
o γa e nea ly pa allel, on a e age, o he line ha passes h ough he
endpoin s o γ.
In o de o analyze his u he , le z( ), ∈R,a≤ ≤b,bea
pa ame e iza ion o γby a cleng h. This means ha z( ) should be
1-Lipschi z, so ha
|z(s)−z( )|≤|s− |(2.15)
o all s, ∈[a, b], and ha |z( )|= 1 o almos all , whe e z( ) deno es
he de i a i e o z( ). Se
ζ=z(b)−z(a)
b−a=1
b−ab
a
z( )d .(2.16)
Le us compu e
1
b−ab
a|z(s)−ζ|2ds,(2.17)
Real Analysis, Quan i a i e Topology, e c. 281
which con ols he a e age oscilla ion o z(s). Le ·,· deno e he s an-
da d inne p oduc on R2, so ha
|x−y|2=x−y,x −y=x, x−2x, y+y,y
=|x|2−2x, y+|y|2
(2.18)
o all x, y ∈R2. Applying his wi h x=z(s), y=ζ, we ge ha
1
b−ab
a|z(s)−ζ|2ds =1−21
b−ab
az(s),ζds +|ζ|2,(2.19)
since |z(s)|= 1 a.e., and ζdoes no depend on s. The middle e m on
he igh side educes o
2ζ,ζ,(2.20)
because o (2.16). Thus (2.19) yields
1
b−ab
a|z(s)−ζ|2ds =1−2|ζ|2+|ζ|2=1−|ζ|2.(2.21)
On he o he hand, |z(b)−z(a)|is he same as he dis ance be ween he
endpoin s o γ, and b−ais he same as he leng h o γ, since z( )is
he pa ame e iza ion o γby a cleng h. Thus |ζ|is exac ly he a io o
he dis ance be ween he endpoin s o γ o he leng h o γ, by (2.16),
and 1 −|ζ|2is a dimensionless quan i y which is small exac ly when
he leng h o γand he dis ance be ween i s endpoin s a e close o each
o he (p opo iona ely). In his case (2.21) p o ides p ecise in o ma ion
abou he way ha z(s) is app oxima ely a cons an on a e age. (These
compu a ions ollow ones in [CoiMe2].)
One can use hese esul s o cu es o looking a mappings om
R2(o Rn) o i sel , by conside ing images o segmen s unde he map-
pings. This does no seem o gi e he p ope bounds in (2.11), in e ms
o dependence on δ, hough. In his ega d, see John’s pape [Joh].
(Compa e also wi h Appendix A.) No e ha o cu es by hemsel es,
he compu a ions abo e a e qui e sha p, as indica ed by he equali y in
(2.21). See also [CoiMe2].
The exponen ial decay o P(λ) equi es mo e wo k. A basic poin is
ha exponen ial decay bounds can be de i ed in a e y gene al way once
one knows (2.11) o all disks Din he plane. This is a amous esul o
John and Ni enbe g [JohN], which will be discussed u he in Sec ion 3.
In he p esen si ua ion, ha ing es ima es like (2.11) o all disks D
(and wi h uni o m bounds) is qui e na u al, and is essen ially au oma ic,
because o he in a iances o he condi ion (2.1) unde ansla ions and
dila ions. In o he wo ds, once one has an es ima e like (2.11) o some
282 S. Semmes
fixed disk Dand all mappings which sa is y (2.1), one can conclude
ha he same es ima e wo ks o all disks D, because o in a iance unde
ansla ions and dila ions.
3. The ma hema ics o good beha io much o he ime,
and he BMO ame o mind
Le us s a anew o he momen , and conside he ollowing ques ion
in analysis. Le hbe a eal- alued unc ion on R2. Le ∆ deno e he
Laplace ope a o , gi en by
∆= ∂2
∂x2
1
+∂2
∂x2
2
,(3.1)
whe e x1,x2a e he s anda d coo dina es on R2. To wha ex en does
he beha io o ∆hcon ol he beha io o he o he second de i a i es
o h?
O cou se i is easy o make examples whe e ∆h anishes a a poin
bu he o he second de i a i es do no anish a he same poin . Le
us ins ead look o ways in which he o e all beha io o ∆hcan con ol
he o e all beha io o he o he second de i a i es.
He e is a basic example o such a esul . Le us assume ( o simplici y)
ha his smoo h and ha i has compac suppo , and le us w i e ∂1
and ∂2 o ∂/∂x1and ∂/∂x2, espec i ely. Then
R2|∂1∂2h(x)|2dx ≤R2|∆h(x)|2dx.(3.2)
This is a well-known ac , and i can be de i ed as ollows. We begin
wi h he iden i y
R2
∂1∂2h(x)∂1∂2h(x)dx =R2
∂2
1h(x)∂2
2h(x)dx,(3.3)
which uses wo in eg a ions by pa s. On he o he hand,
(3.4) R2|∆h(x)|2dx =R2
(∂2
1h(x)+∂2
2h(x))2dx
=R2
(∂2
1h(x))2+2∂2
1h(x)∂2
2h(x)+(∂2
2h(x))2dx.
Real Analysis, Quan i a i e Topology, e c. 283
Combining his wi h (3.3) we ge ha
(3.5) R2|∆h(x)|2dx −2R2|∂1∂2h(x)|2dx
=R2
(∂2
1h(x))2+(∂2
2h(x))2dx.
This implies (3.2), and wi h an ex a ac o o 2 on he le -hand side,
because he igh side o (3.5) is nonnega i e. (One can imp o e his o
ge a ac o o 4 on he le side o (3.2), using he igh -hand side o
(3.5).)
In sho , he L2no m o ∆halways bounds he L2no m o ∂1∂2h.
The e a e simila bounds o Lpno ms when 1 <p<∞. Specifically,
o each pin (1,∞), he e is a cons an C(p) such ha
R2|∂1∂2h(x)|pdx ≤C(p)R2|∆h(x)|pdx(3.6)
whene e his a smoo h unc ion wi h compac suppo . This is a ypical
example o a “Calde ´on-Zygmund inequali y”, as in [Duo], [Ga cR],
[Ga n], [Jou], [Sa ], [S e1], [S e2], [S eW], [S T], [To c]. Such
inequali ies do no wo k o p=1o ∞, and he p=∞case is like
he ques ion o sup emum es ima es in Sec ion 2. No e ha he p=
1 and p=∞cases a e closely connec ed o each o he , because o
duali y (o spaces and ope a o s); he ope a o s ∆ and ∂1∂2he e a e
equal o hei own ansposes, wi h espec o he s anda d bilinea o m
on unc ions on R2(defined by aking he in eg al o he p oduc o wo
gi en unc ions). In a modes ly diffe en di ec ion, he e a e classical
esul s which gi e bounds in e ms o he no m o H¨olde con inuous
(o Lipschi z) unc ions o o de α, o e e y α∈(0,1), ins ead o he
Lpno m. To be explici , gi en α, his no m o a unc ion gon R2can
be desc ibed as he smalles cons an Asuch ha
|g(x)−g(y)|≤A|x−y|α
(3.7)
o all x, y ∈R2. One can iew his as a p=∞si ua ion, like he L∞
no m o g, bu wi h a posi i e o de αo smoo hness, unlike L∞. The e
is a a ie y o o he no ms and spaces which one can conside , and o
which he e a e esul s abou es ima es along he lines o (3.6), bu o
he no m in ques ion ins ead o he Lpno m.
The p=∞ e sion o (3.6) would say ha he e is a cons an Csuch
ha
sup
x∈R2|∂1∂2h(x)|≤Csup
x∈R2|∆h(x)|(3.8)
284 S. Semmes
whene e his smoo h and has compac suppo . In o de o see ha
his is no he case, conside he unc ion h(x) gi en by
h(x)=x1x2log(x2
1+x2
2),(3.9)
x=(x1,x
2). I is no ha d o compu e ∆hand ∂1∂2hexplici ly, and o
see ha ∆his bounded while ∂1∂2his no . Indeed,
∂1∂2h(x) = log(x2
1+x2
2) + bounded e ms,(3.10)
while he loga i hm does no su i e in ∆h, because ∆(x1x2)≡0.
This choice o his nei he smoo h no compac ly suppo ed, bu hese
de ec s can be co ec ed easily. Fo smoo hness we can conside ins ead
h(x)=x1x2log(x2
1+x2
2+'),(3.11)
whe e '>0, and hen look a wha happens as '→0. To make he
suppo compac we can simply mul iply by a fixed cu -off unc ion ha
does no anish a he o igin. Wi h hese modifica ions we s ill ge a
singula i y a he o igin as '→0, and we see ha (3.8) canno be ue
(wi h a fixed cons an C ha does no depend on h).
This is exac ly analogous o wha happened in Sec ion 2, i.e., wi h a
uni o m bound going in bu no coming ou . Ins ead o a uni o m bound
o he ou pu , we also ha e a subs i u e in e ms o “mean oscilla ion”,
jus as be o e. To be p ecise, le Dbe any disk in R2o adius , and
conside he quan i y
1
π 2D|∂1∂2h(x)−A e ageD(∂1∂2h)|dx,(3.12)
whe e “A e ageD∂1∂2h” is he a e age o ∂1∂2ho e he disk D, i.e.,
A e ageD(∂1∂2h)= 1
π 2D
∂1∂2h(u)du.(3.13)
Ins ead o (3.8), i is ue ha he e is a cons an C>0 so ha
1
π 2D|∂1∂2h(x)−A e ageD(∂1∂2h)|dx ≤Csup
x∈R2|∆h(x)|(3.14)
o e e y disk Din R2o adius and e e y smoo h unc ion hwi h
compac suppo . This is no oo ha d o p o e; oughly speaking, he
poin is o “localize” he L2es ima e ha we had be o e. (Resul s o his
na u e a e discussed in [Duo], [Ga cR], [Ga n], [Jou], [Sa ], [S e2],
[To c]. The e a e also eplacemen s o (3.6) o p= 1, and e en o
0<p<1 (!); in his connec ion, see [S e1], [S eW], [S T], in addi ion
o he e e ences jus men ioned.)
Real Analysis, Quan i a i e Topology, e c. 285
Le us o malize his es ima e by defining a new space o unc ions,
namely he space BMO o unc ions o bounded mean oscilla ion, in o-
duced by John and Ni enbe g in [JohN]. A locally-in eg able unc ion g
on R2is said o lie in BMO i he e is a nonnega i e numbe ksuch ha
1
π 2D|g(x)−A e ageD(g)|dx ≤k(3.15)
o e e y disk Din R2o adius . In his case we se
g∗=sup
D
1
π 2D|g(x)−A e ageD(g)|dx,(3.16)
wi h he sup emum aken o e all disks Din R2. This is he same as
saying ha g∗is he smalles numbe k ha sa isfies (3.15). One
e e s o g∗as he “BMO no m o g”, bu no ice ha g∗= 0 when
gis equal o a cons an almos e e ywhe e. (The con e se is also ue.)
This defini ion may look a li le c azy, bu i wo ks qui e well in
p ac ice. Le us e o mula e (3.14) by saying ha he e is a cons an C
so ha
∂1∂2h∗≤C∆h∞,(3.17)
whe e φ∞deno es he L∞no m o a gi en unc ion φ. In o he wo ds,
al hough he L∞no m o ∂1∂2his no con olled ( o all h)by heL∞
no m o ∆h, he BMO no m o ∂1∂2his con olled by he L∞no m o
∆h.
Simila ly, one o he main poin s in Sec ion 2 can be e o mula ed as
saying ha i a mapping :R2→R2dis o s dis ances by only a small
amoun , as in (2.1), hen he BMO no m d ∗o he diffe en ial o is
small (and wi h p ecise es ima es being a ailable).
In Sec ion 2 we men ioned a s onge es ima e wi h exponen ial decay
in he measu e o ce ain “bad” se s. This wo ks o all BMO unc ions,
and can be gi en as ollows. Suppose ha gis a BMO unc ion on R2
wi h g∗≤1, and le Dbe a disk in R2wi h adius . As in (2.12),
conside he “dis ibu ion unc ion” P(λ) defined by
P(λ) = P obabili y({x∈D:|g(x)−A e ageD(g)|≥λ}),(3.18)
whe e “P obabili y” means Lebesgue measu e di ided by he a ea π 2
o D. Unde hese condi ions, he e is a uni e sal bound o P(λ) wi h
exponen ial decay, i.e., an inequali y o he o m
P(λ)≤B−λ o λ≥1,(3.19)
whe e Bis a posi i e numbe g ea e han 1, and Bdoes no depend on
go D. This is a heo em o John and Ni enbe g [JohN].
286 S. Semmes
Al hough we ha e es ic ed ou sel es o R2he e o simplici y, e e y-
hing goes o e in a na u al way o Euclidean spaces o a bi a y dimen-
sion. In ac , he e is a much mo e gene al amewo k o “spaces o ho-
mogeneous ype” in which basic p ope ies o BMO (and o he aspec s
o eal- a iable ha monic analysis) ca y o e . See [CoiW1], [CoiW2],
and compa e also wi h [Ga cR], [S e2]. This amewo k includes ce -
ain Ca no spaces ha a ise in se e al complex a iables, like he uni
sphe e in Cnwi h he app op ia e (noneuclidean) me ic.
The exponen ial decay bound in (3.19) helps o make p ecise he idea
ha BMO unc ions a e e y close o being bounded (which would co -
espond o ha ing P(λ) = 0 o all sufficien ly la ge λ). The exponen ial
a e o decay implies ha BMO unc ions lie in Lplocally o all fini e p,
bu i is qui e a bi s onge han ha .
A basic example o a BMO unc ion is log |x|. This is no ha d o
check, and i shows ha exponen ial decay in (3.19) is sha p, i.e., one
does no ha e supe exponen ial decay in gene al. This example also
fi s wi h (3.10), and wi h he “ o a ional” pa o he diffe en ial o he
mapping in (2.10).
In gene al, BMO unc ions can be much mo e complica ed han he
loga i hm. Roughly speaking, he o al “size” o he unboundedness
is no wo se han o he loga i hm, as in (3.19), bu he a angemen
o he singula i ies can be mo e in ica e, jus as one can make much
mo e complex singula examples han in (3.9) and (2.10). The e a e a
lo o ools a ailable in ha monic analysis o unde s anding how BMO
unc ions beha e. (See [Duo], [Ga cR], [Ga n], [Jou], [Sa ], [S e2],
[S T], [To c], o ins ance.)
BMO unc ions show up all o e he place. One can e o mula e he
basic scena io in his sec ion wi h he Laplacian and ∂1∂2by saying ha
he pseudodiffe en ial o singula in eg al ope a o
∂1∂2
∆
(3.20)
maps L∞ o BMO, and his holds o simila ope a o s (o o de 0) much
mo e gene ally (as in he e e ences abo e). This will be discussed a bi
u he in Appendix A. No e ha he nonlinea p oblem in Sec ion 2 has
a na u al linea iza ion which alls in o his ub ic. (See Appendix A.)
Sobole embeddings p o ide ano he class o linea p oblems in which
BMO comes up na u ally. One migh wish ha a unc ion gon Rn ha
sa isfies ∇g∈Ln(Rn) (in he sense o dis ibu ions) we e bounded o
con inuous, bu nei he o hese a e ue in gene al, when n>1. How-
e e , such a unc ion gis always in BMO, and in he subspace VMO
Real Analysis, Quan i a i e Topology, e c. 287
(“ anishing mean oscilla ion”), in which he measu emen s o mean os-
cilla ion (as in he le side o (3.15) when n= 2) end o 0 as he adius
goes o 0. This is a well-known analogue o con inui y in he con ex o
BMO. (See [B eN], [Ga cR], [Ga n], [Sa ], [Sem8], [S e2], [To c].)
BMO a ises in a lo o nonlinea p oblems, in addi ion o he one in
Sec ion 2. Fo ins ance, he e a e ci cums ances in which one migh wish
ha he de i a i e o a con o mal mapping in he complex plane we e
bounded, and i is no , bu he e a e na u al es ima es in e ms o BMO.
Mo e p ecisely, i is BMO o he loga i hm o he de i a i e ha comes
up mos na u ally. This is closely ela ed o BMO condi ions o angen s
o cu es unde ce ain geome ic condi ions. See [CoiMe1], [CoiMe2],
[CoiMe3], [Da i1], [JeK1], [Pom1], [Pom2], [Pom3], [Sa ], o in-
s ance. Some basic compu a ions ela ed o he la e we e gi en in
Sec ion 2, nea he end. In gene al dimensions (la ge han 1), BMO
shows up na u ally as he loga i hm o he densi y o ha monic mea-
su e o Lipschi z domains, and o he loga i hm o Jacobians o qua-
sicon o mal mappings. See [Dah1], [Dah2], [JeK2], [Geh2], [Rei],
[S e2] and he e e ences he ein. In all dimensions, he e a e in e es -
ing classes o “weigh s”, posi i e unc ions which one can use as densi ies
o modifica ions o Lebesgue measu e, whose loga i hms lie in BMO,
and which in ac co espond o open subse s o BMO ( o eal- alued
unc ions). These weigh s ha e good p ope ies conce ning Lpbounded-
ness o singula in eg al and o he ope a o s, and hey also show up in
o he si ua ions, in connec ion wi h con o mal mappings in he plane,
ha monic measu e, and Jacobians o quasicon o mal mappings in pa -
icula , as abo e. See [Duo], [Ga cR], [Ga n], [Jou], [Sa ], [S e2],
[S T], [To c] o in o ma ion abou hese classes o weigh s.
The e is a simple eason o BMO unc ions o a ise equen ly as
some kind o loga i hm. In many nonlinea p oblems he e is a symme-
y which pe mi s one o mul iply some quan i y by a cons an wi hou
changing any hing in a significan way. (E.g., hink o escaling o o-
a ing a domain, o a mapping, o mul iplying a weigh by a posi i e
cons an .) A he le el o he loga i hm his in a iance is con e ed in o
a eedom o add cons an s, and his is some hing ha BMO accommo-
da es au oma ically.
To summa ize a bi , he e a e a lo o si ua ions in which one has
some unc ion ha one would like o be bounded, bu i is no , and o
which BMO p o ides a good subs i u e. One may no expec a fi s o
ha e o ake measu e heo y in o accoun , bu hen i comes up on i s
own, o wo ks in a na u al o easonable way.
288 S. Semmes
Be o e lea ing his sec ion, le us e u n o he John-Ni enbe g heo-
em, i.e., he exponen ial decay es ima e in (3.19). How migh one y
o p o e his? The fi s main poin is ha one canno p o e (3.19) o a
pa icula disk Dusing only a bound like (3.15) o ha one disk. Tha
would only gi e a a e o decay on he o de o 1/λ. Ins ead one uses
(3.15) o e and o e again, o many diffe en disks.
He e is a basic s a egy. Assume ha gis a BMO unc ion wi h
g∗≤1. Fi s use (3.15) o Di sel (wi h k= 1) o ob ain ha he
se o poin s xin Dsuch ha
|g(x)−A e ageD(g)|≥10,(3.21)
is p e y small (in e ms o p obabili y). On he bad se whe e his
happens, y o make a good co e ing by smalle disks on which one can
apply he same ype o a gumen . The idea is o hen show ha he se
o poin s xin Dwhich sa is y
|g(x)−A e ageD(g)|≥10+10(3.22)
is significan ly smalle s ill, and by a defini e p opo ion. I one can
epea his o e e , hen one can ge exponen ial decay as in (3.19). Mo e
p ecisely, a each s age he size o he de ia ion o g(x) om A e ageD(g)
will inc ease by he addi ion o 10, while he dec ease in he measu e o
he bad se will dec ease mul iplica i ely.
This s a egy is oughly co ec in spi i , bu o ca y i ou one has
o be mo e ca e ul in he choice o “bad” se a each s age, and in he
ansi ion om one s age o he nex . In pa icula , one should y o
con ol he diffe ence be ween he a e age o go e one disk and o e one
o he smalle disks c ea ed in he nex s ep o he p ocess. As a p ac i-
cal ma e , i is simple o wo k wi h cubes ins ead o disks, o he way
ha hey can be decomposed e enly in o smalle pieces. The ac ual con-
s uc ion used is he “Calde ´on-Zygmund decomposi ion”, which i sel
has a lo o o he applica ions. See [JohN], [Duo], [Ga cR], [Ga n],
[Jou], [Sa ], [Sem8], [S e2], [S T], [To c] o mo e in o ma ion.
4. Quan i a i e opology, and calculus on singula spaces
One o he nice ea u es o Euclidean spaces is ha i is easy o wo k
wi h unc ions, de i a i es, and in eg als. He e is a basic example o his.
Le be a eal- alued unc ion on Rnwhich is con inuously diffe en iable
and has compac suppo , and fix a poin x∈Rn. Then
| (x)|≤ 1
νnRn
1
|x−y|n−1|∇ (y)|dy,(4.1)
Real Analysis, Quan i a i e Topology, e c. 289
whe e νndeno es he (n−1)-dimensional olume o he uni sphe e in
Rn, and dy e e s o o dina y n-dimensional olume.
This inequali y p o ides a way o say ha he alues o a unc ion a e
con olled by a e ages o i s de i a i e. In his espec i is like Sobole
and isope ime ic inequali ies, o which we shall e u n in a momen .
To p o e (4.1) one can p oceed as ollows (as on p. 125 o [S e1]).
Le be any elemen o Rnwi h | |= 1. Then
(x)=−∞
0
∂
∂ (x+ )d ,(4.2)
by he undamen al heo em o calculus. Thus
| (x)|≤∞
0|∇ (x+ )|d .(4.3)
This is ue o e e y in he uni sphe e o Rn, and by a e aging o e
hese ’s one can de i e (4.1) om (4.3).
To pu his in o pe spec i e, i is help ul o look a a si ua ion whe e
analogous inequali ies make sense bu ail o hold. Imagine ha one is
in e es ed in inequali ies like (4.1), bu o 2-dimensional su aces in R3
ins ead o Euclidean spaces hemsel es. Le Sbe a smoo hly embedded
2-dimensional submani old o R3which looks like a 2-plane wi h a bubble
a ached o i . Specifically, le us s a wi h he union o a 2-plane P
and a s anda d ( ound) 2-dimensional sphe e Σ which is angen o P
a a single poin z. Then cu ou a li le neighbo hood o z, and glue
in a small “neck” as a b idge be ween he plane and he sphe e o ge a
smoo h su ace S.
I he neck in Sis e y small compa ed o he size o Σ, hen his is
bad o an inequali y like (4.1). Indeed, le xbe he poin on Σ which is
exac ly opposi e om z, and conside a smoo h unc ion which is equal
o 1 on mos o Σ (and a xin pa icula ) and equal o 0 on mos o P.
Mo e p ecisely, le us choose so ha i s g adien is concen a ed nea
he b idge be ween Σ and P.I makes he ansi ion om anishing
o being 1 in a easonable manne , hen he in eg al o |∇ |on Swill
be e y small. This is no ha d o check, and i is bad o ha ing an
inequali y like (4.1), since he le -hand side would be 1 and he igh -
hand side would be small. In pa icula , one could no ha e uni o m
bounds ha would wo k o a bi a ily small b idges be ween Pand Σ.
The inequali y (4.1) is a ela i e o he usual Sobole and isope ime ic
inequali ies, which say he ollowing. Fix a dimension nagain, and an
exponen p ha sa isfies 1 ≤p<n. Define qby 1/q =1/p −1/n,
so ha p<q<∞. The Sobole inequali ies asse he exis ence o a
296 S. Semmes
would like o find a mapping πx:M {x}→Sn−1which is opologically
nondegene a e and sa isfies
|dπx(u)|≤Kd(u, x)−1
(4.18)
o some cons an Kand all u∈M {x}. No e ha now he no m o
he diffe en ial o πxin ol es he Riemannian me ic on M. Fo he
opological nondegene acy o πx, le us ask ha i ha e nonze o deg ee
on small sphe es in M ha su ound xin a s anda d way. This makes
sense, because o he a p io i assump ion ha Mbe smoo h.
I one can p oduce such a mapping πx, hen one can de i e (4.8) as a
consequence, using he same kind o a gumen wi h diffe en ial o ms as
abo e. One can also find enough cu es in he fibe s o πx, wi h con ol
on he way ha hei a cleng h measu es a e dis ibu ed in M, h ough
he use o co-a ea es ima es. Fo his he opological nondegene acy o
πxis needed o showing ha he fibe s o πxconnec x o infini y in M.
In he con ex o con o mal de o ma ions o Rn,asin[Da iS1], such
mappings πxcan be ob ained as pe u ba ions o he s anda d mapping
in (4.13). This is desc ibed in [Sem7]. Fo Theo em 4.11, he me hod
o [Sem6] does no use mappings qui e like πx, bu a “s abilized” e sion
om which one can d aw simila conclusions. In his s abilized e sion
one looks o mappings om M o Sn(ins ead o Sn−1) which a e con-
s an ou side o a (gi en) ball, opologically non i ial (in he sense o
nonze o deg ee), and which sa is y sui able bounds on hei diffe en ials.
These mappings a e like snapsho s o pieces o M, and one has o mo e
hem a ound in a con olled manne . This means mo ing hem bo h in
e ms o loca ion ( he cen e o he suppo ing ball) and scale ( he adius
o he ball).
A his s age he hypo heses o Theo em 4.11 may make mo e sense.
Exis ence o mappings like he ones desc ibed abo e is a s anda d ma e
in opology, excep o he ques ion o uni o m bounds. The hypo heses
o Theo em 4.11 ( he doubling condi ion and local linea con ac abil-
i y) a e also in he na u e o quan i a i e opology. No e, howe e , ha
he kind o bounds in ol ed in he hypo heses o he heo em and he
cons uc ion o mappings in o sphe es a e somewha diffe en om each
o he , wi h bounds on he diffe en ials being c ucial o he la e , while
con ol o e moduli o con inui y does no come up in he o me . (The
local linea con ac abili y condi ion es ic s he o e all dis ances by
which poin s a e displaced in he con ac ions, bu no he sizes o he
smalle -scale oscilla ions, as in a modulus o con inui y.) In he end he
Real Analysis, Quan i a i e Topology, e c. 297
bounds o he diffe en ials come abou because he hypo heses o The-
o em 4.11 pe mi one o educe a ious cons uc ions and compa isons
o fini e models o con olled complexi y.
In he p oo o Theo em 4.11 he e a e h ee ela ed pieces o in o ma-
ion ha come ou , namely (1) es ima es o he beha io o unc ions
on ou space Min e ms o hei de i a i es, as in (4.8), (2) amilies o
cu es in Mwhich a e well-dis ibu ed in e ms o a cleng h measu e,
and (3) mappings o sphe es wi h ce ain es ima es and nondegene acy
p ope ies. These h ee kinds o in o ma ion a e closely linked, h ough
a ious duali ies, bu o some ex en hey also ha e hei own li es. Each
would be immedia e i Mhad a bilipschi z pa ame e iza ion by Rn,bu
in ac hey a e mo e obus han ha , and much easie o e i y.
Indeed, one o he o iginal mo i a ions o [Da iS1] was he p ob-
lem o de e mining which con o mal de o ma ions o Rnlead o me ic
spaces ( h ough he geodesic dis ance) which a e bilipschi z equi alen
o Rn. The de o ma ions a e allowed o be nonsmoo h he e, bu his
does no ma e oo much, because o he na u al scale-in a iance o
he p oblem, and because one seeks uni o m bounds. This p oblem is
he same in essence as asking which (posi i e) unc ions on Rna ise as
he Jacobian o a quasicon o mal mapping, modulo mul iplica ion by a
posi i e unc ion which is bounded and bounded away om 0.
Some na u al necessa y condi ions a e known o hese ques ions, wi h
a p incipal ing edien coming om [Geh2]. I was na u al o wonde
whe he he necessa y condi ions we e also sufficien . As a es o his,
[Da iS1] looked a he Sobole and ela ed inequali ies ha would ol-
low i he necessa y condi ions we e sufficien . These inequali ies could
be s a ed di ec ly in e ms o he da a o he p oblems, he con o mal
ac o o p ospec i e Jacobian. The conclusion o [Da iS1] was ha
hese inequali ies could be de i ed di ec ly om he condi ions on he
da a, independen ly o whe he hese condi ions we e sufficien o he
exis ence o bilipschi z/quasicon o mal mappings as abo e.
In [Sem5] i was shown ha he candida e condi ions a e no suffi-
cien o he exis ence o such mappings, a leas in dimensions 3 and
highe . (Dimension 2 emains open.) The simples coun e examples
in ol ed conside a ions o localized undamen al g oups, in much he
same ashion as in Sec ion 1. (Ano he class o coun e examples we e
based on a diffe en mechanism, al hough hese did no s a in dimen-
sion 3.) These coun e examples a e all pe ec ly well-beha ed in e ms
o he doubling and local linea con ac abili y p ope ies, and in ac
a e much be e han ha .
298 S. Semmes
Pa o he bo om line he e is ha spaces can ha e geome y which
beha es qui e well o many pu poses e en i hey do no beha e so well
in e ms o pa ame e iza ions.
Fo some o he aspec s o “quan i a i e opology”, see [Ale], [AleV1],
[AleV2], [A 1], [A 2], [BloW], [ChaF], [Che], [Fe 1], [Fe 2], [Fe 3],
[Fe 4], [Geh1], [G o1], [G o2], [G o3], [HeiY], [HeiS], [Luu], [Pe 1],
[Pe 2], [TukV], [V¨ai2], [V¨ai3], [V¨ai4]. Rela ed ma e s o Sobole
and o he inequali ies on non-smoo h spaces come up in [HeiKo2],
[HeiKo3], [HeiKo+], in connec ion wi h he beha io o quasicon o -
mal mappings.
5. Uni o m ec ifiabili y
A basic ac in opology is ha he e a e spaces which a e mani old
ac o s bu no mani olds. Tha is, he e a e opological spaces Msuch
ha M×Ris a mani old (locally homeomo phic o a Euclidean space)
bu Mis no . This can e en happen o fini e polyhed a, because o
he double-suspension esul s o Edwa ds and Cannon. See [Da e2],
[Edw2], [Ki ] o mo e in o ma ion.
Uni o m ec ifiabili y is a no ion o con olled geome y ha ades
opology o es ima es. I ole a es some amoun o singula i ies, like
holes and c ossings, and a oids some common difficul ies wi h homeo-
mo phisms, such as mani old ac o s.
The p ecise defini ion is sligh ly echnical, and elies on measu e he-
o y in a c ucial way. In many espec s i is analogous o he no ion o
BMO om Sec ion 3. The ollowing is a p elimina y concep ha helps
o se he s age.
Defini ion 5.1 (Ahl o s egula i y).Fix nand d, wi h na posi i e in-
ege and 0 <d≤n. A se Econ ained in Rnis said o be (Ahl o s)
egula o dimension di i is closed, and i he e is a posi i e Bo el
measu e µsuppo ed on Eand a cons an C>0 such ha
C−1 d≤µ(B(x, )) ≤C
d
(5.2)
o all x∈Eand 0 < ≤diam E. He e B(x, ) deno es he (open) ball
wi h cen e xand adius .
Roughly speaking, his defini ion asks ha Ebeha e like o dina y
Euclidean space in e ms o he dis ibu ion o i s mass. No ice ha
d-planes sa is y his condi ion au oma ically, wi h µequal o he o -
dina y d-dimensional olume. The same is ue o compac smoo h
mani olds, and fini e polyhed a which a e gi en as unions o d-dimen-
sional simplices (i.e., wi h no lowe -dimensional pieces s icking off in an
Real Analysis, Quan i a i e Topology, e c. 299
isola ed manne ). The e a e also plen y o “ ac al” examples, like sel -
simila Can o se s and snowflake cu es. In pa icula , he dimension d
can be any (posi i e) eal numbe .
A basic ac is ha i Eis egula and µis as in Defini ion 5.1, hen µis
p ac ically he same as d-dimensional Hausdo ff measu e Hd es ic ed
o E. Specifically, µand Hda e each bounded by cons an mul iples
o he o he when applied o subse s o E. This is no ha d o p o e,
and i shows ha µis essen ially unique. Defini ion 5.1 could ha e been
o mula ed di ec ly in e ms o Hausdo ff measu e, bu he e sion abo e
is a bi mo e elemen a y.
Le us ecall he defini ion o a bilipschi z mapping. Le Abe a se
in Rn, and le be a mapping om A o some o he se in Rn.Wesay
ha is k-bilipschi z, whe e kis a posi i e numbe , i
k−1|x−y|≤| (x)− (y)|≤k|x−y|(5.3)
o all x, y ∈A.
Defini ion 5.4 (Uni o m ec ifiabili y).Le Ebe a subse o Rnwhich
is Ahl o s egula o dimension d, whe e dis a posi i e in ege , d<n,
and le µbe a posi i e measu e on Eas in Defini ion 5.1. Then Eis
uni o mly ec ifiable i he e exis s a posi i e cons an kso ha o each
x∈Eand each >0 wi h ≤diam E he e is a closed subse Ao
E∩B(x, ) such ha
µ(A)≥9
10 ·µ(E∩B(x, ))(5.5)
and
he e is a k-bilipschi z mapping om Ain o Rd.(5.6)
In o he wo ds, inside o each “snapsho ” E∩B(x, )o E he e should
be a la ge subse , wi h a leas 90% o he poin s, which is bilipschi z
equi alen o a subse o Rd, and wi h a uni o m bound on he bilipschi z
cons an . This is like asking o a con olled pa ame e iza ion, excep
ha we allow o holes and singula i ies.
Defini ion 5.4 should be compa ed wi h he classical no ion o (coun -
able) ec ifiabili y, in which one asks ha Ebe co e ed, excep o a se
o measu e 0, by a coun able union o se s, each o which is bilipschi z
equi alen o a subse o Rd. Uni o m ec ifiabili y implies his condi-
ion, bu i is s onge , because i p o ides quan i a i e in o ma ion a
defini e scales, while he classical no ion eally only gi es asymp o ic in-
o ma ion as one zooms in a almos any poin . See [Fal], [Fed], [Ma ]
o mo e in o ma ion abou classical ec ifiabili y.
300 S. Semmes
No mally one would be much happie o simply ha e bilipschi z coo -
dina es ou igh , wi hou ha ing o allow o bad se s o small measu e
whe e his does no wo k. In p ac ice bilipschi z coo dina es simply do
no exis in many si ua ions whe e one migh o he wise hope o ha e
hem. This is illus a ed by he double-suspension sphe es o Edwa ds
and Cannon [Can1], [Can3], [Da e2], [Edw2], and he obse a ions
abou hem in [SieS]. Fu he examples a e gi en in [Sem4], [Sem5].
The use o a bi a y scales and loca ions is an impo an pa o he
s o y he e, and is e y simila o he concep o BMO. A he le el o
a single snapsho , a fixed ball B(x, ) cen e ed on E, he bad se may
seem p e y wild, as no hing is said abou wha goes on he e in (5.5) o
(5.6). Howe e , uni o m ec ifiabili y, like BMO, applies o all snapsho s
equally, and in pa icula o balls in which he bad se is concen a ed.
Thus, inside he bad se , he e a e in ac u he con ols. We shall see
o he mani es a ions o his la e , and he same basic p inciple is used in
he John-Ni enbe g heo em o BMO unc ions (discussed in Sec ion 3).
Uni o m ec ifiabili y p o ides a subs i u e o (comple e) bilipschi z
coo dina es in much he same way ha BMO p o ides a subs i u e o
L∞bounds, as in Sec ion 3. No e ha L∞bounds and bilipschi z coo -
dina es au oma ically en ail uni o m con ol o e all scales and loca ions.
This is ue jus because o he way hey a e defined, i.e., a bounded unc-
ion is bounded in all snapsho s, and wi h a uni o m majo an . Wi h
BMO and uni o m ec ifiabili y he scale-in a iance is imposed by hand.
I may be a li le su p ising ha one can ge any hing new h ough
concep s like BMO and uni o m ec ifiabili y. Fo ins ance, suppose ha
is a locally-in eg able unc ion on Rk, and ha he a e ages
1
ωk kB(z, )| (w)|dw(5.7)
a e uni o mly bounded, independen ly o zand . He e ωkdeno es he
olume o he uni ball in Rk, so ha ωk kis he olume o B(z, ).
This implies ha mus i sel be bounded by he same amoun almos
e e ywhe e on Rk, since
(u) = lim
→0
1
ωk kB(z, )
(w)dw(5.8)
almos e e ywhe e on Rk. Thus a uni o m bound o he size o he
snapsho s does imply a uni o m bound ou igh . Fo BMO he si ua ion
is diffe en because one asks only o a uni o m bound on he mean
oscilla ion in e e y ball. In o he wo ds, one also has he eedom o
make eno maliza ions by addi i e cons an s when mo ing om place o
Real Analysis, Quan i a i e Topology, e c. 301
place, and his gi es enough oom o some unbounded unc ions, like
log |x|. Uni o m ec ifiabili y is like his as well, al hough wi h diffe en
kinds o “ eno maliza ions” a ailable.
These ema ks migh explain why some condi ion like uni o m ec i-
fiabili y could be use ul o na u al, bu why he specific e sion abo e in
pa icula ? Pa o he answe o his is ha nea ly all defini ions o his
na u e a e equi alen o he o mula ion gi en abo e. Fo ins ance, he
9/10 in (5.5) can be eplaced by any numbe s ic ly be ween 0 and 1.
See [Da iS3], [Da iS5] o mo e in o ma ion.
Ano he answe lies in a heme o en a icula ed by Coi man, abou
he way ha ope a o heo y can p o ide a good guide o geome y.
One o he o iginal mo i a ions o uni o m ec ifiabili y came om
he “Calde ´on p og am” [Cal2], conce ning he Lp-boundedness o ce -
ain singula ope a o s on cu es and su aces o minimal smoo hness.
Da id [Da i2], [Da i3], [Da i5] showed ha uni o m ec ifiabili y o a
se Eimplies Lp-boundedness o wide classes o singula ope a o s on E.
(See [Cal1], [Cal2], [CoiDM], [CoiMcM] and he e e ences he ein
o ela ed wo k connec ed o he Calde ´on p og am.) In [Da iS3], a
con e se was es ablished, so ha uni o m ec ifiabili y o an Ahl o s-
egula se Eis ac ually equi alen o he boundedness o a sui able
class o singula in eg al ope a o s (inhe i ed om he ambien Euclid-
ean space Rn). See also [Da iS2], [Da iS5], [Ma MV], [Ma P].
He e is a conc e e s a emen abou uni o m ec ifiabili y in si ua ions
whe e well-beha ed pa ame e iza ions would be na u al bu may no
exis .
Theo em 5.9. Le Ebe a subse o Rnwhich is egula o dimension d.
I Eis also a d-dimensional opological mani old and sa isfies he local
linea con ac abili y condi ion (Defini ion 4.10), hen Eis uni o mly
ec ifiable.
No e ha Ahl o s- egula i y au oma ically implies he doubling con-
di ion (Defini ion 4.9).
Theo em 5.9 has been p o ed by G. Da id and mysel . Now-a-days
we ha e be e echnology, which allows o e sions o his which a e
localized o indi idual “snapsho s”, a he han using all scales and lo-
ca ions a once. See [Da iS8] (wi h some o he ema ks in Sec ion 12.3
o [Da iS8] helping o p o ide a b idge o he p esen o mula ion). We
shall say a bi mo e abou his, nea he end o Subsec ion 5.3.
The equi emen ha Ebe a opological mani old is con enien , bu
weake condi ions could be used. Fo ha ma e , he e a e na u al
a ia ions o local linea con ac abili y oo.
302 S. Semmes
One can hink o Theo em 5.9 and ela ed esul s in he ollowing
e ms. Gi en a compac se K, uppe bounds o he d-dimensional
Hausdo ff measu e o K oge he wi h lowe bounds o he d-dimen-
sional opology o Kshould lead o s ong in o ma ion abou he geo-
me ic beha io o K. See [Da iS6], [Da iS8], [Sem3] o mo e on
his.
To unde s and be e wha Theo em 5.9 means, le us begin by ob-
se ing ha he hypo heses o Theo em 5.9 would hold au oma ically i
Ewe e bilipschi z equi alen o Rd,o i Ewe e compac and admi ed
bilipschi z local coo dina es om Rd. Unde hese condi ions, a es o
he hypo heses o Theo em 5.9 on Ecan be con e ed in o a simila es
on Rd, whe e i can hen be esol ed in a s aigh o wa d manne .
A simila a gumen shows ha he hypo heses o Theo em 5.9 a e
“bilipschi z in a ian ”. Mo e p ecisely, i Fis ano he subse o Rn
which is bilipschi z equi alen o E, and i he hypo heses o Theo em 5.9
holds o one o Eand F, hen i au oma ically holds o he o he .
Since he exis ence o bilipschi z coo dina es implies he hypo heses
o Theo em 5.9, we canno ask o mo e han ha in he conclusions.
In o he wo ds, bilipschi z coo dina es a e a he high end o wha one
can hope o in he con ex o Theo em 5.9. The hypo heses o Theo-
em 5.9 do in ac ule ou a lo o basic obs uc ions o he exis ence
o bilipschi z coo dina es, like cusps, ac al beha io , sel -in e sec ions
and app oxima e sel -in e sec ions, and bubbles wi h e y small necks.
(Compa e wi h Sec ion 4, especially Theo em 4.11 and he discussion o
i s p oo and consequences.) None heless, i can easily happen ha a
se Esa isfies he hypo heses o Theo em 5.9 bu does no admi bilip-
schi z local coo dina es. Double-suspension sphe es p o ide spec acula
coun e examples o his (using he obse a ions o [SieS]). Addi ional
coun e examples a e gi en in [Sem4], [Sem5].
We should pe haps emphasize ha he assump ion o being a opo-
logical mani old in Theo em 5.9 does no in ol e bounds. By con as ,
uni o m ec ifiabili y does in ol e bounds, which is pa o he poin . In
he con ex o Theo em 5.9, he p oo shows ha he uni o m ec ifiabil-
i y cons an s o he conclusion a e con olled in e ms o he cons an s
ha a e implici in he hypo heses, i.e., in Ahl o s- egula i y, he linea
con ac abili y condi ion, and he dimension.
I bilipschi z coo dina es a e a he high end o wha one could hope
o , wha happens i one asks o less? Wha i one asks o homeomo -
phic local coo dina es wi h some con ol, bu no as much? Fo ins ance,
ins ead o bounding he “ a e” o con inui y h ough Lipschi z condi ions
Real Analysis, Quan i a i e Topology, e c. 303
like
| (x)− (y)|≤C|x−y|(5.10)
( o some Cand all x,yin he domain o ), one could wo k wi h H¨olde
con inui y condi ions, which ha e he o m
| (x)− (y)|≤C|x−y|γ.(5.11)
He e γis a posi i e numbe , some imes called he H¨olde “exponen ”.
As usual, (5.11) is supposed o hold simul aneously o all xand yin he
domain o , and wi h a fixed cons an C. When xand ya e close o
each o he and γis less han 1, his ype o condi ion is s ic ly weake
han ha o being Lipschi z. Jus as (x)=|x|is a s anda d example o
a Lipschi z unc ion ha is no diffe en iable a he o igin, g(x)=|x|γ
is a basic example o a unc ion ha is H¨olde con inuous o o de γ,
γ≤1, bu no o any o de la ge han γ, in any neighbo hood o he
o igin.
Ins ead o local coo dina es which a e bilipschi z, one could conside
ones ha a e “bi-H¨olde ”, i.e., H¨olde con inuous and wi h H¨olde con-
inuous in e se. I u ns ou ha double-suspension sphe es do no ad-
mi bi-H¨olde local coo dina es when he H¨olde exponen γlies abo e
an explici h eshold. Specifically, i Pis an n-dimensional polyhe-
d on which is he double-suspension o an (n−2)-dimensional homology
sphe e ha is no simply connec ed, hen he e a e poin s in P(along
he “suspension ci cle”) o which bi-H¨olde local coo dina es o expo-
nen γ>1/(n−2) do no exis . This comes om he same a gumen
as in [SieS]. Mo e p ecisely, a ound hese poin s in P, he e do no
exis homeomo phic local coo dina es om subse s o Rn o which he
in e se mapping is H¨olde con inuous o o de γ>1/(n−2) (wi hou
equi ing a H¨olde condi ion o he mapping i sel ).
Gi en any posi i e numbe a, he e a e examples in [Sem5] so ha
local coo dina es (a some poin s) canno ha e hei in e ses be H¨olde
con inuous o o de a. These examples do admi bi-H¨olde local coo -
dina es (wi h a smalle exponen ), and e en “quasisymme ic” [TukV]
coo dina es, and hey sa is y he hypo heses o Theo em 5.9. In [Sem4]
he e a e examples which sa is y he hypo heses o Theo em 5.9, bu o
which no uni o m modulus o con inui y o local coo dina e mappings
and hei in e ses is possible (o e all scales and loca ions).
304 S. Semmes
5.1. Smoo hness o Lipschi z and bilipschi z mappings.
Ano he aspec o uni o m ec ifiabili y is ha i p o ides he same
amoun o “smoo hness” as when he e is a global bilipschi z pa ame-
e iza ion. To make his p ecise, le us fi s look a he smoo hness o
Lipschi z and bilipschi z mappings.
A mapping :Rd→Rnis Lipschi z i he e is a cons an Cso ha
(5.10) holds o all x,yin Rd. The space o Lipschi z mappings is a
bi simple han he space o bilipschi z mappings, because he o me
is a ec o space (and e en a Banach space) while he la e is no . Fo
he pu poses o “smoo hness” p ope ies, hough, he e is no eally any
diffe ence be ween he wo. Bilipschi z mappings a e always Lipschi z,
and any hing ha can happen wi h Lipschi z mappings can also happen
wi h bilipschi z mappings (by adding new componen s, o conside ing
x+h(x) when h(x) has Lipschi z no m less han 1 o ge a bilipschi z
mapping).
One should also no wo y oo much abou he diffe ence be ween
Lipschi z mappings which a e defined on all o Rd, and ones ha a e
only defined on a subse . Lipschi z mappings in o Rn ha a e defined
on a subse o Rdcan always be ex ended o Lipschi z mappings on
all o Rd. This is a s anda d ac . The e a e also ex ension esul s o
bilipschi z mappings, i one pe mi s onesel o eplace he image Rnwi h
a Euclidean space o la ge dimension (which is no oo se ious in he
p esen con ex ).
Fo conside a ions o “smoo hness” we migh as well es ic ou a -
en ion o unc ions which a e eal- alued, since he Rn- alued case can
always be educed o ha .
Two basic ac s abou Lipschi z unc ions on Rda e ha hey a e
diffe en iable almos e e ywhe e (wi h espec o Lebesgue measu e),
and ha o each η>0 hey can be modified on se s o Lebesgue measu e
less han η(depending on he unc ion) in such a way as o become
con inuously diffe en iable e e ywhe e. See [Fed].
These a e well-known esul s, bu hey do no ell he whole s o y.
They a e no quan i a i e; hey say a lo abou he asymp o ic beha io
(on a e age) o Lipschi z mappings a e y small scales, bu hey do no
say any hing abou wha happens a scales o defini e size.
To make his p ecise, le a Lipschi z mapping :Rd→Rbe gi en,
and fix a poin x∈Rdand a adius >0. We wan o measu e how
well is app oxima ed by an affine unc ion on he ball B(x, ). To do
Real Analysis, Quan i a i e Topology, e c. 305
his we define he quan i y α(x, )by
α(x, ) = in
A∈A sup
y∈B(x, )
−1| (y)−A(y)|.(5.12)
He e Adeno es he ( ec o space) o affine unc ions on Rd. The pa on
he igh side o (5.12) wi h jus he sup emum (and no he infimum)
measu es how well he pa icula affine unc ion Aapp oxima es inside
B(x, ), and hen he infimum gi es us he bes app oxima ion by any
affine unc ion o a pa icula choice o xand . The ac o o −1
makes α(x, ) scale p ope ly, and be dimensionless. In pa icula , α(x, )
is uni o mly bounded in xand when is Lipschi z, because we can
ake A(y) o be he cons an unc ion equal o he alue o a x.
The smallness o α(x, ) p o ides a mani es a ion o he smoo hness
o . Fo unc ions which a e wice-con inuously diffe en iable one can
ge es ima es like
α(x, )=O( ),(5.13)
using Taylo ’s heo em. I 0 <δ<1, hen es ima es like
α(x, )=O( δ)(5.14)
(locally uni o mly in x) co espond o H¨olde con inui y o he g adien
o o o de δ. Diffe en iabili y almos e e ywhe e o implies ha
lim
→0α(x, ) = 0 o almos e e y x.(5.15)
This does no say any hing abou any pa icula , because one does no
know how long one migh ha e o wai be o e he limi ing beha io kicks
in.
He e is a simple example. Le us ake d= 1, and conside he unc ion
gρ(x)=ρ·sin(x/ρ)(5.16)
on R. He e ρis any posi i e numbe . Now, gρ(x) is Lipschi z wi h no m 1
no ma e how ρis chosen. This is no ha d o check; o ins ance, one
can ake he de i a i e o ge ha
g
ρ(x) = cos(x/ρ)(5.17)
so ha |g
ρ(x)|≤1 e e ywhe e. This implies ha
|gρ(u)−gρ( )|≤|u− |(5.18)
o all uand (and all ρ), because o he mean- alue heo em, o he
undamen al heo em o calculus. (One also has ha |g
ρ(x)|= 1 a some
poin s, so ha he Lipschi z no m is always equal o 1.)
312 S. Semmes
sufficien o imply he uni o m ec ifiabili y o he se E, a leas i Eis
Ahl o s- egula o dimension d. This was p o ed in [Da iS5].
In he con ex o unc ions, his ype o h esholding condi ion is oo
weak, in ha one can ha e he α(x, )’s going o 0 uni o mly as →0 o
unc ions which a e diffe en iable almos nowhe e, as men ioned in Sub-
sec ion 5.1. Simila ly, he e a e Ahl o s- egula se s which a e “ o ally
un ec ifiable” (in he sense o [Fal], [Fed], [Ma ]) and ha e he β(x, )’s
ending o 0 uni o mly as →0. (See [Da iS3].) Fo he bβ’s he s o y
is simply diffe en . On he o he hand, he ac ha sui able h eshold-
ing condi ions on he bβ’s a e sufficien o imply uni o m ec ifiabili y
elies hea ily on he assump ion ha Ebe Ahl o s- egula , while mass
bounds a e pa o he conclusion ( a he han he hypo hesis) in Jones’
esul s, and no coun e pa o he mass bounds a e included in he abo e-
men ioned examples o unc ions. One does ha e mass bounds o he
examples in [Da iS3] (o o ally-un ec ifiable Ahl o s- egula se s o
which he β(x, )’s end o 0 uni o mly as →0), and he e he issue is
mo e in he size o he holes in he se . The bβ’s, by defini ion, con ol
he sizes o holes. No e ha his esul o he bβ’s does ha e an eceden s
o he classical no ion o (coun able) ec ifiabili y, as in [Ma ].
The e a e a numbe o a ian s o he bβ’s, in which one makes com-
pa isons wi h o he collec ions o se s besides d-planes, like unions o
d-planes, o ins ance. See [Da iS5].
Pe haps he s onges o mula ion o smoo hness o uni o mly ec ifi-
able se s is he exis ence o a “Co ona decomposi ion”. This is a geome -
ic e sion o he in o ma ion ha one can ge abou a Lipschi z unc ion
om he me hods o Ca leson’s Co ona cons uc ion (as men ioned in
Subsec ion 5.1). Roughly speaking, in his condi ion one con ols no
only how o en Eis well-app oxima ed by a d-plane, bu how as he
d-planes u n as well. This can also be o mula ed in e ms o good
app oxima ions o Eby fla Lipschi z g aphs.
Al hough a bi echnical, he exis ence o a Co ona decomposi ion is
pe haps he mos use ul way o managing he complexi y o a uni o mly
ec ifiable se . Once one has a Co ona decomposi ion, i is gene ally
p e y easy o de i e wha e e else one would like o know. Con e sely,
in p ac ice he exis ence o a Co ona decomposi ion can be a good place
o s a i one wan s o p o e ha a se is uni o mly ec ifiable.
In ac , he e is a gene al p ocedu e o finding a Co ona decomposi-
ion when i exis s, and one which is ai ly simple (and e y simila o
Ca leson’s Co ona cons uc ion). The difficul pa is o show ha his
p ocedu e wo ks in he igh way, wi h he co ec es ima es. Specifi-
cally, i is a s opping- ime a gumen , and one does no wan o ha e o
Real Analysis, Quan i a i e Topology, e c. 313
s op oo o en. This is a nice poin , because in gene al i is no so easy o
build some hing like a good pa ame e iza ion o a se , e en i one knows
a p io i ha i exis s. In his con ex , he e a e in p inciple me hods o
doing his.
See [Da iS2], [Da iS3], [Da iS5], [Da iS7], [Sem1] o mo e in-
o ma ion abou Co ona decomposi ions o uni o mly ec ifiable se s and
he way ha hey can be used. The pape [Da iS7] is w i en in such
a way as o y o con ey some o he basic concep s and cons uc ions
wi hou wo ying abou why he heo ems a e ue (which is much mo e
complica ed). In pa icula , he basic p ocedu e o finding Co ona de-
composi ions when hey exis is discussed. See [Ga nJ], [Jon1], [Jon3]
o some o he si ua ions in which Ca leson’s Co ona cons uc ion is used
geome ically.
5.3. A class o a ia ional p oblems.
Uni o m ec ifiabili y is a p e y obus condi ion. I one has a se
which looks oughly as hough i ough o be uni o mly ec ifiable, hen
he e is a good chance ha i is. This as opposed o se s which look
oughly as hough hey should admi a well-beha ed (homeomo phic)
pa ame e iza ion, and do no (as discussed be o e).
In his subsec ion we would like o b iefly men ion a esul o his
ype, conce ning a minimal su ace p oblem wi h nonsmoo h coefficien s.
Le g(x) be a Bo el measu able unc ion on Rn, and assume ha gis
posi i e, bounded, and bounded away om 0, so ha
0<m≤g(x)≤M(5.28)
o some cons an s m,Mand all x∈Rn. Le Q0,Q1be a pai o
(closed) cubes in Rn, wi h sides pa allel o he axes, and assume ha
Q0is con ained in he in e io o Q1.
Le Ube an open subse o Q1which con ains he in e io o Q0.
Conside an in eg al like ∂U
g(x)dνU(x),(5.29)
whe e dνUdeno es he measu e ha desc ibes he (n−1)-dimensional
olume o subse s o ∂U. This would be defined as in calculus when ∂U
is a leas a li le bi smoo h (like C1), bu in gene al one has o be mo e
ca e ul. One can simply ake o dνU he es ic ion o (n−1)-dimensional
Hausdo ff measu e o ∂U, bu o echnical easons i is o en be e o
define dνUusing dis ibu ional de i a i es o he cha ac e is ic unc ion
o U,asin[Giu]. Fo his one would wo k wi h se s Uwhich ha e “fini e
314 S. Semmes
pe ime e ”, which means exac ly ha he dis ibu ional fi s de i a i es
o he cha ac e is ic unc ion o Ua e measu es o fini e mass.
He e is one way in which his kind o unc ional, and he minimiza ion
o his kind o unc ional, can come up. Le Fbe a closed subse o Rn.
Imagine ha one is pa icula ly in e es ed in domains Uwhich ha e
hei bounda y con ained in F, o e y nea ly so. On he o he hand,
one migh also wish o limi i egula i ies in he beha io o he bounda y
o U. Fo his ype o si ua ion one could choose gso ha i is much
smalle on F han on he complemen o F, and hen look o minimize s
o (5.29) o find domains wi h a good balance be ween he beha io o
∂U and he desi e o ha e i be con ained (as much as possible) in F.
(See [Da iS6] o an example o his.)
When do minimize s o (5.29) exis , and how do hey beha e? I one
wo ks wi h se s o fini e pe ime e , and i he unc ion gis lowe semi-
con inuous, hen one can ob ain he exis ence o minimize s h ough
s anda d echniques (as in [Giu]). Tha is, one akes limi s o minimizing
sequences o (5.29) using weak compac ness, and one uses he lowe
semi-con inui y o g o ge lowe semi-con inui y o (5.29) wi h espec
o sui able con e gence o he U’s. The la e ensu es ha he limi o he
minimizing sequence is ac ually a minimum. No e ha he “obs acle”
condi ions ha Ucon ain he in e io o Q0and be con ained in Q1
p e en s he minimiza ion om collapsing in o some hing i ial.
As o he beha io o minimize s o (5.29), one canno expec much
in he way o smoo hness in gene al. Fo ins ance, i he bounda y o U
can be ep esen ed locally as he g aph o a Lipschi z unc ion, hen U
in ac minimizes (5.29) o a sui able choice o g. Specifically, one can
ake g o be a sufficien ly small posi i e cons an on ∂U, and o be equal
o 1 e e ywhe e else. Tha such a choice o gwo ks is no e y ha d o
es ablish, and mo e p ecise esul s a e gi en in [Da iS6].
Con e sely, minimize s o (5.29) a e always Ahl o s- egula se s o
dimension n−1, and uni o mly ec ifiable. This is shown in [Da iS6],
along wi h some addi ional geome ic in o ma ion which is sufficien o
cha ac e ize he class o se s Uwhich occu as minimize s o unc ionals
o he o m (5.29) (wi h gbounded and bounded away om 0). I a se U
a ises as he minimize o some g, i is also a minimize wi h gchosen
as abo e, i.e., a small posi i e cons an on ∂U and equal o 1 e e ywhe e
else.
The same egula i y esul s wo k o a sui able class o “quasimini-
mize s” o he usual a ea unc ional, and one ha includes minimize s
o (5.29) as a special case.
Real Analysis, Quan i a i e Topology, e c. 315
Uni o m ec ifiabili y p o ides a na u al le el o s uc u e o si ua-
ions like his, whe e s onge o ms o smoo hness canno be expec ed,
bu quan i a i e bounds a e easonable o seek. No e ha p ope ies
o o dina y ec ifiabili y always hold o bounda ies o se s o fini e
pe ime e , ega dless o any minimizing o quasiminimizing p ope ies.
See [Giu].
Analogous esul s abou egula i y wo k o se s o highe codimension
as well, al hough his case is mo e complica ed echnically. See [Da iS8]
o mo e in o ma ion. One can use his amewo k o minimiza ion (wi h
espec o nonsmoo h coefficien unc ions g) as a ool o s udying he
s uc u e o se s in Rnwi h uppe bounds on hei d-dimensional Haus-
do ff measu e and lowe bounds o hei d-dimensional opology. This
b ings one back o Theo em 5.9 and ela ed ques ions, and in pa icula
mo e “localized” e sions o i .
To pu i ano he way, minimiza ion o unc ionals like hese can
p o ide use ul means o ob aining “exis ence esul s” o app oxima e
pa ame e iza ions wi h good beha io , h ough uni o m ec ifiabili y.
See [Da iS6], [Da iS8]. Pa o he mo i a ion o his came om an
ea lie a gumen o Mo el and Solimini [Mo eS]. Thei a gumen con-
ce ned he exis ence o cu es con aining a gi en se , wi h good p op-
e ies in e ms o he dis ibu ion o he a c-leng h measu e o hese
cu es, unde mo e localized condi ions on he gi en se (a all loca ions
and scales). See Lemma 16.27 on p. 207 o [Mo eS].
Appendix A. Fou ie ans o m calcula ions
I φ(x) is an in eg able unc ion on Rn, hen i s Fou ie ans o m
φ(ξ)
is defined ( o ξ∈Rn)by
φ(ξ)=Rn
eix,ξφ(x)dx.(A.1)
He e x, ξdeno es he usual inne p oduc o x, ξ ∈Rn, and i=√−1.
O en one makes sligh ly diffe en con en ions o his defini ion —wi h
some ex a ac o s o πa ound, o ins ance— bu we shall no bo he
wi h his.
A key ea u e o he Fou ie ans o m is ha i diagonalizes diffe -
en ial ope a o s. Specifically, i ∂kdeno es he ope a o ∂/∂xkon Rn,
hen
(∂kφ)
(ξ)=iξ
k
φ(ξ),(A.2)
i.e., diffe en ia ion is con e ed in o me e mul iplica ion. Fo his one
should ei he make some diffe en iabili y assump ions on φ, so ha he
316 S. Semmes
le side can be defined in pa icula , o one should in e p e his equa ion
in he sense o empe ed dis ibu ions on Rn. The Fou ie ans o m
also ca ies ou his diagonaliza ion in a con olled manne . Tha is,
he e is an explici in e sion o mula (which looks a lo like he Fou ie
ans o m i sel ), and he Fou ie ans o m p ese es he L2no m o
he unc ion φ, excep o a mul iplica i e cons an , by he Planche el
heo em. See [Duo], [S eW], [To c] o hese and o he basic ac s
abou he Fou ie ans o m.
Using Planche el’s heo em, i is e y easy o gi e ano he p oo o he
L2es ima e (3.2) om Sec ion 3, and o de i e many o he inequali ies
o a simila na u e. One can also use he Fou ie ans o m o gi e a
p ecise defini ion o he ope a o R=∂j∂k/∆, whe e ∆ is he Laplace
ope a o n
=1 ∂2
. Specifically, one can define i h ough he equa ion
(Rφ)
(ξ)=ξjξk
|ξ|2
φ(ξ).(A.3)
I m(ξ) is any bounded unc ion on Rn, hen
(Tφ)
(ξ)=m(ξ)
φ(ξ)(A.4)
defines a bounded ope a o on L2(Rn). In gene al hese ope a o s a e
no bounded on Lp o any o he alue o p, bu his is ue o many
o he ope a o s ha a ise na u ally in analysis. Fo ins ance, suppose
ha m(ξ) is homogeneous o deg ee 0, so ha
m( ξ)=m(ξ) when >0,(A.5)
and ha m(ξ) is smoo h away om he o igin. Then he associa ed
ope a o Tis bounded on Lp o all pwi h 1 <p<∞. See [Duo],
[Ga cR], [Jou], [S e1], [S eW], [S T], [To c]. No e ha his c i e ion
applies o he specific choice o m(ξ) in (A.3) abo e.
Fo a mul iplie ope a o as in (A.4) o be bounded on L1o L∞
is e en mo e excep ional han o Lpboundedness when 1 <p<∞.
(See [S eW], [To c].) Fo ins ance, i mis homogeneous, as abo e, and
no cons an , hen he co esponding ope a o canno be bounded on
L1o L∞. Howe e , i mis homogeneous and smoo h away om he
o igin, hen he ope a o Tin (A.4) does de e mine a bounded ope a o
om L∞in o BMO. See [Duo], [Ga cR], [Ga n], [Jou], [Sa ], [S e2],
[To c]. In ac , Tde e mines a bounded ope a o om BMO o i sel .
Real Analysis, Quan i a i e Topology, e c. 317
He e is ano he example. Le φnow be a mapping om R2 o i sel ,
wi h componen s φ1,φ2. Conside he diffe en ial dφ o φas a ma ix-
alued unc ion, namely,
∂1φ1∂1φ2
∂2φ1∂2φ2.(A.6)
(Le us assume ha φis smoo h enough ha he diffe en ial is a leas
some kind o unc ion when aken in he sense o dis ibu ions, al hough
one can pe ec ly well hink o dφ as a ma ix- alued dis ibu ion.) Le A
and Sdeno e he an isymme ic and symme ic pa s o dφ, espec i ely,
so ha
A=dφ −dφ
2,S=dφ +dφ
2,(A.7)
whe e dφ deno es he anspose o dφ.
In his case o 2×2 ma ices, he an isymme ic pa A eally con ains
only one piece o in o ma ion, namely
∂1φ2−∂2φ1.(A.8)
I is no ha d o check ha his unc ion can be econs uc ed om he
en ies o S h ough ope a o s o he o m (A.4), using unc ions mwhich
a e homogeneous o deg ee 0 and smoo h away om he o igin. Fo his
one should add some mild condi ions on φ, like compac suppo , o
a oid he possibili y ha S anishes iden ically bu Adoes no .
Unde hese condi ions, we conclude ha he Lpno m o Ais always
bounded by a cons an mul iple o he Lpno m o S,1<p<∞, and
ha he BMO no m o Ais con olled by he L∞(o BMO) no m o S.
(Fo he case o BMO no ms, he possibili y ha S anishes bu Adoes
no causes no ouble, because Awill be cons an in ha case.)
This example is eally a “linea ized” e sion o he p oblem discussed
in Sec ion 2. Specifically, le us hink o :R2→R2as being o he
o m
(x)=x+'φ(x),(A.9)
whe e 'is a small pa ame e . The ex en o which dis o s dis ances
is go e ned by he ma ix- alued unc ion d d , which we can w i e ou
as
d d =I+4'S+'2dφ dφ.(A.10)
Thus he linea e m in 'is go e ned by S, while Acon ols he leading
beha io in 'o he “ o a ional” pa o d .
318 S. Semmes
Appendix B. Mappings wi h b anching
In gene al, he e can be a lo o ouble wi h exis ence and complexi y
o homeomo phisms (wi h pa icula p ope ies, like specified domain
and ange). I one allows mappings wi h b anching, hen he s o y can
be e y diffe en .
As a basic example o his, he e is a classical esul o igina ing wi h
Alexande o he effec ha any o ien ed pseudomani old o dimension n
admi s an o ien a ion-p ese ing b anched co e ing o e he n-sphe e.
Le us s a e his mo e ca e ully, and hen see how i is p o ed.
Le Mbe a fini e polyhed on. We assume ha Mis gi en as a fini e
union o n-dimensional simplices ha mee only in hei bounda y aces
(so ha Mis eally a simplicial complex). To be a pseudomani old means
ha e e y (n−1)-dimensional ace in Ma ises as he bounda y ace o
exac ly wo n-dimensional simplices. In effec his says ha Mlooks
like a mani old away om i s codimension-2 skele on ( he co espond-
ing s a emen o he codimension-1 skele on being au oma ic). Fo he
p esen pu poses i would be enough o ask ha e e y (n−1)-dimen-
sional ace in Ma ise as he bounda y ace o a mos wo n-dimensional
simplices, which would be like a “pseudomani old wi h bounda y”.
An o ien a ion o an n-dimensional pseudomani old Mmeans a
choice o o ien a ion (in he usual sense) o each o he cons i uen n-di-
mensional simplices in M, wi h compa ibili y o o ien a ions o adjacen
n-dimensional simplices along he common (n−1)-dimensional ace. In
e ms o algeb aic opology, his means ha he sum o he n-dimen-
sional simplices in M, wi h hei o ien a ions, defines an n-dimensional
cycle on M.
Fo he pu poses o he Alexande - ype esul , i will be con enien
o hink o he n-sphe e as consis ing o wo s anda d simplices S1and
S2glued oge he along he bounda y. This is no qui e a polyhed on in
he usual (affine) sense, bu one could easily epai his by subdi iding
S1o S2. We also assume ha S1and S2ha e been o ien ed, and ha e
opposi e o ien a ions ela i e o hei common bounda y.
To define a mapping om M o he n-sphe e one would like o simply
iden i y each o he cons i uen n-dimensional simplices in Mwi h S1o
S2in a sui able manne . Un o una ely, his does no wo k, e en when
n= 2, bu he p oblem can be fixed using a ba ycen ic subdi ision
o M. Recall ha he ba ycen e o a simplex (embedded in some ec o
space) is he poin in he in e io o he simplex which is he a e age
o he e ices o he simplex. The se o ba ycen e s o Mmeans he
se o ba ycen e s o all o he cons i uen simplices in M( iewed as a
Real Analysis, Quan i a i e Topology, e c. 319
simplicial complex), o all dimensions, including 0. In pa icula , he se
o ba ycen e s o Mincludes he e ices o M(which a e hemsel es
0-dimensional simplices, and hei own ba ycen e s). The ba ycen ic
subdi ision o Mis a efinemen o Mas a simplicial complex whose
e ices a e exac ly he se o ba ycen e s o M. In o he wo ds, he
se Mas a whole does no change, jus i s decomposi ion in o simplices,
which is eplaced by a fine decomposi ion.
He e is a p ecise desc ip ion o he simplices in he ba ycen ic sub-
di ision o M. Le s0,s
1,...,s
kbe a fini e sequence o simplices in
M, wi h each sian i-dimensional simplex which is a ace o si+1 (when
i<k). Le b(si) deno e he ba ycen e o si. Then b(s0),b(s1),...,b(sk)
a e affinely independen , and hence de e mine a k-dimensional simplex.
The simplices ha a ise in his manne a e p ecisely he ones used o
he ba ycen ic subdi ision o M. (See p. 123 o [Spa] o mo e de ails.)
Le
Vdeno e he se o all e ices in he ba ycen ic subdi ision
o M. This is he same as he se o poin s which a ise as ba ycen e s o
simplices in he o iginal e sion o M, and in pa icula we ha e a na u al
mapping om
V o he in ege s {0,1,...,n}, defined by associa ing o
each poin bin
V he dimension o he simplex om which i was de i ed.
I Tis a k-dimensional simplex in he ba ycen ic subdi ision o
M, hen he mapping om
V o {0,1,...,n}jus desc ibed induces
a one- o-one co espondence be ween he k+ 1 e ices o Tand he
se {0,1,...,k}. This ollows easily om he defini ions.
We a e now eady o define ou mapping om M o he n-sphe e.
The e a e exac ly n+ 1 e ices in ou ealiza ion o he n-sphe e as he
gluing o S1and S2. Le us iden i y hese e ices wi h he in ege s om 0
o n. Thus ou mapping om
V o {0,1,...,n}can now be in e p e ed
as a mapping om he e ices o he ba ycen ic subdi ision o M o
he e ices o he n-sphe e.
This mapping be ween e ices admi s a canonical linea ex ension o
each k-dimensional simplex, k<n, in he ba ycen ic subdi ision o M.
Fo he n-dimensional simplices he ex ension is uniquely de e mined
once one chooses S1o S2 o he image o he simplex. Because o
he o ien a ions, he e is only one na u al choice o S1o S2 o each
n-dimensional simplex T, namely he one so ha he linea mapping
om Ton o Sjis o ien a ion-p ese ing.
In he end we ge a mapping om he ba ycen ic subdi ision o M
o he n-sphe e which p ese es o ien a ions and which defines an affine
isomo phism om each n-dimensional simplex Tin he domain on o one
o S1and S2. This uses he ac ha ou ini ial mapping be ween e ices
320 S. Semmes
was always one- o-one on he se o e ices in any gi en simplex in he
domain, by cons uc ion.
This comple es he p oo . We should emphasize ha he singula i ies
o he mapping om M o he n-sphe e —i.e., he places whe e i ails o
be a local homeomo phism— a e confined o he codimension-2 skele on
o he ba ycen ic subdi ision o M. This is because o he o ien a ion
and pseudomani old condi ions, which ensu e ha i a poin in Mlies
in he in e io o an (n−1)-dimensional simplex in he ba ycen ic sub-
di ision o M, hen he ( wo) adjacen n-simplices a ha poin a e no
sen o he same Sjin he image.
The idea o b anching also makes sense o mappings ha a e no
piecewise-linea , and he e a e well-de eloped no ions o “con olled
geome y” in his case, as wi h he classes o quasi egula mappings and
mappings o bounded leng h dis o ion. See [HeiKiM], [Ma RiV1],
[Ma RiV2], [Ma RiV3], [Ma V], [Res], [Ric1], [V¨ai1], [Vuo], o in-
s ance. In [HeiR1], [HeiR2] he e a e examples whe e b anching maps
o con olled geome y can be cons uc ed bu sui able homeomo phisms
ei he do no exis o mus dis o dis ances mo e se e ely.
Sulli an [Sul2], [Sul3] has p oposed some mechanisms by which he
exis ence o local (con olled) b anching maps can be deduced, and some
ideas o s udying obs uc ions o con olled homeomo phic coo dina es.
See [Gu +], [HeiKi], [Ma RyV] o some ecen esul s abou
b anching and egula i y condi ions unde which i does no occu . A
b oade and mo e de ailed discussion o mappings wi h b anching can
be ound in [HeiR2]. Fo some eal- a iable conside a ions o map-
pings which may b anch bu enjoy subs an ial geome ic p ope ies,
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Depa men o Ma hema ics
Rice Uni e si y
Box 1892
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