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Rigidi y and almos igidi y o Sobole inequali ies on compac spaces wi h lowe Ricci
cu a u e bounds
© The Au ho (s) 2022.
Published e sion
Nobili, F ancesco; Violo, I an Yu i
Nobili, F., & Violo, I. Y. (2022). Rigidi y and almos igidi y o Sobole inequali ies on compac
spaces wi h lowe Ricci cu a u e bounds. Calculus o Va ia ions and Pa ial Di e en ial
Equa ions, 61(5), A icle 180. h ps://doi.o g/10.1007/s00526-022-02284-7
2022
Calc. Va . (2022) 61:180
h ps://doi.o g/10.1007/s00526-022-02284-7
Calculus o Va ia ions
Rigidi y and almos igidi y o Sobole inequali ies on
compac spaces wi h lowe Ricci cu a u e bounds
F ancesco Nobili1
·I an Yu i Violo1
Recei ed: 5 Augus 2021 / Accep ed: 7 June 2022
© The Au ho (s) 2022
Abs ac
We p o e ha i Mis a closed n-dimensional Riemannian mani old, n≥3, wi h Ric ≥n−1
and o which he op imal cons an in he c i ical Sobole inequali y equals he one o he n-
dimensionalsphe eSn, henMisisome ic oSn.Analmos - igidi y esul isalsoes ablished,
saying ha i equali y is almos achie ed, hen Mis close in he measu e G omo –Hausdo
sense oasphe icalsuspension.Theses a emen sa eob ainedin heRCD-se ingo (possibly
non-smoo h) me ic measu e spaces sa is ying syn he ic lowe Ricci cu a u e bounds. An
independen esul o ou analysis is he cha ac e iza ion o he bes cons an in he Sobole
inequali y on any compac CD space, ex ending o he non-smoo h se ing a classical esul
by Aubin. Ou a gumen s a e based on a new concen a ion compac ness esul o mGH-
con e ging sequences o RCD spaces and on a Pólya–Szeg˝o inequali y o Euclidean- ype in
CD spaces. As an applica ion o he echnical ools de eloped we p o e bo h an exis ence
esul o he Yamabe equa ion and he con inui y o he gene alized Yamabe cons an unde
measu e G omo –Hausdo con e gence, in he RCD-se ing.
Ma hema ics Subjec Classi ica ion 53C24 ·46E36
Con en s
1 In oduc ion ................................................
1.1 Bes cons an in he Sobole inequali y on compac CD spaces ...................
1.2 Main igidi y and almos igidi y esul s in compac RCD spaces ..................
1.3 Addi ional esul s and applica ion o he Yamabe equa ion ......................
1.4 P oo -ou line o he igidi y o Aop
q.................................
2 P elimina ies ................................................
2.1 Basic no a ions ............................................
Communica ed by A. Mondino.
BF ancesco Nobili
[email p o ec ed]
I an Yu i Violo
i an.y[email p o ec ed]
1Depa men o Ma hema ics and S a is ics, Uni e si y o Jy äskylä, P.O. Box 35 (MaD),, FI 40014
jy äskylän, Finland
0123456789().: V,- ol 123
180 Page 2 o 65 F. Nobili, I. Y. Violo
2.2 Calculus on me ic measu e spaces ..................................
2.2.1 Sobole spaces .........................................
2.2.2 Func ions o bounded a ia ions and se s o ini e pe ime e .................
2.3 CD(K,N)and RCD(K,N)spaces ..................................
2.3.1 Main de ini ions and p ope ies .................................
2.3.2 Sobole –Poinca é inequali y on CD(K,N)spaces ......................
2.3.3 Con e gence and compac ness unde mGH-con e gence ...................
2.4 Pólya–Szeg˝o inequali y ........................................
3 Uppe bound o αp............................................
3.1 Pólya–Szeg˝o inequali y o Euclidean- ype ..............................
3.2 Local Sobole inequali y .......................................
3.3 P oo o he uppe bound .......................................
4 Lowe bound on αp............................................
4.1 Blow-up analysis o Sobole cons an s ................................
4.2 Sha p and igid Sobole inequali ies unde Euclidean olume g ow h ................
5 The cons an Aop
qin me ic measu e spaces ................................
5.1 Uppe bound on Aop
qin e ms o Ricci bounds ............................
5.2 Lowe bound on Aop
qin e ms o he i s eigen alue ........................
5.3 Lowe bound on Aop
qin e ms o he diame e ............................
6 Rigidi y o Aop
q..............................................
6.1 Concen a ion Compac ness ......................................
6.2 Quan i a i e linea iza ion .......................................
6.3 P oo o he igidi y ..........................................
7 Almos igidi y o Aop ..........................................
7.1 Beha io a concen a ion poin s ...................................
7.2 Con inui y o Aop unde mGH-con e gence .............................
7.3 P oo o he almos - igidi y ......................................
8 Applica ion: The Yamabe equa ion on RCD(K,N)spaces ........................
Capaci y and quasi con inuous unc ions .................................
8.1 Exis ence o solu ions o he Yamabe equa ion on compac RCD spaces ..............
8.2 Con inui y o λSunde mGH-con e gence ..............................
Re e ences ...................................................
1 In oduc ion
The s anda d Sobole inequali y in sha p o m eads as
uLp∗(Rn)≤Eucl(n,p)∇uLp(Rn),∀u∈W1,p(Rn), (1.1)
whe e p∈(1,n),p∗:= pn
n−pis he Sobole conjuga e exponen and Eucl(n,p)is he
smalles posi i e cons an o which he inequali y (1.1) is alid. I s p ecise alue (see (2.2)
below) was compu ed independen ly by Aubin [20] and Talen i [94](seealso[41]).
In he se ing o compac Riemannian mani olds, he p esence o cons an unc ions in
he Sobole space immedia ely shows ha an inequali y o he kind o (1.1) mus ail. Ye ,
Sobole embeddings a e ce ainly alid also in his con ex and hey can be exp essed by
calling in o play he ull Sobole no m:
up
Lp∗(M)≤A∇up
Lp(M)+Bup
Lp(M),∀u∈W1,p(M), ()
whe e Misa compac n-dimensional Riemannian mani old and A,B>0.F om he p esence
o he wo pa ame e s A,B, i is no s aigh o wa d which is he no ion o bes cons an s
in his case. The issue o de ining and de e mining he bes cons an s in () has been he
cen al ole o he celeb a ed AB-p og am, we e e o [59] o a ho ough p esen a ion o
123
Rigidi y and almos igidi y o Sobole … Page 3 o 65 180
his opic (see also [46]). The s a ing poin o his p og am is he de ini ion o he ollowing
wo di e en no ions o “bes Sobole cons an s”:
αp(M):= in {A:() holds o some B},β
p(M):= in {B:() holds o some A}.
Then he i s na u al p oblem is o de e mine he alue o αp(M)and βp(M).I is a he
easy o see ha
βp(M)=Vol(M)p/p∗−1,
indeed cons an unc ions gi e au oma ically βp(M)≥Vol(M)p/p∗−1, while he o he
inequali y ollows om he Sobole –Poinca é inequali y (see, e.g. [59,Sec . 4.1]). I is ins ead
mo e sub le o de e mine whe he βp(M)is a ained, in he sense ha he in imum in i s
de ini ion is ac ually a minimum. This is ue o p=2 and due o Bak y [23](seealso
P oposi ion 5.1), bu ac ually alse o p>2 (see e.g. [59,P op. 4.1]).
Conce ning ins ead he alue o αp(M), i u ns ou o be ela ed o he sha p cons an in
he Euclidean Sobole inequali y (1.1). Mo e p ecisely Aubin in [20](seealso[59]) showed
ha on any compac n-dimensional Riemannian mani old Mwi h n≥2, we ha e
αp(M)=Eucl(n,p)p∀p∈(1,n). (1.2)
We poin ou ha i is a ha d ask o show ha αp(M)is a ained, namely ha he e exis s
some B>0 o which () holds wi h A=αp(M)and B. This has been e i ied o p=2
in [60], answe ing a i ma i ely o a conjec u e o Aubin.
On he o he hand, knowing he alue o βp(M)(and ha i is a ained o p=2),wecan
de ine a u he no ion o op imal-cons an A, “ ela i e” o B=β2(M).Mo ep eciselywe
de ine
Aop
2∗(M):= Vol(M)1−2/2∗·in {A:() o p=2 holds wi h Aand B=Vol(M)2/2∗−1}.
Fo he sake o gene ali y we will ac ually conside Aop also in he so-called subc i ical
case, meaning ha we enla ge he class o Sobole inequali ies and conside o e e y q∈
(2,2∗]
u2
Lq(M)≤A∇u2
L2(M)+Vol(M)2/q−1u2
L2(M),∀u∈W1,2(M), ()
o some cons an A≥0. Then we de ine
Aop
q(M):= Vol(M)1−2/q·in {A:() holds}.
No e ha he in imum abo e is always a minimum and ha Vol(M)2/q−1is he “minimal B”
ha we can ake in ().
Rema k 1.1 We b ing o he a en ion o he eade he eno maliza ion ac o Vol(M)1−2/q
in he de ini ion o Aop
q(M). This is usually no p esen in he li e a u e conce ning he AB-
p og am (see e.g. [59]), howe e his choice will allow us o ha e cleane inequali ies. This
also makes Aop
qin a ian unde escalings o he olume measu e o M.
One o he main ques ions ha we will in es iga e in his no e conce ns he alue o
Aop
q(M). So a Aop
q(M)is known explici ly only in he case o Snand was i s ly compu ed
by Aubin in [19] in he case o q=2∗and by Beckne in [27] o a gene al q:
Aop
q(Sn)=q−2
n,∀n≥3.(1.3)
123
180 Page 4 o 65 F. Nobili, I. Y. Violo
Aubin also exhibi ed a amily o non-cons an unc ions ha achie e equali y in () wi h
A=Aop
2∗(Sn). Fo a gene al mani old Mins ead i can be p o ed ha
Aop
q(M)≤C(K,D,N), (1.4)
whe e K∈Ris a lowe bound on he Ricci cu a u e o M,Nis an uppe bound on he
dimension and D∈R+an uppe bound on i s diame e . This ollows om he Sobole –
Poinca éinequali y combined wi haninequali y by Bak y(seee.g. [46,Theo em 4.4] andalso
Sec . 5.1). On he o he hand, o posi i e Ricci cu a u e we ha e he ollowing celeb a ed
compa ison esul o iginally p o en in [66](seealso[24,76] o he case o a gene al q):
Theo em 1.2 Le M be an n-dimensional Riemannian mani old, n ≥3, wi h Ric ≥n−1.
Then, o e e y q ∈(2,2∗], i holds
Aop
q(M)≤Aop
q(Sn). (1.5)
One o he main consequence o he esul s in his no e is he cha ac e iza ion o he equali y
in (1.5), in pa icula we show:
Theo em 1.3 Equali y in (1.5)holds o some q ∈(2,2∗]i and only i M is isome ic o Sn.
I is impo an o poin ou ha he no el y o he abo e esul is ha i co e s he case q=2∗.
Indeed, o q<2∗, Theo em 1.3 was al eady es ablished (see e.g. [24,Rema k 6.8.5]) and
ollows om an imp o emen (only o q<2∗)o (1.5) due o [50] in ol ing he spec al
gap (see Rema k 6.9 o mo e de ails). On he o he hand, up o ou knowledge, his is he
i s ime ha i appea s in he c i ical case q=2∗.
I is also wo h o compa e Theo em 1.3 wi h he igidi y esul in [75] o he Sobole
inequali y on mani olds wi h non-nega i e Ricci cu a u e (and la e imp o ed in [97], see
also [26]). In [75]i isp o ed ha i (1.1) is alid on a non-compac mani old wi h non-
nega i e Ricci cu a u e, hen he mani old mus be he Euclidean space. He e ins ead we
conside compac mani olds and he igidi y is ob ained in compa ison wi h he Sobole
inequali y on he sphe e. Fo his eason, ou a gumen s will also be subs an ially di e en
om he ones in [75,97]. Ne e heless, we will also deal wi h he o me ypes o igidi y in
Co olla y 1.14 below.
Theo em 1.3 will be p o ed in he con ex o me ic measu e spaces wi h syn he ic Ricci
cu a u e bounds. One o he main easons o app oach he p oblem in his mo e gene al
se ing is ha i will allow us o cha ac e ize also he “almos -equali y” in (1.5) (see Theo-
em 1.10 below). Indeed, as we will see, in his case we need o compa e he mani old M o
a class o singula spaces, a he han o he ound sphe e.
1.1 Bes cons an in he Sobole inequali y on compac CD spaces
The no ion o me ic measu e spaces wi h syn he ic Ricci cu a u e bounds o igina ed in he
independen seminal wo ks o [92,93]and[80], whe e he celeb a ed cu a u e-dimension
condi ion CD(K,N)was in oduced. He e K∈Ris a lowe bound o he Ricci cu a u e
and N∈[1,∞] is an uppe bound on he dimension. The de ini ion is gi en ia op imal
anspo , by equi ing some con exi y p ope ies o en opy unc ionals (see De ini ion 2.5
below).
The p oo o he igidi y (and almos igidi y) o Aop
qin he case q=2∗, will o ce us o
s udy also he alue o αpin he con ex o CD-spaces. The connec ion o his wi h he p oo
123

Rigidi y and almos igidi y o Sobole … Page 5 o 65 180
o Theo em 1.9 will be explained owa ds he end o Sec . 1.4, whe e we p o ide a ske ch o
he p oo yielding he main igidi y heo em.
Le hen (X,d,m)be a CD(K,N)space wi h N∈(1,∞).Fo anyp∈(1,N)se
p∗:= Np
N−pand, in he same ashion o (), we conside :
up
Lp∗(m)≤A|Du|p
Lp(m)+Bup
Lp(m),∀u∈W1,p(X). (1.6)
We a e hen in e es ed in he minimal A o which (1.6) holds. In o he wo ds we se (wi h
he usual con en ion ha he in is ∞when no Aexis s):
αp(X):= in {A:(1.6)holds o some B}.(1.7)
We will be able o compu e he alue o αp(X) o e e y compac CD(K,N)space X,
ex ending he esul o Aubin o Riemannian mani olds (see (1.2) abo e). Be o e passing
o he ac ual s a emen , i is use ul o explain i s he in ui ion behind i and he geome ical
meaning o he cons an αp(X). The ough idea is ha i s alue is igh ly linked o he local
s uc u e o he space. Indeed, he key obse a ion is ha αp(X)is in a ian unde escaling
o he o m (X,d/ ,m/ N). Fo example, since mani olds a e locally Euclidean, i is no
su p ising ha in (1.2) he op imal Euclidean–Sobole cons an appea s. On he o he hand,
CD(K,N)spaces ha e a mo e singula local beha io and addi ional pa ame e s mus be
aken in o accoun . In pa icula he alue o αp(X) u ns ou o be ela ed o he Bishop–
G omo densi y:
(0,+∞]  θN(x):= lim
→0+
m(B (x))
ωN N,x∈X,
whe e ωNis he olume o he Euclidean uni ball (see (2.1) o non in ege N). Ou esul
is hen he ollowing:
Theo em 1.4 Le (X,d,m)be a compac CD(K,N)space o some K ∈Rand N ∈(1,∞).
Then o e e y p ∈(1,N)
αp(X)=Eucl(N,p)
minx∈XθN(x)1
Np
.(1.8)
We poin ou ha , since X is compac , minx∈XθN(x)always exis s because θNis lowe
semicon inuous (see Sec . 2.3.1).
Rema k 1.5 No e ha i X is a n-dimensional Riemannian mani old, θn(x)=1 o e e y
x∈X, hence in his case (1.8) (wi h N=n) is exac ly Aubin’s esul in (1.2). Recall also
ha he e Nneeds no o be an in ege and hus Eucl(N,p)has o be de ined o a bi a y
N∈(1,∞)(see (2.2)). 
Rema k 1.6 We a e no assuming (X,d,m) o be eno malized. In pa icula obse e ha i
we escale he e e ence measu e mas c·m, henαpge s mul iplied by c−p/N, which is in
acco dance wi h he scaling in (1.8). 
Rema k 1.7 Theo em 1.4 gi es non- i ial in o ma ion e en in he “collapsed” case, i.e.
when θN=+∞in a se o posi i e (o e en ull) measu e. Indeed, o ha e αp(X)>0i is
su icien ha θN(x)<+∞a a single poin x∈X. As an example, conside he model space
([0,π],|.|,sinN−1L1)which is CD(N−1,N)wi h θN(x)<+∞ only o x∈{0,π}.
123
180 Page 6 o 65 F. Nobili, I. Y. Violo
Theo em 1.4 will be p o ed in wo s eps, by he combina ion o an uppe bound (The-
o em 3.13), ob ained ia local Sobole inequali ies (Theo em 3.8), and a lowe bound
(Theo em 4.4) de i ed wi h a blow-up analysis.
We end his pa wi h a ques ion ha na u ally a ises om he alidi y o Theo em 1.4:
Ques ion: Le (X,d,m)be a compac CD(K,N)(o RCD(N,K)) space wi h N∈(1,∞)
and suppose ha α2∗(X)∈(0,∞). Is he e a cons an B<+∞ such ha
up
L2∗(m)≤α2(X)|Du|2
L2(m)+Bu2
L2(m),∀u∈W1,2(X)? (1.9)
This has posi i e answe in he smoo h se ing [60]. Howe e in [59,P oposi ion 5.1] i is
shown ha on a Riemannian mani old Mo dimension n≥4, he scala cu a u e o Mis
bounded abo e by cnB, o a dimensional cons an cn>0. This poin s o a nega i e answe ,
since we a e assuming only a Ricci lowe bound on he space, howe e i is no clea o us
how o p o e o disp o e (1.9).
1.2 Main igidi y and almos igidi y esul s in compac RCD spaces
E en i some o ou esul s will hold o he gene al class o CD(K,N)spaces, ou main ocus
will be he smalle class o spaces sa is ying he Riemannian cu a u e-dimension condi ion
RCD(K,N), which adds o he CD class he linea i y o he hea low (see De ini ion 2.7
below). This no ion appea ed i s in he in ini e dimensional case (N=∞)in[11](seealso
[9] in he case o σ- ini e e e ence measu e) while, in he ini e dimensional case (N<∞), i
was in oduce in [51]. We also men ion he sligh ly weake RCD∗(K,N)condi ion (coming
om he educed cu a u e-dimension condi ion CD∗(K,N)in oduced in [22]) which has
been p o ed in [15,47] o be equi alen o he alidi y o a weak N-dimensional Bochne -
inequali y (see also [12] o he same esul in he in ini e dimensional case). We ecall ha
in he compac case (o mo e gene ally o ini e e e ence measu e) which will be he main
se ing o his no e, he RCD∗(K,N)and he RCD(K,N)condi ions u n ou o be pe ec ly
equi alen a e he wo k in [35]. The main ad an age o us o wo k in he RCD class, as
opposed o he mo e gene al CD class, is ha i enjoys igidi y and s abili y p ope ies ha
a e analogous o he Riemannian mani olds se ing.
To s a e ou main esul s o me ic measu e spaces we need o de ine i s he no ion
o op imal cons an in he Sobole inequali y in he non-smoo h se ing. Gi en a (compac )
RCD(K,N)space (o mo e gene ally a CD(K,N)space) (X,d,m), o someK∈R,
N∈(2,∞),wese 2
∗:= 2N/(N−2)and conside he analogous o ():
u2
Lq(m)≤A|Du|2
L2(m)+m(X)2/q−1u2
L2(m),∀u∈W1,2(X), (1.10)
o q∈(2,2∗]and a cons an A≥0. Then we de ine
Aop
q(X):= m(X)1−2/q·in {A:(1.10)holds},
wi h he con en ion ha Aop
q(X)=∞when no Aexis s. No e ha Aop
q(X), when is ini e,
is ac ually a minimum. Obse e also ha , as in he smoo h case, he e is a eno maliza ion
ac o m(X)1−2/qin he de ini ion. Howe e , being no es ic i e, we will mainly wo k
asking m(X)=1 so ha he alue o Aop
q(X)is equi alen o he non- eno malized one.
Rema kably in his mo e gene al amewo k, a compa ison analogous o (1.5) holds.
123
Rigidi y and almos igidi y o Sobole … Page 7 o 65 180
Theo em 1.8 ([36]) Le (X,d,m)be an essen ially non-b anching CD(N−1,N)space,
N∈(2,∞). Then, o e e y q ∈(2,2∗]
Aop
q(X)≤q−2
N.(1.11)
The essen ially nonb anching condi ion is a echnical p ope y o mass anspo a ion ha ,
oughly said, equi es a sui able nonb anching p ope y o anspo a ion geodesics. I was
in oduced in [88] whe e i was shown ha i is sa is ied in he RCD(K,N)-class. We also
men ion ha Theo em 1.8 in he RCD case was p e iously ob ained in [86]. Obse e also ha ,
whene e Nis an in ege and hanks o (1.3), o a N-dimensional Riemannian mani olds
(1.11) is exac ly (1.5) and in pa icula Theo em 1.8 gene alizes Theo em 1.2.
We can now s a e ou main igidi y esul in he se ing o me ic measu e spaces.
Theo em 1.9 (Rigidi y o Aop
q)Le (X,d,m)be an RCD(N−1,N)space o some N ∈
(2,∞)and le q ∈(2,2∗]. Then, equali y holds in (1.11)i and only i (X,d,m)is isomo phic
o a sphe ical suspension, i.e. he e exis s an RCD(N−2,N−1)space (Z,dZ,mZ)such
ha (X,d,m)≃[0,π]×N−1
sin Z.
Di e en ly om he smoo h case, in he mo e abs ac se ing o RCD spaces he abo e
esul is ins ead new o all q. As an icipa ed abo e, we can also p o e an “almos - igidi y”
s a emen linked o he almos -equali y case in (1.11) (see Sec . 2.3.3 o he no ion o
measu e-G omo –Hausdo con e gence and dis ance dmGH.).
Theo em 1.10 (Almos - igidi y o Aop
q)Fo e e y N ∈(2,∞),q ∈(2,2∗]and e e y
ε>0, he e exis s δ:= δ(N,ε,q)>0such ha he ollowing holds. Le (X,d,m)be an
RCD(N−1,N)space wi h m(X)=1and suppose ha
Aop
q(X)≥(q−2)
N−δ,
Then, he e exis s a sphe ical suspension (Y,dY,mY)(i.e. he e exis s an RCD(N−2,N−1)
space (Z,dZ,mZ)so ha Yis isomo phic as a me ic measu e space o [0,π]×N−1
sin Z)such
ha
dmGH((X,d,m), (Y,dY,mY)) < ε.
Rema k 1.11 We b ie ly poin ou wo impo an ac s conce ning he wo abo e s a emen s.
(i) In he smoo h se ing, o q<2∗, he almos igidi y ollows “di ec ly” om he sha pe
e sion o (1.5) ci ed abo e (see Rema k 6.9 o he explici s a emen ) and using he
almos - igidi y o he 2-spec al gap [36,38]. Ne e heless, we a e no awa e o any such
s a emen in he li e a u e and anyhow, ou p oo does no ely on any imp o ed e sion
o (1.5).
(ii) The key ea u e o Theo ems 1.9 and 1.10 is ha hey include he “c i ical” exponen .
Indeed, he di e ence be ween he “subc i ical” case q<2∗and q=2∗is no only
echnical bu a majo issue linked o he lack o compac ness in he Sobole embedding.
As i will be clea in he sequel, he p oo o he c i ical case equi es se e al addi ional
a gumen s ha cons i u e he hea o his no e. 
The almos - igidi y esul con ained in Theo em 1.10 will be ac ually a consequence
o a s onge s a emen , ha is he con inui y o Aop
qunde measu e G omo –Hausdo
con e gence. Mo e p ecisely we will p o e he ollowing:
123
180 Page 8 o 65 F. Nobili, I. Y. Violo
Theo em 1.12 (Con inui yo Aop
qunde mGH-con e gence)Le (Xn,dn,mn),n ∈N∪{∞},
be a sequence o compac RCD(K,N)-spaces wi h mn(Xn)=1and o some K ∈R,
N∈(2,∞)so ha Xn
mGH
→X∞. Then, Aop
q(X∞)=limnAop
q(Xn), o e e y q ∈(2,2∗].
1.3 Addi ional esul s and applica ion o he Yamabe equa ion
Euclidean- ype Pólya–Szeg˝o inequali y on CD(K,N)spaces. Wewillde elopaPólya–Szeg˝o
inequali y (see Sec . 3.1), which is oughly a Euclidean- a ian o he Pólya–Szeg˝o inequali y
o CD(K,N)spaces, K>0,de i edin[84]. The main ea u e o his inequali y is ha i
holds on a bi a y CD(K,N)spaces, K∈R, bu assumes he alidi y o an isope ime ic
inequali y o he ype
Pe (E)≥CIsopm(E)N−1
N,∀E⊂Bo el,
o some ⊂X open and whe e CIsop is a posi i e cons an independen o E. Fo ou
pu poses his Pólya–Szeg˝o inequali y will be used o de i e local Sobole inequali ies o
Euclidean- ype (see Theo em 3.8), howe e i allows us o ob ain also sha p Sobole inequal-
i ies unde Euclidean- olume g ow h assump ion.
Sha p and igid Sobole inequali ies unde Euclidean- olume g ow h. As a by-p oduc o ou
analysis, we achie e sha p Sobole inequali ies on CD(0,N)spaces wi h Euclidean- olume
g ow h. We ecall ha a CD(0,N)space (X,d,m)has Euclidean- olume g ow h i
AV R(X):= lim
R→+∞
m(BR(x0))
ωNRN>0,
o some (and hus any) x0∈X. We will p o e he ollowing.
Theo em 1.13 Le (X,d,m)be a CD(0,N)space o some N ∈(1,∞)and wi h Euclidean
olume g ow h. Then, o e e y p ∈(1,N), i holds
uLp∗(m)≤Eucl(N,p)AV R(X)−1
N|Du|Lp(m),∀u∈LIPc(X). (1.12)
Mo eo e (1.12)is sha p.
Thisex endsa esul ecen ly de i edin[26]in he case o Riemannianmani oldsand answe s
posi i ely o a ques ion posed in [26,Sec. 5.2].
Combining Theo em 1.13 wi h he olume igidi y o non-collapsed RCD spaces in
[53] and he esul s in [40,Appendix A] (see also [68,Theo em 3.5]) we immedia ely ge
he ollowing opological igidi y which ex ends o he non-smoo h se ing he esul s o
Riemannian mani olds in [75,97]. Recall ha an RCD(K,N)space (X,d,m)is said o
be non-collapsed (see De ini ion 2.13)i m=HN, heN-dimensional Hausdo measu e
( his no ion has been in oduced in [53], see also [72] and inspi ed by [40]).
Co olla y 1.14 (Topological- igidi y o Sobole embeddings) Fo e e y N ∈N,p∈(1,N)
and ε>0 he e exis s δ>0such ha he ollowing holds. Le (X,d,HN)be an RCD(0,N)
space wi h Euclidean olume g ow h and such ha
uLp∗(m)≤(Eucl(N,p)+δ)|Du|Lp(m),∀u∈LIPc(X). (1.13)
Then Xis homeomo phic o RNand dGH(B (x), B (0N)) ≤ε o e e y x ∈Xand >0.
123
Rigidi y and almos igidi y o Sobole … Page 15 o 65 180
Fo a Bo el se E⊂X o ini e measu e we also de ine i s Minkowski con en as:
m+(E)=lim
δ→0+
m(Eδ)−m(E)
δ,
whe e Eδ:= {x∈X:d(x,E)<δ}. In gene al we only ha e Pe (E)≤m+(E).
We ecall ha he ollowing coa ea o mula is alid a e [82,P oposi ion 4.2].
Theo em 2.4 (Coa ea o mula) Le (X,d,m)be a locally compac me ic measu e space and
∈BV(X). Then he se { > }is o ini e pe ime e o a.e. ∈Rand gi en any Bo el
unc ion g :X→[0,∞), i holds ha
ˆ{s≤u< }
gd|D |=ˆ
sˆgdPe ({ > },·)d ,∀s, ∈[0,∞), s< .(2.8)
2.3 CD(K,N)and RCD(K,N)spaces
2.3.1 Main de ini ions and p ope ies
In his no e, as an icipa ed in he in oduc ion, we will wo k in he gene al amewo k o
me ic measu e spaces (X,d,m)sa is ying syn he ic Ricci cu a u e lowe bounds. Fo
comple eness, we b ie ly ecall he de ini ion and he key p ope ies ha we will need.
The i s no ion o syn he ic Ricci lowe bounds was gi en independen ly in he seminal
pape s [80]and[92,93] whe e he au ho s in oduced he celeb a ed cu a u e dimension
condi ion. We epo he e i s de ini ion only in ini e dimension N∈[1,∞), gi en in e m
o con exi y p ope ies o he N-Rényi-en opy unc ional UN:P2(X)→[−∞,0]de ined
by
UN(μ|m):= −ˆρ1−1
Ndm,i μ=ρm+μs,
whe e μ∈P2(X)and μsis singula wi h espec o m. In his no e, since op imal anspo a-
ion plays a mino ole, we shall assume he eade o be amilia wi h Op imal T anspo and
he Wasse s ein Space (P2(X), W2)and we e e o [96] o a sys ema ic discussion (see
also [8]).
We s a ecalling he de ini ion o dis o ion coe icien s. Fo e e y K∈R,N∈
[0,∞), ∈[0,1]se
σ( )
K,N(θ) := ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
+∞,i Kθ2≥Nπ2,
sin( θ√K/N)
sin(θ√K/N),i 0 <Kθ2<Nπ2,
,i Kθ2<0andN=0o i Kθ2=0,
sinh( θ√−K/N)
sinh(θ√−K/N),i Kθ2≤0andN>0.
Se also, o N>1, τ( )
K,N(θ) := 1
Nσ( )
K,N−1(θ)1−1
Nwhile τ( )
K,1(θ) = i K≤0and
τ( )
K,1(θ) =∞i K>0.
De ini ion 2.5 (CD(K,N)-spaces) Le K∈Rand N∈[1,∞). A me ic measu e space
(X,d,m)sa is ies he cu a u e dimension condi ion CD(K,N)i , o e e y μ0,μ
1∈
P2(X)absolu ely con inuous wi h bounded suppo s, he e exis s a dynamical op imal ans-
e ence plan π∈P(Geo(X)) be ween μ0,μ
1so ha : o e e y ∈[0,1]and N≥N,we
123

180 Page 16 o 65 F. Nobili, I. Y. Violo
ha e μ := (e )π=ρ mand
UN(μ |m)≤−ˆτ(1− )
K,N(d(γ1,γ
0))ρ0(γ0)−1
N+τ( )
K,N(d(γ1,γ
0))ρ1(γ1))−1
Ndπ(γ).
(2.9)
We ecall he also he no ion o one-dimensional model space o he CD(N−1,N)
condi ion:
De ini ion 2.6 (One dimensional model space) Fo e e y N>1wede ineIN:=
([0,π],|.|,mN),whe e|.|is he Euclidean dis ance es ic ed on [0,π]and
mN:= 1
cNsinN−1L1|[0,π],
wi h cN:= ´[0,π]sin( )N−1d .
Toencode amo e“Riemannian”beha io o hespace,and o ule ou Finsle spaceswhich
a e allowed by he CD condi ion, i was in oduced in [11] he so-called RCD condi ion in
he in ini e dimensional case (see also [55] o he case o σ- ini e e e ence measu e). In his
no e howe e we will only wo k in ini e dimensional RCD-spaces in oduced in [51].
De ini ion 2.7 (RCD(K,N)-spaces) Le K∈Rand N∈[1,∞). A me ic measu e space
(X,d,m)is an RCD(K,N)-space, p o ided i is an in ini esimal Hilbe ian CD(K,N)-
space.
Rema k 2.8 Spaces sa is ying he CD(K,N)(and hus also he RCD(K,N)) condi ion, sup-
po a (1,1)-local Poinca é inequali y (see [87]) and by he Bishop–G omo inequali y below
hey a e locally-doubling, he e o e om he esul s in [39] we know ha he minimal weak
uppe g adien is independen on he exponen p(see also [54]). Fo his eason, o ligh en
he no a ion, in his se ing we will simply w i e |D | o ∈W1,p(X)and call i simply
minimal weak uppe g adien o .
We s a by ecalling some use ul p ope ies abou hese spaces ha a e going o be used
in he sequel.
On CD(K,N)spaces he Bishop–G omo inequali y holds (see [93]):
m(BR(x))
K,N(R)≤m(B (x))
K,N( ), o any 0 < <R≤πN−1
K+and any x∈X,(2.10)
whe e he quan i ies K,N( ),N∈[1,∞)K∈Ra e de ined as
K,N( ):= σN−1ˆ
0|sK,N( )|N−1d ,
and sK,N( )is de ined as sin  K
N−1,i K>0, sinh  |K|
N−1,i K<0and i K=0.
In pa icula CD(K,N)spaces a e uni o mly locally doubling and hus p ope , i.e. closed
and bounded se s a e also compac . We also no e ha in he case K=0 his implies ha he
limi
AV R(X):= lim
→+∞
m(B (x))
ωN N
123
Rigidi y and almos igidi y o Sobole … Page 17 o 65 180
exis s ini e and does no dependon he poin x∈X.Wecall hequan i y AV R(X)asymp o ic
olume a io o X and i AV R(X)>0 we say ha X has Euclidean- olume g ow h.Akey
ole in he no e will be played by he ollowing quan i ies:
θN, (x):= m(B (x))
ωN N,θ
N(x):= lim
→0+θN, (x), ∀ >0,x∈X.
Obse e ha he abo e limi exis s hanks o he Bishop–G omo inequali y and he ac ha
lim →0+ωN N
K,N( )=1 o e e yK∈R,N∈[1,∞), which in pa icula g an s ha
θN(x)=lim
→0
m(B (x))
K,N( )=sup
>0
m(B (x))
K,N( ).(2.11)
This and he ac ha m(∂ B (x)) =0 o e e y >0andx∈X (which ollows om he
Bishop–G omo inequali y), implies ha θN(x)is a lowe -semicon inuous unc ion o x.
The e o e, when X is compac , he e exis s minx∈XθN(x).
Nex we ecall he B unn–Minkowski inequali y.
Theo em 2.9 ([93]) Le (X,d,m)be a CD(K,N)space wi h N ∈[1,∞),K∈R. Fo any
couple o Bo el se s A0,A1⊂Xi holds ha
m(A )1
N≥σ(1− )
K,N(θ)m(A0)1
N+σ( )
K,N(θ)m(A1)1
N,∀ ∈[0,1],(2.12)
whe e A := {γ :γgeodesic such ha γ0∈A0,γ
1∈A1}and
θ:= in (x0,x1)∈A0×A1d(x0,x1), i K ≥0,
sup(x0,x1)∈A0×A1d(x0,x1), i K <0,
We ema k ha (2.12) is ac ually weake han he s a emen appea ing in [93] and i holds
o he (a p io i) la ge class o CD∗(K,N)spaces (see [22]).
We epo he Bonne –Mye s diame e -compa ison heo em o CD-spaces om [93]:
(X,d,m)is a CD(K,N)space, o some K>0⇒diam(X)≤πN−1
K,(2.13)
The Lichne owi z 2-spec al gap inequali y is alid also in he CD-se ing. To s a e i we
ecall he no ion o i s non- i ial Neumann eigen alue o he Laplacian (o 2-spec al gap)
in me ic measu e spaces.
De ini ion 2.10 Le (X,d,m)be a me ic measu e space wi h ini e measu e. We de ine he
i s non i ial 2-eigen alue λ1,2(X)as he non-nega i e numbe gi en by
λ1,2(X):= in ´|D |2
2dm
´| |2dm: ∈LIP(X)∩L2(m), = 0,ˆ dm=0.(2.14)
Clea ly, in ligh o [10], in he abo e de ini ion one can equi alen ly ake he in imum among
all ∈W1,2(X). In he sequel will use his ac wi hou u he no ice.
Then he spec al-gap inequali y as p o en in [80](seealso[67]) says ha :
λ1,2(X)≥N, o e e y CD(N−1,N)-space X,
wi h N anging in (1,∞).
We conclude his pa ecalling some igidi y and s abili y s a emen s o RCD(K,N)
spaces and o his goal we need o de ine he no ion o sphe ical suspension o e a me ic
measu e space. Fo any N∈[1,∞) he N-sphe ical suspension o e a me ic measu e space
123
180 Page 18 o 65 F. Nobili, I. Y. Violo
(Z,mZ,dZ)is de ined o be he space ([0,π]×N
sin Z):= Z×[0,π]/(Z×{0,π})endowed
wi h he ollowing dis ance and measu e
d(( ,z), (s,z)) := cos−1cos(s)cos( )+sin(s)sin( )cos dZ(z,z)∧π,
m:= sinN−1( )d ⊗mZ.
I u ns ou ha he RCD condi ion is s able unde he ac ion o aking sphe ical suspensions,
mo e p ecisely i has been p o en in [71] ha
[0,π]×N
sin Z,N≥2 is a RCD(N−1,N)space i and only i
diam(Z)≤πand Z is an RCD(N−2,N−1)space, (2.15)
We can now ecall he wo main igidi y s a emen s ha we will use in he no e: he
maximal diame e heo em and he Oba a heo em o RCD(K,N)spaces:
Theo em 2.11 ([70]) Le (X,d,m)be an RCD(N−1,N)space wi h and N ∈[2,∞)
and suppose ha diam(X)=π.Then(X,d,m)is isomo phic o a sphe ical suspension,
i.e. he e exis s an RCD(N−2,N−1)space (Z,dZ,mZ)wi h diam(Z)≤πsa is ying
X≃[0,π]×N
sin Z.
Theo em 2.12 ([71]) Le (X,d,m)be an RCD(N−1,N)space wi h and N ∈[2,∞)
and suppose ha λ1,2(X)=N. Then (X,d,m)is isomo phic o a sphe ical suspension,
i.e. he e exis s an RCD(N−2,N−1)space (Z,dZ,mZ)wi h diam(Z)≤πsa is ying
X≃[0,π]×N
sin Z.
Weend his pa by ecalling he de ini iono “non-collapsed” RCD-spaces, which ex ends
he no ion o non-collapsed Ricci-limi s in oduced in [40].
De ini ion 2.13 ([53])We say ha (X,d,m)is a non-collapsed RCD(K,N)space, o some
K∈R,N∈N, p o ided i is RCD(K,N)and m=HN,whe eHNis he N-dimensional
Hausdo measu e.
This class o spaces enjoys ex a egula i y wi h espec o he gene al RCD-class and a e a
sui able se ing o de i e he opological igidi y esul s o his no e. He e we jus men ion
ha i θNis ini e m-a.e. (o equi alen ly i mHN), hen up o a cons an mul iplica i e
ac o , mequals HNand he space is non-collapsed. This has been p o ed i s in [62] o
compac spaces and hen in [31] in he gene al case sol ing a conjec u e o [53](seealso
[65] o an accoun on u he conjec u es a ound his opic).
2.3.2 Sobole –Poinca é inequali y on CD(K,N)spaces
A well-es ablished ac which goes back o he seminal wo k [58], is ha a (1,p)-Poinca é
inequali y on a doubling me ic measu e space, imp o es o a (q,p)-Poinca é inequali y wi h
q>1.On CD(K,N)spaces his ansla es in he ollowing esul .
Theo em 2.14 ((p∗,p)-Poinca é inequali y) Le (X,d,m)be a CD(K,N)space o some
N∈(1,∞),K ∈R.Fixalsop∈(1,N)and 0>0. Then, o e e y B (x)⊂Xwi h
≤ 0i holds
 B (x)|u−uB (x)|p∗dm1
p∗≤C(K,N,p, 0)  B2 (x)|Du|pdm1
p,∀u∈LIP(X),
(2.16)
whe e p∗:= pN/(N−p)and uB (x):= ´B (x)udm.
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Rigidi y and almos igidi y o Sobole … Page 19 o 65 180
P oo F om [87] we ha e ha X suppo s a s ong (1,1)-Poinca é inequali y, in pa icula i
also suppo s a s ong (1,p)-Poinca é inequali y o e e y p∈[1,∞), by Hölde inequali y.
Mo eo e , o e e y x0∈X, ≤ 0and x∈B 0(x0), om he Bishop–G omo inequali y
(2.10) i holds ha
m(B (x))
m(B 0(x0)) ≥C(K,N, 0)
0N
.
Then (2.16) ollows om [58,Theo em 5.1] (see also [29,Theo em 4.21]). 
We end his pa ecalling he sha p Sobole -inequali y on he Nmodel space IN(see
De . 2.6) o N∈(2,∞)(see e.g. [76]):
u2
Lq(mN)≤q−2
N|Du|2
L2(mN)+u2
L2(mN),∀u∈W1,2([0,π],|.|,mN), (2.17)
o e e y q∈(2,2∗], wi h 2∗=2N/(N−2).
2.3.3 Con e gence and compac ness unde mGH-con e gence
We ecall he e he no ion o poin ed-measu e G omo Hausdo con e gence (pmGH con-
e gence o sho ). Le us say ha he de ini ion we will adop is no he classical one (see
e.g. [34,57]), bu i is equi alen in he case o a sequence o uni o mly locally doubling
me ic measu e spaces, hanks o he esul s in [55]. I will be con enien o conside in his
sec ion he se ¯
N:= N∪{∞}. Recall also ha a poin ed me ic measu e space is a quad uple
(X,d,m,x)consis ing o a me ic measu e space (X,d,m)and a poin x∈X.
De ini ion 2.15 (Poin ed measu e G omo –Hausdo con e gence) Wesay ha he sequence
(Xn,dn,mn,xn),n∈N, o poin ed me ic measu e spaces, poin ed measu e G omo –
Hausdo -con e ges (pmGH-con e ges in sho ) o (X∞,d∞,m∞,x∞),i he eexis
isome ic embeddings ιn:Xn→(Z,dZ),n∈¯
N, in o a common me ic space (Z,dZ)
such ha
(ιn)mn(ι∞)m∞in duali y wi h Cbs(Z)and ιn(xn)→ι∞(x∞).
In he case o a sequence o uni o mly locally doubling spaces (as in he case o CD(K,N)-
spaces o ixed K∈R,N<∞) we can also ake (Z,dZ) o be p ope . Mo eo e , again o
a class o uni o mly locally doubling spaces, in [55] i is p o en ha he pmGH-con e gence
is me izable wi h a dis ance which we call dpmGH .
I will be also con enien o adop , hanks o De ini ion 2.15, he so-called ex insic
app oach, whe e he spaces Xna e iden i ied as subse s o a common p ope me ic space
(Z,dZ),Xn⊂Z, supp(mn)=Xn,dZ|Xn×Xn=dn o all n∈¯
N,anddZ(xn,x∞)→0,
mnm∞in duali y wi h Cbs(Z). Any such space (Z,dZ)( oge he wi h an he iden i ica ion
o Xn⊂Z) is called ealiza ion o he con e gence and (in he case o geodesic uni o mly
locally doubling spaces) can be aken so ha dZ
H(BXn
R(xn), BX∞
R(x∞)) →0 o e e yR>
0,whe e dZ
His he Hausdo dis ance in Z. To a oid con usion when dealing wi h his
iden i ica ion, we shall some imes w i e BXn
(x)wi h x∈Xn, >0, o deno e he se
BZ
(x)∩Xn.
A e hewo ksin [11,55,80,92,93] and hanks o he G omo ’s p ecompac ness heo em
[57] we ha e he ollowing p ecompac ness esul .
123
180 Page 20 o 65 F. Nobili, I. Y. Violo
Theo em 2.16 Le (Xn,dn,mn,xn)be a sequence o poin ed CD(Kn,Nn)( esp.
RCD(Kn,Nn)) spaces, n ∈¯
N, wi h m(B1(xn)) ∈[ −1, ], o >1and Kn→K∈
R,Nn→N∈[1,∞). Then, he e exis s a subsequence (nk)and a poin ed CD(K,N)
( esp. RCD(K,N)) space (X∞,d∞,m∞,x∞)sa is ying
lim
k→∞dpmGH (Xnk,dnk,mnk,xnk), (X∞,d∞,m∞,x∞)=0.
We will be equen ly conside he case o compac (wi h uni o mly bounded diame e )
me ic measu e spaces which is he na u al se ing o he Sobole embedding o his no e,
o which we can educe he abo e con e gence o he so-called measu e G omo Hausdo
con e gence, mGH-con e gence o sho , whe e we simply igno e he con e gence o he
base poin s. Also in his case, on e e y class o uni o mly doubling me ic measu e spaces
wi h uni o mly bounded diame e , he mGH-con e gence can be me ized by a dis ance ha
we deno e by dmGH.The ex insic app oach applies e ba im as well, wi h he excep ion
ha he common ambien space Z can be also aken o be compac .
We now ecall some s abili y and con e gence esul s o unc ions along pmGH-
con e gence. Fo addi ional de ails and analogous esul s we e e o [13,55,63]. Fo b e i y
easons in wha ollows we ix a sequence o poin ed CD(K,N)spaces (Xn,dn,mn,xn),
o n∈¯
N,so ha X
n
pmGH
→X∞.
De ini ion 2.17 Le p∈(1,∞), we say ha
(i) n∈Lp(mn)con e ges L p-weak o ∞∈Lp(m∞), p o ided supn∈N nLp(mn)<∞
and nmn ∞m∞in Cbs(Z),
(ii) n∈Lp(mn)con e ges L p-s ong o ∞∈Lp(m∞), p o ided i con e ges Lp-weak
and limn nLp(mn)≤ ∞Lp(m∞),
(iii) n∈W1,2(Xn)con e ges W1,2-weak o ∞∈W1,2(X)p o ided i con e ges L2-weak
and supn∈N|D n|L2(mn)<∞,
(i ) n∈W1,2(Xn)con e ges W1,2-s ong o ∞∈W1,2(X)p o ided i con e ges L2-
s ong and |D n|L2(mn)→|D ∞|L2(m∞).
Mo eo e , we say ha nis uni o mly bounded in Lpi supn nLp(mn)<∞.In he
ollowing s a emen we collec a lis o use ul p ope ies o Lp-con e gence.
P oposi ion 2.18 (P ope ies o Lp-con e gence) Fo all p ∈(1,∞), i holds
(i) I ncon e ges L p-s ong o ∞, henϕ( n)con e ges L p-s ong o ϕ( ∞) o e e y
ϕ∈LIP(R)wi h ϕ(0)=0,
(ii) I n( esp. gn)con e ges L p-s ong o ∞( esp. g∞), hen n+gncon e ges L p-s ong
o ∞+g∞,
(iii) i ncon e ges L p-weak o , hen  ∞Lp(m∞)≤limn nLp(mn),
(i ) suppose ha supn nLp(mn)<+∞, hen up o a subsequence ncon e ges L p-weak
o some ∞∈Lp(m∞),
( ) I ncon e ges L p-s ong ( esp. L p-weak) o ∞, henϕ ncon e ges L p-s ong ( esp.
Lp-weak) o ϕ ∞, o all ϕ∈Cb(Z),
( i) o e e y ∈Lp(m∞) he e exis s a sequence n∈Lp(mn)con e ging L p-s ong o
,
( ii) i na e non-nega i e and con e ge in L p-s ong o , hen o e e y q ∈(1,∞),
p/q
ncon e ge Lq-s ong o p/q,
( iii) Fix p,q∈(1,∞] so ha p <q.I he sequence ( n)is uni o mly bounded in Lqand
con e ges L p-s ong o ∞, hen i con e ges also L -s ong o ∞ o e e y ∈[p,q),
123

Rigidi y and almos igidi y o Sobole … Page 21 o 65 180
P oo Fo he p oo o he i ems (i)–( ) we e e o [13,P op. 3.3]. ( i)can ins ead be
ound in [55](seealso[63]). ( ii) ollows immedia ely om he cha ac e iza ion o Lp-
s ong con e gence ia con e gence o g aph (see e.g. [13,Rema k 3.2]). Fo ( iii), he
case q=∞ ollows immedia ely om i em (i)(see also [13,e) o P op. 3.3 ]), hence we
can assume q<+∞.Fix ∈[p,q). Clea ly om he Hölde inequali y nis uni o mly
bounded in L , hence by de ini ion ncon e ges L -weakly o ∞. Mo eo e om i em (iii)
we known ha ∞∈L (m∞), he e o e by unca ion and diagonaliza ion we can suppose
ha ∈L∞(m∞). F om ( i) hen he e exis s a sequence gn∈L (mn)con e ging o ∞
in L -s ong and by i em i)we can also assume ha gna e uni o mly bounded in L∞.Then,
om ( iii)in he case q=∞we ha e ha gncon e ge also in Lp-s ong o ∞.Then by
(ii)we ha e ha gn− ncon e ges o 0 in Lp-s ong and in pa icula  n−gnLp(mn)→0.
Finally by he Hölde inequali y (since n,gna e bo h uni o mly bounded in Lq)weha e ha
 n−gnL (mn)→0. In pa icula limn nL (mn)=limngnL (mn)= ∞L (m∞),
which concludes he p oo . 
We now pass o some con e gence and s abili y esul s ela ed o Sobole spaces. We s a
wi h he ollowing gene alized e sion o he compac embedding o W1,2→L2( epo ed
he e speci ically o compac me ic measu e spaces):
P oposi ion 2.19 ([55]) Suppose ha Xn,n ∈¯
Na e compac and assume ha ( n)∈
W1,2(Xn)a e uni o mly bounded in W 1,2,i.e.supn nW1,2(Xn)<+∞.Then( n)has
aL
2-s ongly con e gen subsequence.
We ecall he -con e gences o he 2-Cheege ene gies p o en in [55]:
◦-lim: o e e y n∈L2(mn)L2-s ong con e ging o ∞∈L2(m∞), i holds
ˆ|D ∞|2dm∞≤lim
n→∞ˆ|D n|2dmn;(2.18)
◦-lim: o e e y ∞∈L2(m∞), he e exis s a sequence n∈L2(mn)con e ging
L2-s ong o ∞so ha
lim
n→∞ˆ|D n|2dmn≤ˆ|D ∞|2dm∞.(2.19)
We will also need he -lim inequali y also o he p-Cheege ene gies as p o ed in
[13,Theo em 8.1]: o e e y p∈(1,∞)and e e y ∞∈Lp(m∞), he e exis s n∈Lp(mn)
con e ging Lp-s ong o ∞so ha
lim
n→∞ˆ|D n|pdmn≤ˆ|D ∞|pdm∞.
The abo e is s a ed in [13] only o a sequence o RCD(K,∞)spaces, bu i easily seen ha
he p oo wo ks wi hou modi ica ion also in he case o CD(K,∞)spaces.
We end his pa ecalling a well known con inui y esul o he spec al gap (see [55]and
[14]): i Xn,n∈¯
N, a e all compac i holds
λ1,2(X∞)=lim
n→∞λ1,2(Xn). (2.20)
We men ion ha he con inui y o he spec al gap was p e iously ob ained in he se ing o
Ricci-limi spaces by Cheege and Colding [40].
123
180 Page 22 o 65 F. Nobili, I. Y. Violo
2.4 Pólya–Szego inequali y
The Pólya–Szeg˝o inequali y, namely he ac ha he Di ichle ene gy dec eases unde
dec easing ea angemen s, da es back o Fabe and K ahn and was successi ely o mal-
ized in [85]. La e , in [28], his collec ion o ideas was b ough o he con ex o mani olds
wi h Ricci lowe bounds o achie e applica ions conce ning he igidi y o he 2-spec al gap.
Conce ning he opic o his manusc ip , he said inequali y has e ealed e ec i e in [66]in
he p oo o Theo em 1.2.
In his pa we ecall he Pólya–Szeg˝o inequali y o essen ially nonb anching CD(K,N)
spaces p o en in [84]. We will also collec some addi ional echnical esul s and de ini ions
om [84] ha will be used in Sec . 3.1 o p o e a Euclidean- a ian o his inequali y.
De ini ion 2.20 (Dis ibu ion unc ion) Le (X,d,m)be a compac me ic measu e space,
⊆X an open se wi h m() < +∞and u:→[0,+∞)a non-nega i e Bo el unc ion.
We de ine μ:[0,+∞)→[0,m()], he dis ibu ion unc ion o u,as
μ( ):= m({u> }). (2.21)
Fo uand μas abo e, we le u#be he gene alized in e se o μ,de inedby
u#(s):= ess sup ui s=0,
in { :μ( )<s}i s>0.
I can be checked ha u#is non-inc easing and le -con inuous.
Then, gi en ⊆Xan open se and u:→[0,+∞)a non-nega i e Bo el unc ion,
we de ine he mono one ea angemen in o IN=([0,π],|.|,mN)(see De ini ion 2.6)as
ollows: i s , we conside >0so ha m() =mN([0, ])and de ine ∗:= [0, ], hen
we de ine he mono one ea angemen unc ion u∗
N:∗→R+as
u∗
N(x):= u#(mN([0,x])), ∀x∈[0, ].
In he sequel, whene e uand a e ixed, ∗and u∗
Nwill be implici ly de ined as abo e.
Theo em 2.21 (Pólya–Szeg˝o inequali y, [84]) Le (X,d,m)be an essen ially non b aching
CD(N−1,N)space o some N ∈(1,∞)and ⊆Xbe open. Then, o e e y p ∈
(1,∞), he mono one ea angemen in INmaps L p() ( esp. W1,p
0())in oLp(∗)( esp.
W1,p(∗)) and sa is ies:
uLp() =u∗
NLp(∗),∀u∈Lp() (2.22)
ˆ|Du|pdm≥ˆ∗|Du∗
N|pdmN,∀u∈W1,p
0(). (2.23)
We will also need he ollowing igidi y o he Pólya–Szeg˝o inequali y p o en in
[84,Theo em 5.4].
Theo em 2.22 Le (X,d,m)be an RCD(N−1,N)space o some N ∈[2,∞)wi h m(X)=
1and p ∈(1,∞).Le ⊂Xbe an open se and assume ha he e exis s a non-nega i e
and non-cons an unc ion u ∈W1,p
0() achie ing equali y in (2.23).
Then (X,d,m)is isomo phic o a sphe ical suspension, i.e. he e exis s an RCD(N−
2,N−1)space (Z,dZ,mZ)wi h mZ(Z)=1so ha X≃[0,π]×N
sin Z.
123
Rigidi y and almos igidi y o Sobole … Page 23 o 65 180
Rema k 2.23 Obse e ha in Theo em 2.22 we did no assume ha m() < 1, assump ion
ha is ac ually p esen in Theo em 5.4 o [84]. This is in en ional, since we will need o apply
Theo em 2.22 p ecisely in he case =X. This is possible since he a gumen s in [84]wo k
also in he case =X wi hou modi ica ion. The only pa whe e he a gumen does no
co e explici ly he case =X is he p oo o he app oxima ion Lemma 3.6 in [84], which
howe e can be easily adap ed (see Lemma 2.24 below). 
The ollowing echnical esul will be needed in Sec . 3.1. We include a ske ch o he
a gumen in he case =X, o u he jus i y he alidi y o Theo em 2.22 also in his case
(see he abo e Rema k).
Lemma 2.24 (App oxima ion wi h non- anishing g adien s) Le (X,d,m)be a CD(K,N)
me ic measu e space wi h N <+∞, and le ⊂Xbe open wi h m() < +∞.Then
o any non-nega i e u ∈LIPc() he e exis s a sequence o non-nega i e un∈LIPc()
sa is ying |Dun|1= 0m-a.e. in {un>0}and such ha un→uinW
1,p(X).
P oo The case = X has been p o en in [84,Lemma 3.6 and Co olla y 3.7]. The p oo
p esen ed he e, as i is w i en, doesno co e he case =X wi h X compac and supp(u)=
X. Howe e , he a gumen can be easily adap ed by conside ing a sequence εn→0such
ha m({lip(un)=εn})=0 and aking
un:= u+εn ,
wi h (x):= d(x0,x), o an a bi a y ixed poin x0∈X. Since ∈LIP(X)and lip( ) =
1m-a.e. in X, a guing exac ly as in [84,Lemma 3.6] we ge ha un→uin W1,p(X)
and lip(un)= 0m-a.e. in {un>0}. To ge he claimed non- anishing o |Dun|1,asin
[84,Co olla y 3.7] we use he exis ence o a cons an c>0 such ha
|Du|1≥clip(u), m-a.e.,
o e e y u∈LIPloc(X), which holds om he esul s in [16] and he ac ha CD(K,N)
spaces a e locally doubling and suppo s a local-Poinca é inequali y. 
Lemma 2.25 (De i a i e o he dis ibu ion unc ion, ([84])) Le (X,d,m)be a me ic mea-
su e space and le ⊆X be an open subse wi h m() < +∞. Assume ha u ∈LIPc()
is non-nega i e and |Du|1(x)= 0 o m-a.e. x ∈{u>0}. Then i s dis ibu ion unc ion
μ:[0,+∞)→[0,m()], de ined in (2.21), is absolu ely con inuous. Mo eo e i holds
μ( )=−ˆ1
|Du|1dPe ({u> },·)a.e.,(2.24)
whe e he quan i y 1/|Du|1is de ined o be 0whene e |Du|1=0.
3 Uppe bound o ˛p
To p o e an uppe bound o αpwe will need o de i e a Sobole inequali y o he ype (1.6) o
some explici A. This will be achie ed by p o ing i s a class o local Sobole -inequali ies
(see Theo em 3.8) and hen “pa ch” hem oge he (see Theo em 1.8) o ob ain he desi ed
globalinequali y. The local-Sobole inequali ies willbe achie ed h ough a Euclidean Pólya–
Szeg˝o symme iza ion inequali y (Theo em 3.6).
123
180 Page 24 o 65 F. Nobili, I. Y. Violo
3.1 Pólya–Szego inequali y o Euclidean- ype
The goal o his sec ion is o p o e a Euclidean- a ian o he Pólya–Szeg˝o inequali y o
CD(K,N)spaces de i ed in [84] (unde essen ially nonb anching assump ion, see also
Sec . 2.4). The main di e ence is ha ou inequali y holds o a bi a y K∈Rand assumes
heap io i alidi y o aEuclidean- ypeisope ime icinequali y, while heonein[84] equi es
K>0 and i is based on he Lé y-G omo isope ime ic inequali y o he CD(K,N)con-
di ion. As opposed o Sec . 2.4, whe e he symme iza ion has as a ge he model space
o he CD(K,N)condi ion wi h K>0, we will use a no ion o symme iza ion ha
li es in he weigh ed hal line ([0,∞), |.|, N−1L1). I should be ema ked ha , in gene al,
he e is no a na u al cu a u e model space o symme ize unc ions de ined on an a bi a y
CD(K,N)-space wi h K≤0. This is because he e is no a unique model-space o he
Lé y–G omo isope ime ic inequali y in he case K≤0(see[81]). The e o e, i is unclea
in his high-gene ali y whe e he ea angemen s should li e. Fo his eason we will equip he
me ic measu e spaces unde conside a ion wi h a (possibly local) isope ime ic inequali y
o Euclidean- ype:
Pe (E)≥Cm(E)N−1
N,
o N>1andCa non-nega i e cons an .
We s a wi h he de ini ion o Euclidean model space (I0,N,|.|,m0,N),N∈(1,∞):
I0,N:= [0,∞), m0,N:= σN−1 N−1L1,
whe e |.|is he Euclidean dis ance. Nex , we de ine he Euclidean mono one ea angemen .
De ini ion 3.1 (Euclidean mono one ea angemen ) Le (X,d,m)beame icmeasu e
space and ⊂X be open wi h m() < +∞. Fo any Bo el unc ion u:→R+,we
de ine ∗:= [0, ]wi h m0,N([0, ])=m() (i.e. N=ω−1
Nm()) and he mono one
ea angemen u∗
0,N:∗→R+by
u∗
0,N(x):= u#(m0,N([0,x])) =u#(ωNxN), ∀x∈∗,
whe e u#is he gene alized in e se o he dis ibu ion unc ion o u, as de ined in Sec . 2.4.
In he sequel, whene e we ix and u:→[0,∞), hese ∗and he ea angemen
u∗
0,Na e au oma ically de ined as abo e.
P oposi ion 3.2 Le (X,d,m)be a me ic measu e space and ⊂Xbe open and bounded
wi h m() < +∞.Le u:→[0,+∞)be Bo el and le u∗
0,N:∗→[0,+∞)be i s
mono one ea angemen .
Then, u and u∗
0,Nha e he same dis ibu ion unc ion. Mo eo e
uLp() =u∗
0,NLp(∗),∀1≤p<+∞,(3.1)
and he adial dec easing ea angemen ope a o L p() u→ u∗
0,N∈Lp(∗)is
con inuous.
The p oo o he abo e p oposi ion is classical, ollowing e.g. [69], wi h s aigh o wa d
modi ica ion o he me ic measu e se ing (see also [84]). Obse e also ha , gi en u∈
Lp(), i s mono one ea angemen mus be de ined by ixing a Bo el ep esen a i e o u.
Howe e , his choice does no a ec he ou come objec u∗
0,N, as clea ly he dis ibu ion
unc ion μ( )o uis independen o he ep esen a i e.
123
Rigidi y and almos igidi y o Sobole … Page 31 o 65 180
Theo em 3.13 (Uppe bound on αp)Le (X,d,m)be a compac CD(K,N)space, o some
N∈(1,∞),K ∈R. Then, o e e y ε>0and e e y p ∈(1,N), he e exis s a cons an
B=B(ε, p,X)>0such ha
up
Lp∗(m)≤Eucl(N,p)p
minXθN(x)p/N+ε|Du|p
Lp(m)+Bup
Lp(m),∀u∈LIP(X).
(3.10)
P oo We s a claiming ha he ollowing local e sion o (3.10) holds: o any x∈Xand
e e y ε>0 he e exis s = (ε, x)>0andC=C(ε, p,x)<+∞ such ha
up
Lp∗(m)≤Eucl(N,p)p
miny∈XθN(y)p/N+ε|Du|p
Lp(m)+Cup
Lp(m),∀u∈LIPc(B (x)).
(3.11)
To show he abo e we obse e i s ha in he case ha θN(x)=+∞,(3.11) ollows
immedia ely om (3.9) o small enough. We a e le wi h he case 0 <θ
N(x)<+∞.We
s a by ixing ε∈(0,1/2). F om he de ini ion o θN(x), he e exis s = (x,ε)so ha
o e e y ∈(0, )i holds θN, (x)∈((1−ε)θN(x), (1+ε)θN(x)). In pa icula we ha e
ha θN, (x)
θN,R(x)≤4 o e e y ,R∈(0, ). We a e he e o e in posi ion o apply Theo em 3.8
and deduce ha he e exis s δ=δ(ε, N)so ha o e e y ,R∈(0, ∧δN/K−), wi h
<δR, he ollowing inequali y holds o e e y u∈LIPc(B (x))
up
Lp∗(m)
(3.6)
≤(1+ε)pEucl(N,p)p
θN,R(x)p/N|Du|p
Lp(m)
≤(1+ε)p
(1−ε)p/N
Eucl(N,p)p
minXθN(x)p/N|Du|p
Lp(m),
whe e in he second inequali y we ha e used θN,R(x)≥(1−ε)θN(x). The e o e (3.11) (wi h
C=0) ollows om he abo e p o ided we choose εsmall enough.
Since X is compac we can ex ac a ini e co e ing o balls {Bi}M
i=1 om he co e ing
∪x∈XB (ε,x)/2(x).Wealsose C:= maxiCiand
A:= Eucl(N,p)p
minXθN(x)p/N+ε.
We claim ha he e exis s a pa i ion o uni y made o unc ions {ϕi}M
i=1such ha ϕi∈
LIPc(2Bi),0≤ϕi≤1andϕ1/p
i∈LIPc(2Bi) o all i, ha ing deno ed 2Bi, he ball o
wice he adius. To build such pa i ion o uni y we can a gue as ollows: s a conside ing
unc ions ψi∈LIPc(2Bi), such ha 0 ≤ψi≤1andψi≥1inBi.Thenwe ixβ>pand
ake
ϕi:= ψβ
i
M
j=1ψβ
j
.
Since by cons uc ion M
j=1ψβ
j≥1e e ywhe eonX,weha e ha ϕ1/p
i∈LIPc(2Bi).
Finally i is clea ha M
i=1ϕi=1.
We a e now eady o p o e (3.10). Fix u∈LIP(X)and obse e ha
up
Lp∗(m)=


i
ϕi|u|p

Lp∗/p(m)≤
i
ϕi|u|p
Lp∗/p(m)=
i
ϕ1/p
i|u|

p
Lp∗(m).
(3.12)
123

180 Page 32 o 65 F. Nobili, I. Y. Violo
Since ϕ1/p
i|u|∈LIPc(2Bi)we can apply (3.11) o ob ain
up
Lp∗(m)≤
M

i=1
Aˆ|Dϕ1/p
i||u|+|Du|ϕ1/p
ipdm+Cˆϕi|u|pdm
≤
M

i=1
Aˆϕi|Du|p+c1|Du|p−1ϕ
p−1
p
i|Dϕ1/p
i||u|+c2|Dϕ1/p
i|p|u|pdm
+Cˆϕi|u|pdm,
whe e c1,c2≥0 a e such ha (1+ )p≤1+c1 +c2 p o all ≥0.Recalling ha he
unc ions 0 ≤ϕ1/p
i≤1 a e Lipschi z we ob ain
up
Lp∗(m)≤Aˆ|Du|pdm+˜
Cˆ|Du|p−1|u|dm+˜
Cˆ|u|pdm,
whe e ˜
C=˜
C(p,M,L),Lbegin he maximum o he Lipschi z cons an s o he unc ions
ϕ1/p
i.Finally om he Young inequali y we ha e o e e y δ>0
ˆ|Du|p−1|u|dm≤pδ
p
p−1
p−1ˆ|Du|pdm+1
pδpˆ|u|pdm,∀δ>0
and plugging his es ima e abo e, choosing δsmall enough (bu independen o u), we ob ain
ha
up
Lp∗(m)≤(A+ε) ˆ|Du|pdm+Cˆ|u|pdm,
o some C=C(ε, L,M,p).Sinceε>0andu∈LIP(X)we e a bi a y, his concludes
he p oo . 
4 Lowe bound on ˛p
he ough idea o he lowe bound on αpis ha , when θN(x)<+∞ he space nea xhas
a conical s uc u e, hence he cons an in he Sobole inequali y canno be be e han he
one o he angen s uc u es o he unde lying space. This will be o malized wi h a blow-up
a gumen combined wi h a s abili y esul o he Sobole cons an s.
4.1 Blow-up analysis o Sobole cons an s
Fo con enience, we in oduce he ollowing no a ion: whene e in a me ic measu e space
(X,d,m)i holds ha
up
Lq(m)≤A|Du|pp
Lp(m)+Bup
Lp(m),∀u∈W1,p(X).
o some cons an s A,B>0 and exponen s 1 <p<q, we will say ha X suppo s a
(q,p)-Sobole inequali y wi h cons an s A,B. This con en ion will be used o en he e,
and some imes in he subsequen sec ions, wi hou u he no ice.
We make p ecise he scaling enjoyed by he Sobole inequali ies unde conside a ion. I
is immedia e o check ha i a space (X,d,m)suppo s a (p∗,p)-Sobole o p∈(1,N)
123
Rigidi y and almos igidi y o Sobole … Page 33 o 65 180
and p∗:= pN
N−pwi h cons an s A,B, hen o e e y >0weha e
(X,d/ ,m/ N)suppo s a (p∗,p)−Sobole wi h cons an s A,B p.(4.1)
We pass o he s abili y o Sobole embeddings unde pmGH-con e gence (see also
[64,Thm. 3.1] o a simila esul o Ricci-limi s).
Lemma 4.1 (pmGH-S abili y o Sobole cons an s) Le (Xn,dn,mn,xn),n ∈¯
N,bea
sequence o CD(K,N)spaces o some K ∈R,N ∈(1,∞)wi h Xn
pmGH
→X∞. Sup-
pose Xnsuppo a (q,p)-Sobole inequali y o 1<p<q wi h cons an s A,B. Then also
X∞suppo s a (q,p)-Sobole inequali y wi h he same cons an s A,B.
P oo Fix u∈LIPc(X∞), om he -lim inequali y o he Chpene gy, he e exis s a
sequence un∈W1,p(X∞)such ha uncon e ges in Lp-s ong o uand limn´|Du|pdmn≤
´|Du|pdm∞. In pa icula
lim
nunp
Lq(mn)≤lim
n→∞ A|Dun|p
Lp(mn)+Bunp
Lp(mn)
≤A|Du|p
Lp(m∞)+Bup
Lp(m∞)<+∞.
The e o e uncon e ge also Lq-weak o u. F om he lowe semicon inui y o he Lq-no m
wi h espec o Lq-weak con e gence and he a bi a iness o u∈LIPc(X∞) he conclusion
ollows. 
The ollowing esul is a consequence o he exis ence o he disin eg a ion and can be
ound o example in [42,Co olla y 3.8].
Lemma 4.2 Le (X,d,m)be a CD(0,N)space wi h N ∈[1,∞). Suppose ha o some
x0∈Xi holds ha m(B (x0))
ωN N=1 o e e y ∈(0,∞), hen
ˆϕ(d(x0,x)) dm=σN−1ˆ∞
0
ϕ( ) N−1d ,∀ϕ∈Cc([0,∞]).
Lemma 4.3 Le (X,d,m)be a CD(0,N)space, N ∈(1,∞),p∈(1,N)and se p∗:= pN
N−p.
Suppose ha o some x0∈Xi holds ha m(B (x0))
ωN N=1 o e e y ∈(0,∞). Then he e
exis s a sequence o non-cons an unc ions un∈LIPc(X)sa is ying
lim
nunLp∗(m)
|Dun|Lp(m)≥Eucl(N,p).
P oo Le :[0,∞)→[0,∞), ∈C∞(0,∞), be an ex emal unc ion o he Bliss
inequali y (3.8) as gi en by Lemma 3.11. I can be easily shown ha we can app oxi-
ma e wi h unc ions n∈LIPc([0,∞)) so ha  nLp∗(hNL1)→ Lp∗(hNnL1)and
 
nLp(hNL1)→ Lp(hNL1),whe ehNL1=σN−1 N−1L1.Fo example we can ake
n:= ϕn(ub)wi h ϕn∈LIP[0,∞),ϕn≥0, ϕn( )≤| |,Lip(ϕn)≤2, ϕn( )= in
[2/n,∞)and supp(ϕn)⊂[1/n,∞). The claimed app oxima ion o he no ms hen ollows
immedia ely om he ac ha is dec easing and anishing a in ini y. The e o e we ha e
lim
n nLp∗(hnL1)
 
nLp(hnL1)=Eucl(N,p). (4.2)
We can now de ine un:= n◦dx0,whe edx0(·):= d(x0,·). We clea ly ha e ha un∈
LIPc(X)and om he chain ule also ha |Dun|=| 
n|◦dx0|Ddx0|≤| 
n|◦dx0m-a.e., since
123
180 Page 34 o 65 F. Nobili, I. Y. Violo
dx0is 1-Lipschi z. Hence applying Lemma 4.2 we ob ain unLp∗(m)= nLp∗(hNL1)and
|Dun|Lp(m)≤ 
nLp(hNL1). This combined wi h (4.2) (up o passing o a subsequence)
gi es he conclusion. 
Theo em 4.4 (Lowe bound on he Sobole cons an ) Le (X,d,m)be a CD(K,N)space,
K∈R,N∈(1,∞) ha suppo s a (p∗,p)-Sobole inequali y o p ∈(1,N)wi h con-
s an s A,B,whe e p∗=pN/(N−p). Then
A≥Eucl(N,p)p
θN(x)p
N
,∀x∈X.(4.3)
P oo I θN(x)=∞, he e is no hing o p o e. Hence we can assume ha θN(x)<+∞.
F om he compac ness and s abili y o he CD(K,N)condi ion, he e exis s a sequence
i→0 such ha Xi:= (X,d/ i,m/ iN,x)pmGH-con e ge o a CD(0,N)space
(Y,dY,mY,oY). Mo eo e , om (4.1)weha e ha X
isuppo s a (p∗,p)-Sobole inequal-
i y wi h cons an s A, p
iB. This combined wi h Lemma 4.1 shows ha (Y,dY,mY)suppo s
a(p∗,p)-Sobole inequali y wi h cons an s A,0. Howe e we clea ly ha e ha mYsa is ies
mY(B (oY))
ωN N=θN(x) o e e y >0.The e o e Lemma 4.3, a e a escaling, ensu es ha
A≥Eucl(N,p)p
θN(x)
p
N
,which is wha we wan ed. 
The abo e, oge he wi h Theo em 3.13, p o es ou main esul Theo em 1.4 conce ning
αp(X).
Using Theo em 4.4 we can also p o e he opological igidi y o he Sobole inequali y on
non-collapsed RCD spaces. Mo e p ecisely combining he olume igidi y o non-collapsed
RCD spaces ([53,Theo em 1.6]) and he Cheege –Colding’s me ic Rei enbe g’s heo em
([40,Theo em A.1.2]) (see also [68]) we can ob ain he ollowing esul .
Co olla y 4.5 (Mani old- egula i y om almos Euclidean–Sobole inequali y) Fo e e y
K∈R,N∈N,p∈(1,N),α∈(0,1),ε>0 he e exis s δ=δ(K,N,ε,α)such ha he
ollowing holds. Suppose ha (X,d,HN)is a compac RCD(K,N)space sa is ying he
ollowing Sobole inequali y
up
Lp∗(HN)≤(Eucl(N,p)p+δ)|Du|p
Lp(HN)+Bup
Lp(HN),∀u∈W1,p(X),
(4.4)
o some cons an B >0,whe ep
∗:= pN/(N−p).
Then, he e exis s a smoo h N-dimensional Riemannian mani old M and an α-biHölde
homeomo phism F :M→X.
P oo The a gumen is analogous o [68,Theo em 3.1], howe e o comple eness we include
he de ails.
We s a ixing ε>0, N∈N,K∈R,p∈(1,N)and wo numbe s ¯
δ=¯
δ(K,N,p,ε)>
0¯ =¯ (K,N,p,ε)small enough o be chosen la e .
Suppose ha (X,d,HN)is a compac RCD(K,N)space ha suppo s a (p∗,p)-Sobole
inequali y wi h cons an Eucl(N,p)p+δ, B, o someδ≤¯
δand B>0 (i.e. such ha (4.4)
holds). Then om (4.3), i ¯
δ≤Eucl(N,p)p/4,we ha e ha
θN(x)≥1−2δ, ∀x∈X.
The e o e o e e y x∈X he e exis s x∈(0,¯ )such ha HN(B x(x)) ≥(1−3δ) N
xωN.
Mo eo e om he Bishop–G omo inequali y, o e e y y∈Bδ x(x)and e e y s∈(0, x)
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Rigidi y and almos igidi y o Sobole … Page 35 o 65 180
i holds ha
HN(Bs(y))
K,N(s)≥HN(B(1+δ) x(y))
K,N((1+δ) x)≥HN(B x(x))
K,N((1+δ) x)≥(1−3δ) N
xωN
K,N((1+δ) x).(4.5)
Recalling ha lim →0+ωN N
K,N( )=1, om (4.5) we deduce ha i bo h ¯ and ¯
δa e small
enough, wi h espec o K,N,p,ε, hen
HN(Bs(y)) ≥(1−ε)sNωN,∀y∈B x(x), s∈(0, x).
Finally om he compac ness o X he e exis s a ini e numbe o poin s xi,i=1,...,m
such ha X ⊂∪
iB xi(xi).TakingR:= mini xi<¯ we hen ha e
HN(Bs(y)) ≥(1−ε)sNωN,∀y∈X,s∈(0,R).
F om his he conclusion ollows combining he olume igidi y heo em o non-
collapsed RCD spaces ([53,Theo em 1.6]) and he in insic me ic-Rei enbe g’s heo em
([40,Theo em A.1.2]). 
4.2 Sha p and igid Sobole inequali ies unde Euclidean olume g ow h
He e we p o e he sha p Sobole inequali ies on CD(0,N)spaces con ained Theo em 1.13.
The alidi y o he inequali y (1.12) will be de i ed as a consequence o he local-Sobole
inequali ies in Theo em 3.8. The sha pness ins ead ollows om a well known p inciple o
which he alidi y o a Euclidean–Sobole inequali y implies ce ain g ow h on he measu e
o balls. In pa icula we ha e he ollowing esul :
Theo em 4.6 Le (X,d,m)be an CD(0,N),N∈(1,∞)such ha o some p ∈(1,N)and
A>0
uLp∗(m)≤A|Du|Lp(m),∀u∈LIPc(X), (4.6)
whe e p∗:= pN
N−p.ThenXhas Euclidean olume-g ow h and
AV R(X)≥Eucl(N,p)
AN
.(4.7)
On he gene al se ing o CD spaces Theo em 4.6 isp o edin[73](seealso[74] o he
case p=2), ex ending o non-smoo h se ing he same esul s o Riemannian mani olds
due o Ledoux [75] and imp o ed by Xia [97]. We men ion also [45]and[98] o analogous
s a emen s ela ed odi e en classo inequali ies.Inall heci edwo ks hea gumen sdepend
on a he in ica e ODE-compa ison (o igina ed in [75] and inspi ed by he p e ious [25]) and
hea ily ely on he explici knowledge o he ex emal unc ions o he inequali ies. Howe e ,
using he esul s in Sec . 4we a e able o gi e a sho p oo o Theo em 4.6, which uses a
mo e di ec blow-down p ocedu e, ha we belie e being in e es ing on i s own. The main
ad an age o his app oach is ha we will ne e need, as opposed o he ODE-compa ison
app oach, he explici exp ession o ex emals unc ions in he Euclidean Sobole inequali y
(1.1).
P oo o Theo em 4.6 The ac ha m(X)=+∞can be immedia ely seen by plugging in he
Sobole inequali y unc ions uR∈LIPc(X)so ha uR=1inBR(x0)supp(uR)⊂B2R(x0)
and Lip(uR)≤1/Rand sending R→+∞.The ac ha X has Euclidean olume g ow h
123
180 Page 36 o 65 F. Nobili, I. Y. Violo
ollows by conside ing ins ead unc ions uR(·):= (R−dx0(·))+as R→+∞wi h ixed
x0∈X and using he Bishop–G omo inequali y.
I emains o p o e (4.7). We a gue ia blow-down. Le Ri→+∞. F om he Euclidean
olume-g ow h p ope y, up o passing o a non elabeled subsequence, he escaled spaces
(X,d/Ri,m/RN
i,x0),x0∈X, pmGH-con e ge o an CD(0,N)space (Y,dY,mY,oY)
sa is ying mY(BR(oY))
ωN N=AVR (X). Mo eo e combining (4.6) wi h Lemma 4.1 p o es ha Y
sa is ya (p∗,p)-Sobole inequali ywi hcons an s A,0.Then (4.7) ollows om Lemma 4.3.

We can now mo e o he p oo o he sha p Sobole inequali ies unde he Euclidean
olume g ow h assump ion.
P oo o Theo em 1.13 Fix x∈X. F om he de ini ion o AV R(X), o e e y big enough
θN, (x)≤2AV R(X). Fix one o such >0.F om he Bishop–G omo inequali y we also
ha e ha θN,R(x)≥AV R(X) o e e y R>0. In pa icula θN, (x)/θN,R(x)≤2 o
e e y R>0.Hence by Theo em 3.8 ( o K=0) we ha e ha o e e y ε>0, he e exis s
δ=δ(ε) > 0 so ha o e e y R> /δ he ollowing local Euclidean Sobole inequali y
holds:
uLp∗(m)≤(1+ε)Eucl(N,p)θN,R(x)−1
N
||Du|Lp(m),∀u∈LIPc(B (x)).
Taking R→∞we achie e
uLp∗(m)≤(1+ε)Eucl(N,p)AV R(X)−1
N
||Du|Lp(m),∀u∈LIPc(B (x)).
Since εwas chosen a bi a ily and independen o >0, we can i s send ε→0+and hen
→+∞ o achie e he i s pa o he s a emen .
The sha pness o (1.12) ins ead ollows immedia ely om Theo em 4.6.
5 The cons an Aop
qin me ic measu e spaces
In his sec ion we will p o e some uppe and lowe bounds on Aop
qin he case o me ic
measu e spaces. Some o he esul s con ained he e (mo e p ecisely, Sec . 5.3) a e ac ually
no used in o he pa s o he no e, howe e we chose o include hem he e o comple eness
and o gi e a mo e clea pic u e a ound he alue o Aop
q. Le us also ema k ha he esul s
o his pa a e alid o a gene al lowe bound K∈R.
We s a ecalling he de ini ion o Aop
q. In his sec ion we assume ha (X,d,m)is a me ic
measu e space wi h m(X)=1. Fo e e y q∈(2,+∞)we de ine Aop
q(X)∈[0,+∞]as he
minimal cons an sa is ying
u2
Lq(m)≤Aop
q(X)|Du|22
L2(m)+u2
L2(m),∀u∈W1,2(X), (5.1)
wi h he con en ion ha A:= +∞i no such Aexis s. No e ha , since m(X)=1, his is he
same de ini ion gi en igh a e (1.10). In he ollowing sec ions we will p o e h ee ype o
bounds on Aop
q(X): an uppe bound in he case o syn he ic Ricci cu a u e and dimension
bounds; a lowe bound in e ms o he i s non- i ial eigen alue; a lowe bound ela ed o
he diame e .
123

Rigidi y and almos igidi y o Sobole … Page 37 o 65 180
5.1 Uppe bound on Aop
qin e ms o Ricci bounds
He e we p o e a gene aliza ion o he non-smoo h se ing o a well known es ima e on Aop
q
alid on mani olds ( ecall (1.4)). The wo key ing edien s o he p oo a e he Sobole –
Poinca é inequali y and an inequali y due o Bak y:
P oposi ion 5.1 Fo e e y K ∈R,N ∈(2,∞)and D >0 he e exis s a cons an A =
A(K,N,D)>0such ha he ollowing holds. Le (X,d,m)be a compac CD(K,N)
space wi h N ∈(1,∞),K∈R,m(X)=1and diam(X)≤D. Then o e e y q ∈(2,2∗]
we ha e
u2
Lq(m)≤A|Du|2
L2(m)+u2
L2(m),∀u∈W1,2(X)(5.2)
and in pa icula Aop
q(X)≤A(K,N,D).
P oo The p oo is based on he ollowing inequali y: o e e y q∈(2,∞)
ˆ|u|qdm2/q≤(uX)2+(q−1)ˆ|u−uX|qdm2/q∀u∈Lq(m), (5.3)
whe e uX=´udm.See ([23]o [24,P op. 6.2.2] ) o a p oo o his ac . Then (5.2) ollows
combining (5.3) wi h (2.16) and he Jensen inequali y. 
Recall ha o K>0 an explici and sha p uppe bound on Aop
qexis s and has been p o en in
[36] (see Theo em 1.8). The a gumen in [36] elies on he powe ul localiza ion echnique.
Howe e , i is wo h o poin ou ha Theo em 1.8 can also be deduced om he Pólya–Szeg˝o
inequali y p o ed in [84] (see Theo em 2.21) and he Sobole inequali y on he model space
(2.17).
5.2 Lowe bound on Aop
qin e ms o he i s eigen alue
I is well known ha a “ igh -Sobole inequali y” as in (5.1) (i.e. wi h a cons an 1 in on
o uL2when X is no malized wi h uni olume) implies a Poinca é-inequali y (see e.g.
[24,P op. 6.2.2]). This can be eph ased as a lowe bound on Aop
qin e ms o he i s non-
i ial eigen alue:
P oposi ion 5.2 Le (X,d,m)be a me ic measu e space wi h m(X)=1.Then o e e y
q∈(2,+∞)i holds
Aop
q(X)≥q−2
λ1,2(X),(5.4)
(meaning ha i λ1,2(X)=0, hen Aop
q(X)=+∞).
Wewill gi e a de ailed p oo o his esul ,whichamoun s o a linea iza ion p ocedu e. Indeed
a e inemen o he same a gumen will also play a key ole on he igidi y and almos - igidi y
esul s in he sequel (see Sec . 6.2).
We s a wi h an elemen a y linea iza ion-Lemma.
123
180 Page 38 o 65 F. Nobili, I. Y. Violo
Lemma 5.3 Le (X,d,m)be a me ic measu e space wi h m(X)=1and ix q ∈(2,∞).Le
∈L2∩Lq(m)wi h ´ dm=0.Then
ˆ|1+ |qdm2/q−´(1+ )2dm−(q−2)´| |2dm
≤Cq´| |3∧q+| |qdm+´| |qdm2+´| |2dm2,
(5.5)
whe e Cqis a cons an depending only on q.
P oo We s a de ining I:= ´|1+ |qdm−1 and obse e ha
ˆ|1+ |qdm2/q−1−2
qI≤cq|I|2,(5.6)
which ollows om he inequali y ||1+ |2/q−1−2 /q|≤cq 2, ≥0.I emains o
in es iga e he beha io o I.Exploi ing he inequali y ||1+ |q−1−q |≤˜cq(| |2+| |q),
≥0, and he ac ha has ze o mean we ha e he ollowing simple bound
|I|≤˜cqˆ| |2+| |qdm.(5.7)
We will also need a mo e p ecise es ima e o I, which will ollow om he ollowing inequal-
i y
|1+ |q−1−q −q(q−1)
2 2≤Cq(| |3∧q+| |q), ∀ ∈R,(5.8)
ha can be seen using Taylo expansion when | |≤1/2 and elemen a y es ima es in he case
| |≥1/2. Using (5.8) we ob ain ha
I−ˆq +q(q−1)
2| |2dm≤Cqˆ| |3∧q+| |qdm
and since we a e assuming ha has ze o mean, we deduce
I−q(q−1)
2ˆ| |2dm≤Cqˆ| |3∧q+| |qdm.(5.9)
Combining (5.6), (5.7)and(5.9), no ing ha ´(1+ )2dm=1+´ 2dm,we deduce (5.5).

Exploi ing he abo e linea iza ion, we can now p o e he lowe bound on Aop
qin e ms o
he i s eigen alue.
P oo o P oposi ion 5.2 I Aop
q(X)=+∞ he e is no hing o p o e, hence we assume ha
Aop
q(X)<+∞.Le ∈LIP(X)∩L2(m)wi h ´ dm=0and L2(m)=1. Obse e
also ha , since Aop
q(X)<+∞, ∈Lq(X). The e o e applying (5.5) we ob ain
ˆ|1+ε |qdm2/q−ˆ(1+ε )2dm−(q−2)ˆ|ε |2dm=o(ε2),
which combined wi h (5.1)gi es
Aop
q(X)ε2ˆ|D |2
2dm−(q−2)ˆ|ε |2dm≥o(ε2).
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Rigidi y and almos igidi y o Sobole … Page 39 o 65 180
Di iding by ε2and sending ε→0 gi es ha λ1,2(X)≥q−2
Aop
q(X), which concludes he p oo .

5.3 Lowe bound on Aop
qin e ms o he diame e
We s a ecalling he ollowing esul , which was p o ed in [25] in he con ex o Ma ko -
iple and which p oo wo ks wi h s aigh o wa d modi ica ions also in he se ing o me ic
measu e spaces (see also [59] o an exposi ion o he a gumen on Riemannian mani olds).
Fo his eason we shall omi i s p oo . We s ess ha , since his esul and i s consequences
a e used only on his sec ion, he exposi ion o he es o he no e emains sel -con ained.
Theo em 5.4 Le q ∈(2,∞)and de ine N(q):= 2q
q−2.Le (X,d,m)be a compac me ic
measu e wi h diam(X)=π,m(X)=1and suppose ha
uLq(m)≤q−2
N(q)|Du|2
L2+u2
L2(m),∀u∈W1,2(X). (5.10)
Then he e exis s a non-cons an unc ion ∈LIP(X) ealizing equali y in (5.10).
No e ha q=2N(q)/(N(q)−2),so ha inasense“q=2∗(N(q))". Wi h Theo em 5.4
we can now p o e he ollowing lowe bound on Aop
q(X). The p oo uses a scaling a gumen
due o Hebey [59,P oposi ion 5.11].
P oposi ion 5.5 Le (X,d,m)be a compac me ic measu e space wi h m(X)=1and
diam(X)≤π. Then o e e y q ∈(2,∞)i holds
Aop
q(X)≥diam(X)
π2q−2
N(q),(5.11)
whe e N(q)=2q
q−2.
P oo Se D:= diam(X)and, by con adic ion, suppose ha Aop
q(X)<(D
π)2q−2
N(q).De ine
he scaled me ic measu e space
(X,d,m):= (X,1
D/π d,m).
I can be di ec ly checked ha Xsa is ies he hypo heses o Theo em 5.4. Hence he e exis s
a non-cons an unc ion u∈LIP(X)sa is ying (5.10) wi h equali y (in he space X), which
ew i en on he he o iginal space X eads as
uLq(m)=D
π2q−2
N(q)|Du|2
L2(m)+u2
L2(m),
which howe e con adic s he assump ion Aop
2∗(X)<(D
π)2q−2
N(q).
Rema k 5.6 A guing exac ly as in [25], i is possible o p o e ha unde he assump ions
o Theo em 5.4 and assuming X o be also in ini esimal Hilbe ian, he e exis s a unc ion
sa is ying u=N(q)u. F om his, i di ec ly ollows ha equali y in (5.11) (in he case o
an In ini esimally Hilbe ian space) implies he exis ence o a unc ion sa is ying:
u=π
diam(X)2
N(q)u.
Since his is no ele an in he p esen no e, we will no p o ide he de ails o such esul . 
123
180 Page 40 o 65 F. Nobili, I. Y. Violo
6 Rigidi y o Aop
q
6.1 Concen a ion Compac ness
In his sec ion we assume ha (Xn,dn,mn)is a sequence o compac RCD(K,N)spaces,
o some ixed K∈R,N∈(2,∞), which con e ges in mGH- opology o a compac
RCD(K,N)space (X∞,d∞,m∞). We will also adop he ex insic app oach [55] iden i y-
ing Xn,X∞as subse o a common compac me ic space (Z,dZ), wi h supp(mn)=Xn,
supp(m∞)=X∞,mnm∞in duali y wi h Cb(Z)and Xn→X∞in he Hausdo opology
o Z. To ligh en he discussion, we shall no ecall in he ollowing s a emen s hese ac s and
assume (Xn,dn,mn),n∈¯
N=N∪{∞}and (Z,d) o be ixed as jus explained. Also, we
will se 2∗:= 2N/(N−2)wi hou ecalling i s exp ession in he s a emen s.
Ou main goal hen is o p o e he ollowing dicho omy o he beha io o ex emizing
sequence o he Sobole inequali ies, on a ying me ic measu e spaces.
Theo em 6.1 (Concen a ion-compac ness o Sobole -ex emals) Suppose ha mn(Xn),
m∞(X∞)=1and ha Xnsuppo s a (2∗,2)-Sobole inequali y
u2
L2∗(mn)≤A|Du|2
L2(mn)+Bu2
L2(mn),∀u∈W1,2(Xn),
o some cons an s A,B>0.Suppose ha un∈W1,2(Xn)is a sequence o non-ze o
unc ions sa is ying
un2
L2∗(mn)≥An|Dun|2
L2(mn)+Bnun2
L2(mn),
o some sequences An→A, Bn→B.
Then, se ing ˜un:= unun−1
L2∗(mn), he e exis s a non elabeled subsequence such ha
only one o he ollowing holds:
(I) ˜uncon e ges L2∗-s ong o a unc ion u∞∈W1,2(X∞);
(II) ˜unL2(mn)→0and he e exis s x0∈X∞so ha |un|2∗mnδx0in duali y wi h
Cb(Z).
The p inciple behind he concen a ion compac ness echnique is e y gene al and was o igi-
na edin[78,79]. In ou case, since we will wo k in a compac se ing, he lack o compac ness
is o mally due o dila ions o escalings (and no o ansla ions) and he ac ha we deal
wi h he c i ical exponen in he Sobole embedding. The main idea behind he p inciple is
i s o p o e ha in gene al he ailu e o compac ness can only be ealized by concen a ion
on a coun able numbe o poin s. The second s ep is hen o exploi a s ic sub-addi i i y
p ope y o he minimiza ion p oblem o show ha ei he we ha e ull concen a ion a a
single poin o we do no ha e concen a ion a all and hus compac ness.
We s a by p o ing necessa y esul s owa ds he p oo o Theo em 6.1.
A a ian o he ollowing appea s also in [63,P op. 3.27]. Fo he sake o comple eness,
we p o ide he e a comple e p oo .
P oposi ion 6.2 Le p,q∈(1,∞)wi h 1
p+1
q=1.Suppose ha uncon e ges Lq-s ong
o u∞and ha ncon e ges L p-weak o ∞, hen
lim
n→∞ˆun ndmn=ˆu∞ ∞dm∞.
123
Rigidi y and almos igidi y o Sobole … Page 47 o 65 180
(i) he e exis s a non-cons an unc ion u ∈W1,2(X)sa is ying
u2
Lq(m)=Aop
q(X)|Du|2
L2(m)+u2
L2(m),(6.14)
(ii) Aop
q(X)=q−2
λ1,2(X),
(iii) q=2∗and Aop
2∗(X)=α2(X)=Eucl(N,2)2
min θ2/N
N
(see he in oduc ion and (2.2) o he
de ini ion o α2(X)and Eucl(N,2)).
P oo By de ini ion o Aop
q(X) he e exis s a sequence o non-cons an unc ions un∈
LIP(X)such ha QX
q(un)→Aop
q(X)( ecall (6.10)). By scaling we can suppose ha
unL2∗(m)≡1.In pa icula (un)is bounded in W1,2(X). We dis inguish wo cases.
Subc i ical: q<2∗.By compac ness (see P oposi ion 2.19), up o passing o a subse-
quence, un→us ongly in Lq o some unc ion u∈W1,2(X)such ha , om he lowe
semicon inui y o he Cheege ene gy, QX
q(u)=Aop
q(X). I uis non-cons an (i)holds and
we a e done, so suppose ha uis cons an . Then om he eno maliza ion we mus ha e
u≡1.Mo eo e , since unLq(m),unL2(m)→1andQX
q(un)→Aop
q(X), we deduce
ha |Du|2
L2(m)→0.Conside now he unc ions n:= un−1∈LIP(X), which a e non-
cons an and such ha n→0inW1,2(X). We a e he e o e in posi ion o apply Lemma 6.7
and deduce ha
Aop
q(X)=lim
n→∞QX
q(un)=lim
n
(q−2)´ n−´ ndm2dm
´|D n|2dm≤q−2
λ1,2(X).
Combining his wi h (5.4), we ge ha Aop
q(X)=q−2
λ1,2(X), i.e. (ii)is ue and we conclude
he p oo in his case.
C i ical: q=2∗.We apply he concen a ion-compac ness esul in Theo em 6.1 and
deduce ha up o a subsequence: ei he un→uin L2∗(m) o some u∈W1,2(X)o
unL2(m)→0.In he i s case we a gue exac ly as abo e using Lemma 6.7 and deduce
ha ei he (i)o (ii)holds. Hence we a e le o deal wi h he case unL2(m)→0.F om
he de ini ion o α2(X), o e e yε he e exi s Bεso ha a (2∗,2)-Sobole inequali y wi h
cons an s α2(X)+εand Bεis alid. Hence we ha e
QX
2∗(un)|Dun|2
L2(m)+un2
L2(m)=unL2∗(m)≤(α2(X)+ε)|Dun|2
L2(m)
+Bεun2
L2(m),
which gi es
QX
2∗(un)≤(α2(X)+ε) +Bεun2
L2(m)(|Dun|2
L2(m))−1.
Obse ing ha limn|Dun|2
L2(m)>0 (which ollows om he Sobole inequali y,
un2
L2(m)→0andunL2∗(m)=1) and le ing n→+∞we a i e a Aop
2∗(X)≤
(α2(X)+ε). F om he a bi a iness εwe deduce ha Aop
2∗(X)≤α2(X)and he p oo is
concluded (indeed by de ini ion α2(X)≥Aop
2∗(X)is always ue). 
We can inally come o he p oo o he p incipal esul o his no e.
123

180 Page 48 o 65 F. Nobili, I. Y. Violo
P oo o Theo em 1.9 The “i ” implica ion is di ec as any N-sphe ical suspension, X is so
ha Aop
q(X)=q−2
N. This can be seen om he lowe bound in P oposi ion 5.2 ( ecall also
Theo em 2.11) and he uppe bound gi en in Theo em 1.8.
Fo he “only i ’ implica ion, he esul will ollow om h ee di e en igidi y esul s,
one o each o he al e na i es in Theo em 6.8. Up o scaling he e e ence measu es, we
can suppose m(X)=1.
Case 1: i)inTheo em6.8holds.Le ube henon-cons an unc ionsa is ying(6.14).Obse e
ha we can assume ha uis non-nega i e. We aim o apply he Pólya–Szeg˝o inequali y wi h
he model space INas in Sec . 2.4.Le u∗
N:IN→[0,∞]be he mono one- ea angemen o
u. F om he Pólya–Szeg˝o inequali y in Theo em 2.21 we ha e ha u∗
N∈W1,2(IN,|.|,mN),
uLp(m)=u∗
NLp(mN) o bo h p∈{q,2}and ha |Du∗
N|L2(mN)≤|Du|L2(m).
Combining his wi h (2.17)weha e
u2
Lq(m)=u∗
N2
Lq(mN)≤q−2
N|Du∗
N|2
L2(mN)+u∗
N2
L2(mN)
≤q−2
N|Du|2
L2(m)+u2
L2(m)=u2
Lq(m).
The e o e |Du∗
N|L2(mN)=|Du|L2(m)and, since uis non-cons an , we a e in posi ion
o apply he igidi y o he Pólya–Szeg˝o inequali y o Theo em 2.22 and conclude he p oo
in his case.
Case 2: ii)in Theo em 6.8 holds. We immedia ely deduce ha λ1,2(X)=Nand he
conclusion ollows om he Oba a’s igidi y (Theo em 2.11).
Case 3: iii)in Theo em 6.8 holds. F om Theo em 3.13 and he explici exp ession o
Eucl(N,2)(see (2.3)) we ha e ha
2∗−2
N=Aop
2∗(X)=α2(X)=Eucl(N,2)2
minx∈XθN(x)2/N=2∗−2
Nσ2/N
Nminx∈XθN(x)2/N,
he e o e minx∈XθN=σ−1
N.On he o he hand by he Bishop–G omo inequali y and
iden i y (2.11)
1
σN=in
XθN(x)≥m(X)
N−1,N(diam(X)) =1
N−1,N(diam(X)) ,
which, om he de ini ion o N−1,Nand (2.4) o ces diam(X)=π. The conclusion hen
ollows by he igidi y o he maximal diame e (Theo em 2.12). 
Rema k 6.9 The igidi y esul o Aop
q(M)in he subc i ical ange q<2∗was al eady
obse ed in [76] as a consequence o he ollowing sha pe es ima e due o [50]: o any
n-dimensional Riemannian mani olds M,n≥3, wi h Ric ≥n−1 i holds
Aop
q(M)≤(q−2)
κ(θ) ,∀q∈(2,2∗), (6.15)
whe e κ(θ) := θn+(1−θ)λ1,2(M),λ1,2(M)being he i s non i ial eigen alue and θ=
θ(q)∈[0,1]is a sui able in e pola ion pa ame e . The spec al gap inequali y λ1,2(M)≥n
g an s ha he bound (6.15) imp o es he one o (1.5). Fo e e y q∈(2,2∗), he condi ion
Aop
q(M)=Aop
q(Sn)(=(q−2)/n) o ces κ(θ) =nwhich in u n implies λ1,2(M)=n.By
appealing o he classical Oba a’s Theo em, his a gumen co e s he igidi y o Theo em 1.3
o q<2∗. Ne e heless, his does no ex end o he c i ical exponen : mo e p ecisely
θ(q)→1asq→2∗, hence he quan i y κ(θ) ca ies no in o ma ion on he spec al gap in
his case. 
123
Rigidi y and almos igidi y o Sobole … Page 49 o 65 180
7 Almos igidi y o Aop
7.1 Beha io a concen a ion poin s
The ollowing echnical esul will be needed o he almos - igidi y esul and has he ole
o eplacing in he a ying-space case, he Sobole inequali y wi h cons an s α2(X)+ε, Bε
whichweusedin he ixed-spacecaseo he igidi y(see hep oo o Theo em6.8).Indeedi is
no clea how o con ol he cons an Bεin a sequence o mGH-con e ging spaces. The e o e
we need a mo e p ecise local analysis ha ully exploi s he local Sobole inequali ies in
Theo em 3.8 and P oposi ion 3.12.
Lemma 7.1 (Beha io a concen a ion poin s) Le (Xn,dn,mn,xn),n∈¯
N, be a sequence
o RCD(K,N)spaces K ∈R,N ∈(1,∞), so ha Xn
pmGH
→X∞.Fixp∈(1,N),se
p∗:= pN/(N−p)and assume ha un∈LIPc(Xn)is a sequence sa is ying
unp
Lp∗(mn)≥An|Dun|p
Lp(mn)−Bnunp
Ls(mn),(7.1)
o some cons an s An,Bn≥0uni o mly bounded and s >0so ha s ∈[p,p∗). Assume
u he mo e ha un→0s ongly in L p,unLp∗(mn)=1and ha |un|p∗mnδy0 o some
y0∈X∞in duali y wi h Cbs(Z)(whe e (Z,dZ)is a p ope space ealizing he con e gence
in he ex insic app oach). Then
θN(y0)≤Eucl(N,p)N(lim
nAn)−N/p,(7.2)
meaning ha i θN(y0)=+∞, henlimnAn=0.
P oo We subdi ide he p oo in wo cases.
Case 1: θN(y0)<+∞.
Fix ε<θ
N(y0)/4 a bi a y. Since θN, (y0)→θN(y0)as →0+ he e exis s ¯ =¯ (ε)
such ha
|θN, (y0)−θN(y0)|≤ε, ∀ <¯ .(7.3)
Le δ:= δ(2ε, D,N), wi h D=4,be he cons an gi en by Theo em 3.8 and ix wo adii
,R∈(0,¯ )such ha R<δ
N/K−and <δR.Conside now a sequence yn∈Xnsuch
ha yn→y0. F om he con e gence o he measu es mn o m∞we ha e ha θN, (yn)→
θN, (y0)and θN,R(yn)→θN,R(y0). In pa icula by (7.3) he e exis s ¯n=¯n( ,R,ε)such
ha
|θN,R(yn)−θN(y0)|,|θN, (yn)−θN(y0)|≤2ε, ∀n≥¯n.(7.4)
F om he ini ial choice o ε his also implies ha θN, (yn)/θN,R(yn)≤4 o e e yn≥¯n.
We a e in posi ion o apply Theo em 3.8 and ge ha o e e y n≥¯n
 Lp∗(mn)≤(1+2ε)Eucl(N,p)
(θN(y0)−2ε) 1
N|D |Lp(mn),∀ ∈LIPc(B (yn)). (7.5)
Choose ϕ∈LIP(Z)such ha ϕ=1inBZ
/8(y0), supp(ϕ) ⊂BZ
/4(y0)and 0 ≤ϕ≤1. F om
he assump ions, we ha e ha ´ϕ|un|p∗dmn→1, in pa icula up o inc easing ¯ni holds
ha ´ϕ|un|p∗dmn≥1−ε o all n≥¯n. Mo eo e , again up o inc easing ¯n,weha e ha
dZ(yn,y0)≤ /4 o alln≥¯n, he e o e
1−ε≤ˆB /2(yn)|un|p∗dmn,∀n≥¯n.(7.6)
123
180 Page 50 o 65 F. Nobili, I. Y. Violo
Fo e e y nwe choose a cu -o unc ion ϕn∈LIP(Xn)such ha ϕn=1inB /2(yn),
0≤ϕn≤1, supp(ϕn)⊂LIPc(B (yn)) and Lip(ϕn)≤2/ .Plugging he unc ion unϕn∈
LIPc(B (yn)) in (7.5) and using (7.6) we ob ain
(1−ε)
1
p∗≤unϕnLp∗(mn)≤(1+2ε)Eucl(N,p)
(θN(y0)−2ε) 1
N|Dun|Lp(mn)+2
unLp(mn).
(7.7)
Mo eo e ecalling ha unLp∗(mn)=1 and he assump ion (7.1), om (7.7) we each
(1−ε)
1
p∗A1/p
n|Dun|Lp(mn)−Bnunp
Ls(mn)
≤(1+2ε)Eucl(N,p)
(θN(y0)−2ε) 1
N|Dun|Lp(mn)+2
unLp(mn).
We also obse e ha om he assump ion unLp(mn)→0 and he ac ha unLp∗(mn)=
1, we ha e by ( iii)in P oposi ion 2.18 ha unLs(mn)→0.Finally by (7.7)and he
assump ion unLp(mn)→0 i holds ha limn|Dun|Lp(mn)>0.In pa icula o nbig
enough we can di ide by |Dun|Lp(mn) he abo e inequali y and le ing n→+∞we ge
lim
nA1/p
n≤(1+2ε)Eucl(N,p)
(1−ε)1/p∗(θN(y0)−2ε) 1
N
.
F om he a bi a iness o ε, he conclusion ollows.
Case 2: θN(y0)=∞.
The a gumen is simila o Case 1, bu we will use P oposi ion 3.12 ins ead o Theo em 3.8.
Le M>0 be a bi a y. The e exis s ≤1such ha θN, (y0)≥2M. As abo e we choose a
sequence yn→y0.Fo nbig enough we ha e ha
θN, (yn)≥M.(7.8)
Applying P oposi ion 3.12, om (7.8) we ge ha o e e y nbig enough
 p
Lp∗(B (yn)) ≤CK,N,p
Mp
N|D |p
Lp(B (yn)) +Cp,N p
Lp(B (yn))
p/NMp
N
,∀ ∈LIP(Xn).
(7.9)
Obse ing ha (7.6) is s ill sa is ied wi h ε=1/Mand nbig enough, we can epea he
abo e a gumen , using (7.1) and plugging ϕnunin (7.9), whe e ϕnis as abo e. This leads us
o
lim
nA1/p
n≤CK,N,p
(1−1/M)1/p∗M1
N
,
which om he a bi a iness Mimplies he conclusion. 
7.2 Con inui y o Aop unde mGH-con e gence
In Lemma 4.1, we p o ed ha Sobole embeddings a e s able wi h espec o pmGH-
con e gence. A much mo e in ol ed ask i o p o e ha op imal cons an s a e also
con inuous: indeed, i Xn
mGH
→X∞, in gene al Lemma 4.1 ensu es only ha Aop
q(X∞)≤
limnAop
q(Xn). Wi h he concen a ion compac ness ools de eloped in Sec . 6.1, he
123
Rigidi y and almos igidi y o Sobole … Page 51 o 65 180
“quan i a i e-linea iza ion” esul in Lemma 6.7 and he echnical ool de eloped in he p e-
ious sec ion we can now p o e he mGH-con inui y o Aop
q(Xn)as s a ed in Theo em 1.12,
ha we es a e he e o con enience o he eade .
Theo em 7.2 (Con inui y o Aop
qunde mGH-con e gence) Le (Xn,dn,mn)be a sequence,
n∈N∪{∞}, o compac RCD(K,N)-spaces wi h mn(Xn)=1and o some K ∈R,
N∈(2,∞)so ha Xn
mGH
→X∞. Then, Aop
q(X∞)=limnAop
q(Xn), o e e y q ∈(2,2∗].
P oo By de ini ion o Aop
q(Xn), he e exis s sequence o non-nega i e and non-cons an
unc ions un∈LIP(Xn)sa is ying
un2
Lq(mn)≥An|Dun|2
L2(mn)+un2
L2(mn),(7.10)
ha ing se An:= Aop
q(Xn)−1
n. By scaling in a iance, i is no es ic i e o suppose
unLq(mn)=1 o e e yn∈N. Obse e ha hanks o Lemma 4.1 we al eady ha e ha
0<Aop
q(X∞)≤limnAop
q(Xn),hencewe only need o show ha Aop
q(X)≥limnAop
q(Xn).
To his aim, we dis inguish wo cases.
Subc i ical: q<2∗. I is clea ha Anis uni o mly bounded om below whence he
sequence unhas uni o mly bounded W1,2no ms. Then, by P oposi ion 2.19 and he -
lim inequali y o he Ch2ene gy, he e exis s a (no elabeled) subsequence L2-s ongly
con e ging o some u∞∈W1,2(X∞). Mo eo e , since una e bounded in L2∗, heyalso
con e ge o u∞in Lq-s ong and in pa icula u∞2
Lq(m∞)=1. Suppose i s ha he
unc ion u∞is no cons an , hen we ge
1=u∞2
Lq(m∞)≥lim
n→∞ An|Dun|2
L2(mn)+un2
L2(mn)
(2.18)+L2-s ong ≥lim
n→∞ Aop
q(Xn)|Du∞|2
L2(m∞)+u∞2
L2(m∞).
Since u∞is no cons an his in u n yields limnAop
q(Xn)≤Aop
q(X∞)which is wha we
wan ed.
Suppose now ha u∞is cons an . Then, necessa ily u∞=1. De ine now n:= 1−unand
obse e ha  nW1,2(Xn)→0, which ollows om (7.10) and he ac ha unL2(mn)→1.
Mo eo e om (2.20)weha e ha λ1,2(Xn)a e uni o mly bounded below away om ze o.
The e o e we can apply Lemma 6.7 o deduce ( ecall (6.10) o hede .o QX
q)
lim
n→∞ Aop
q(Xn)=lim
n→∞QXn
q(un)
=lim
n→∞
(q−2)´ n−´ ndmn
2dmn
´|D n|2dmn
≤lim
n→∞
(q−2)
λ1,2(Xn)=(q−2)
λ1,2(X∞),(7.11)
ha ing used, in he las inequali y, he con inui y o he 2-spec al gap (2.20). This combined
wi h (5.4) gi es ha limnAop
q(Xn)≤Aop
q(X∞).
C i ical exponen : q=2∗. Obse e ha we a e now in posi ion o in oke Theo em 6.1
and, up o a u he no elabeled subsequence, we jus need o handle one o he wo di e en
si ua ions I),II) occu ing in Theo em 6.1. I he case I) occu s, we a gue exac ly as in he
Subc i ical: q<2∗case, o conclude ha limnAop
q(Xn)≤Aop
q(X∞). Hence we a e le
wi h si ua ion II), whe e he sequence unde elops a concen a ion poin y0∈X∞. Recalling
123
180 Page 52 o 65 F. Nobili, I. Y. Violo
Lemma 7.1, ei he θN(y0)=∞and limnAop
2∗(Xn)=0o θN(y0)<∞. The i s si ua ion
canno happen, since Aop
2∗(X∞)>0. In he second one ea anging in (7.2)weha e
lim
n→∞ Aop
2∗(Xn)(7.2)
≤Eucl(N,2)2
θN(y0)2/N
(1.7)
≤α2(X∞)≤Aop
2∗(X∞).

7.3 P oo o he almos - igidi y
Combining he igidi y esul o Aop
qwi h he con inui y esul p o ed in he p e ious pa
we can now p o e he almos - igidi y esul o Aop
q.
P oo o Theo em 1.10 We a gue by con adic ion, and suppose ha he e exis s ε>0, q∈
(2,2∗]and a sequence (Xn,dn,mn)o RCD(N−1,N)-spaces wi h mn(Xn)=1so ha
dmGH((Xn,dn,mn), (Y,dY,mY)) > ε, (7.12)
o e e y sphe ical suspension (Y,dY,mY)and limnAop
q(Xn)=q−2
N.Theo em 2.16 ( ecall
ha mn(Xn)=1) ensu es ha up o passing o a non- elabeled subsequence we ha e Xn
mGH
→
X∞, o some RCD(N−1,N)-space (X∞,d∞,m∞)wi h m∞(X∞)=1. Hence (7.12)
implies
dmGH((X∞,d∞,m∞), (Y,dY,mY)) ≥ε, (7.13)
o e e y sphe ical suspension (Y,dY,mY). Finally, by Theo em 1.12 we deduce
Aop
q(X∞)=lim
nAop
q(Xn)=q−2
N.
The e o e, by in oking he igidi y Theo em 1.9,wege ha (X∞,d∞,m∞)is isomo phic
o a sphe ical suspension. This con adic s (7.13) and concludes he p oo . 
Rema k 7.3 The esul s o Theo em 1.10 (and he e o e o Theo em 1.9) ex end di ec ly o
he class o RCD(K,N)spaces o some K>0andN≥2 wi h no malized olume.
Conside an RCD(K,N)space (X,d,m)and de ine (X,d,m):= (X,K
N−1d,m)which
is RCD(N−1,N). Then, since Aop
q(X)=K
N−1Aop
q(X), i is s aigh o wa d o se δ=
δ(K,N,ε,q):= N−1
Kδ(N,ε,q)and ex end he a o emen ioned esul s also o a bi a y
K>0. 
8 Applica ion: The Yamabe equa ion on RCD(K,N)spaces
In his sec ion we apply Theo em 1.4 and he concen a ion compac ness esul s o Sec . 6.1
o s udy he Yamabe equa ion o he RCD(K,N)se ing. In pa icula , we p o e an exis ence
esul o he Yamabe equa ion and con inui y o he gene alized Yamabe cons an s unde
mGH-con e gence, ex ending and imp o ing some o he esul s p o ed in [64] in he case
o Ricci limi s. Fo esul s conce ning he Yamabe p oblem and he Yamabe cons an in
non-smoo h spaces see also [1–3,83,83].
We ecall ha he Yamabe p oblem [99] asks i a compac Riemannian mani old admi s a
con o mal me ic wi h cons an scala cu a u e. This has been comple ely sol ed and shown
123

Rigidi y and almos igidi y o Sobole … Page 53 o 65 180
o be ue a e he wo ks o T udinge , Aubin and Schoen [19,89,95]. We also e e o [77]
o an in oduc ion o his p oblem and o a comple e and sel -con ained p oo o his esul .
The Yamabe p oblem u ns ou o be linked o he so-called Yamabe equa ion:
−u+Su=λu2∗−1,λ∈R,S∈L∞(M), (8.1)
whe e 2∗=2n
n−2. Indeed sol ing he Yamabe p oblem is equi alen o ind a non-nega i e
and non-ze o solu ion o (8.1) o someλ∈Rand wi h S =Scal, he scala cu a u e o M.
In his di ec ion, i is ele an o see ha he Yamabe equa ion is he Eule –Lag ange equa ion
o he ollowing unc ional:
Q(u):= ´|Du|2+S|u|2dVol
u2
L2∗
,u∈W1,2(M) {0},
whe e Vol is he olume measu e o M. One hen de ines he Yamabe cons an as he in imum
o he abo e unc ional:
λS(M):= in
u∈W1,2(M) {0}
Q(u).
A c ucial s ep in he solu ion o he Yamabe p oblem is:
Theo em 8.1 ([19,95,99]) Le M be a compac n-dimensional Riemannian mani old sa is-
ying λS(M)<Eucl(n,2)−2. Then he e is a non-ze o solu ion o (8.1)wi h λ=λS(M).
Recall ha Eucl(n,2)deno es he op imal cons an in he sha p Euclidean Sobole inequali y
(1.1). I has also been p o en by Aubin [20](seealso[77]) ha
λS(M)≤Eucl(n,2)−2(8.2)
always holds.
The ele an poin o ou discussion is ha Theo em 8.1 u ns ou o be linked o he
no ion o op imal Sobole cons an α2(M), in pa icula i is ac ually a co olla y o he ac
ha α2(M)=Eucl(n,2)2( ecall (1.2)). Since we gene alized his las esul o se ing
o compac RCD(K,N)-spaces (see Theo em 1.4), i is na u al o ask i an analogue o
Theo em 8.1 holds also in his singula amewo k. We will posi i ely add ess his in his
pa o he no e.
Capaci y and quasi con inuous unc ions
In he nex sec ion we will use he no ions o capaci y and quasi con inuous unc ions. We
b ie ly ecall he e he needed de ini ions and p ope ies.
Gi en a me ic measu e space (X,d,m), hecapaci y o a se E⊂Xisde inedas
Cap(E):= in { 2
W1,2(X): ∈W1,2(X), ≥1m-a.e. in a neighbo hood o E}.
(8.3)
I u ns ou (see, e.g., [43,P oposi ion 1.7]) ha Cap is a submodula ou e measu e on X and
sa is ies m(E)≤Cap(E) o e e y Bo el se E⊂X.
A unc ion :X→Ris said o be quasi-con inuous i o e e y ε>0 he e exis s a se
E⊂X such ha Cap(E)<εand |X Eis con inuous. We deno e by QC(X) he se o all
equi alence classes-up o Cap-a.e. equali y-o quasi-con inuous unc ions.
123
180 Page 54 o 65 F. Nobili, I. Y. Violo
In [43] i has been p o en ha , in si ua ions whe e con inuous unc ions a e dense in
W1,2(X), he e exis s a unique map
QCR :W1,2(X)→L0(Cap)
ha is linea and such ha QCR( )is ( he Cap-a.e. equi alence class o ) a unc ion which is
quasi con inuous and coincides m-a.e. wi h . Recall ha when X is e lexi e, hen Lipschi z
unc ions a e dense in W1,2(X)(see, e.g., [5,P oposi ion 7.6]), hence he map QCR is
a ailable.
We conclude wi h he ollowing con e gence esul con ained in [43]:
n→ s ongly in W1,2(X)"⇒ up o subsequence QCR( n)→QCR( )Cap-a.e..
(8.4)
8.1 Exis ence o solu ions o he Yamabe equa ion on compac RCD spaces
We s a by cla i ying in which sense (8.1) is in ended and, o his aim, we ix (X,d,m)a
compac RCD(K,N)space o some K∈R,N∈(2,∞)wi h m(X)=1. We will also
deno e by 2∗ he Sobole -exponen de ined as 2∗:= 2N/(N−2). We ix a adon measu e
S in X so ha , o some p>N/2, i sa is ies
S≥gm,g∈Lp(m)and S Cap,(8.5)
whe e Cap deno es he capaci y o X as de ined abo e. We also deno e by |S| he o al
a ia ion o S which o ins ance can be cha ac e ized by he o mula S =S++S−,being
S± he Hahn’s decomposi ion o a gene al signed σ-addi i e measu e. The eason o his
mo e gene al choice o S is he ac ha on RCD(K,N)spaces a “scala cu a u e” ha
is bounded is no na u al ( ecall ha o sol e he Yamabe p oblem one would like o ake
S=Scal). Indeed, equi ing only a syn he ic lowe bound on he Ricci cu a u e, i is mo e
desi able o impose only lowe bounds on S.
Recall ha e e y unc ion u∈W1,2(X)has a well de ined and unique quasi con inuous
ep esen a i e QCR(u)de ined Cap-a.e.. In pa icula , hanks o (8.5), he objec QCR(u)
is also de ined S o |S|-a.e.. To a oid hea y no a ion, o any u∈W1,2(X), we shall deno e
in he sequel by ui s quasi-con inuous ep esen a i e wi hou u he no ice.
The goal is hen o discuss posi i e solu ions u∈D()∩L2(|S|)o
−u=λu2∗−1m−uS,λ∈R.(8.6)
Obse e ha i u∈D()⊂W1,2(X), by he Sobole embedding we ha e ha u∈L2∗(m)
and hus, he igh hand side o (8.6) is a well de ined Radon measu e on X.A solu ion o
his equa ion will be deduced wi h a a ia ional app oach as desc ibed abo e. Mo e p ecisely
we de ine he unc ional QS:W1,2(X) {0}→Rde ined as
u→ QS(u):= ´|Du|2dm+´|u|2dS
u2
L2∗(m)
.
Obse e ha since S ≥gm, wi h g∈Lp(m),p>N/2, he in eg al ´|u|2dS exis s, i.e. i s
alue is well de ined. We hen de ine
λS(X):= in {QS(u):u∈W1,2(X) {0}}
=in {QS(u):u∈W1,2(X), uL2∗(m)=1},(8.7)
123
Rigidi y and almos igidi y o Sobole … Page 55 o 65 180
and claim ha
λS(X)∈(−∞,+∞). (8.8)
Indeed, λS(X)<+∞ as can be seen conside ing cons an unc ions. On he o he hand o
e e y u∈W1,2(X)wi h uL2∗(m)=1, Hölde inequali y yields
QS(u)≥−gLp(m)uL2∗(m)=−gLp(m).
The ul ima e goal o his sec ion is o p o e he ollowing:
Theo em 8.2 Le (X,d,m)be a compac RCD(K,N)space o some K ∈R,N∈(2,∞)
wi h m(X)=1and le Sas in (8.5).I
λS(X)< minXθ2/N
N
Eucl(N,2)2,(8.9)
hen he e exis s a non-nega i e and non-ze o u ∈D()∩L2(|S|)which is a minimum o
(8.7)and sa is ies (8.6).
Wes a byshowing ha (8.6) is he Eule –Lag ange equa ion o he minimiza ion p ob-
lem (8.7).
P oposi ion 8.3 Le (X,d,m)be a compac RCD(K,N)-space o some K ∈R,N∈
(2,∞)wi h m(X)=1and le Sbe as in (8.5). Suppose u ∈W1,2(X)∩L2(|S|)is a
minimize o (8.7)sa is ying uL2∗(m)=1.Then
ˆ∇u,∇ dm=−ˆu dS +λS(X)ˆu2∗−1 dm,∀ ∈LIP(X). (8.10)
P oo We conside o e e y ε∈(−1,1)and ∈LIP(X), he unc ion uε:= u+
ε −1
L2∗(m)(u+ε ), whene e u+ε L2∗(m)is no ze o. I can be seen ha o a ixed
hen uεis well de ined a leas o εclose o ze o. Indeed, he ac ha ´|u|2∗,dm=1
g an s ha u+ε L2∗(m)→1asε→0 (see below) and in pa icula u+ε L2∗(m)does
no anish o |ε|small enough. By minimali y we ha e ( ecall also (2.7))
0≤lim
ε↓0
QS(uε)−QS(u)
ε=lim
ε↓0
1
ε1
I2
ε−1λS(X)+2
I2
εˆ∇u,∇ dm+ˆu dS,
whe e Iε:= u+ε L2∗(m). Fu he mo e, om he elemen a y es ima e ||a+εb|q−|a|q|≤
q|εb||a+εb|q−1+|a|q−1,wi h q=2∗,and he ac ha u, ∈L2∗(m),weha e ha
´|u+ε |qm→1asε→0.Thanks o he same es ima es, he domina ed con e gence
heo em g an s ha
lim
ε↓0
1−I2
ε
ε=2
2∗lim
ε↓0ˆ|u|2∗−|u+ε |2∗
εdm=−2ˆu2∗−1 dm.
A guing analogously conside ing ε↑0gi es(8.10). 
We can now p o e Theo em 8.2 which, hanks o he p e ious p oposi ion, amoun s o he
exis ence o a minimize o (8.7). We will do so using he concen a ion-compac ness ools
de eloped in Sec . 6.1, he e employed wi h a ixed space X.
123
180 Page 56 o 65 F. Nobili, I. Y. Violo
P oo (P oo o Theo em 8.2)Le un∈W1,2(X)be such ha QS(un)→λS(X)and
unL2∗(m)=1. We claim ha una e uni o mly bounded in W1,2(X). Indeed, his can
be seen om he es ima e
ˆ|Dun|2+|un|2dm≤ˆ|Dun|2dm+ˆ|un|2dS
+(1+gLp(m))unL2∗(m)=1+QS(un)+gLp(m),
ob ained combining he Hölde inequali y wi h (8.5). Hence, by compac ness (see P opo-
si ion 2.19), up o a no elabeled subsequence, we ha e un→uin L2(m) o some
u∈W1,2(X). Obse e ha , since u∈W1,2(X),uadmi s a quasi-con inuous ep esen-
a i e (s ill deno ed by u) and hus hanks o (8.5) i makes sense o in eg a e u2agains |S|.
We claim ha u∈L2(|S|)and
ˆu2dS ≤lim
nˆu2
ndS.(8.11)
Obse e i s ha , by (8.5), we ha e S−≤|g|m. In pa icula by he Hölde inequali y,
deno ed by p he conjuga e exponen o p,´u2dS−≤gLp(m)u2
L2p<+∞,since
u∈L2∗(m)by he Sobole embedding, hence u∈L2(S−). Mo eo e , again by he Hölde
inequali y, since un→uin L2(m), we ge ha and un→ualso in L2(S−). To p o e (8.11)
i emains o p o e ha ´u2dS+≤limn´u2
ndS+.Obse e i s ha up o passing o a
u he non- elabeled subsequence we can assume ha he igh hand side is ac ually a limi .
F om Mazu ’s lemma he e exis s a sequence (Nn)⊂Nand numbe s (αn,i)Nn
i=n⊂[0,1]
such ha Nn
i=nαni=1 o e e yn∈Nand n:= Nn
i=nαniuicon e ges o us ongly in
W1,2(X). In pa icula om (8.4) up o a subsequence n→ualso Cap-a.e. and hus, since
S+Cap ( ecall (8.5)), also S+-a.e.. The e o e, om Fa ou’s Lemma and he con exi y o
he L2-no m we ha e
uL2(S+)≤lim
n nL2(S+)≤
Nn

i=n
αniuiL2(S)≤lim
nunL2(S+),
since we a e assuming ha he las limi exis s. This p o es he claim.
We now dis inguish wo cases:
Case 1. λS(X)<0. By lowe semicon inui y o he Cheege -ene gy and (8.11)weha e
0>λ
S(X)=lim
nQS(un)≥ˆ|Du|2dm+ˆu2dS.
In pa icula uis no iden ically ze o and by he lowe semicon inui y o he L2∗(m)-no m
we ha e 0 <uL2∗(m)≤1. Mo eo e , om he abo e we ha e ha ´|Du|2dm+´u2dS
is nega i e, hence
λS(X)≥u−2
L2∗(m)ˆ|Du|2dm+ˆu2dS=QS(u−1
L2∗(m)u).
The e o e u−1
L2∗(m)uis a minimize o QS(u).
Case 2. λS(X)≥0. Recall ha he sequence (un)is uni o mly bounded bo h in L2∗(m)
and in W1,2(X). The e o e since X is compac , again up o a subsequence, |Dun|2mμ and
|un|2∗ν o some μ∈M+
b(X)and ν∈P(X)in duali y wi h C(X). By assump ion he e
exis s ε>0 such ha λS(X)< minXθ2/N
N
Eucl(N,2)2+ε=: λ. We ix one o such ε>0 and de ine
123
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