Renata Grimaldi Stefano

10septembre2009

RenataGrimaldi,StefanoNardulli,andPierrePansu

Semianalyticityofisoperimetricprofiles

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
Inthistalk,
M
isacompactrealanalyticRiemannianmanifold,
ifitisnototherwisespecified.Weareconcernedwiththe
regularityofthe
isoperimetricprofile
of
M
.
Itisshownthat,indimensions
<
8,isoperimetricprofilesof
compactrealanalyticRiemannianmanifoldsaresemi-analytic.

Abstract

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
Given0
<
v
<
vol
(
M
)
,considerallintegralcurrentsin
M
with
volume
v
.Define
I
M
(
v
)
astheleastupperboundofthe
boundaryvolumesofsuchcurrents.Inthisway,onegetsa
function
I
M
:(
0
,
vol
(
M
))
→
R
+
calledthe
isoperimetricprofile
of
M
.Infact,foreach0
<
v
<
vol
(
M
)
,thereexistcurrentsin
M
withvolume
v
andboundaryvolume
I
M
(
v
)
.Suchminimizing
currentswillbecalled
bubbles
,forshort.

Definitionoftheisoperimetricprofilefunction

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
S√
4
π
v
for0
<
v
≤
4
π,
I
M
(
v
)=
4
π
for4
π
≤
v
≤
4
π
(
π
−
1
)
,
p4
π
(
4
π
2
−
v
)
for4
π
(
π
−
1
)
≤
v
<
4
π
2
.

Hereisatypicalexample.Let
S
denotethecircleoflength2
π
.
Let
M
=
S
×
S
.Thentheisoperimetricprofileof
M
iseasily
computedtobe

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaime
Thisisprovenasfollows.In2dimensions,theboundariesof
thesebubblesaresmooth,theyhaveconstantgeodesic
curvature,thereforetheylifttodisjointunionsofcirclesof
equalradiiorlinesin
R
2
=
M
˜
.Itfollowsthatbubblesareeither
rounddisksorannuliboundedbyparallelgeodesics,or
complementsofsuch.Thereremainstominimizeboundary
lengthamongthesethreefamilies.

Sketchofproof

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
Theanswerisyesmodulosomesupplementaryassumption.
Thishasbeenprovenin[Pan98]indimension2.

Forgeneralrealanalyticmanifolds,isittruethatbubblesfall
intofinitelymanyanalyticfamilies,andthattheprofileis
piecewiseanalytic?

Question

Firstquestion

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
LetMbeacompactrealanalyticRiemannianmanifold.There
exists
>
0
suchthatI
M
isrealanalyticon
(
0
,
)
.

Theorem(Grimaldi-N.-Pansu,2009)

First,inaneighborhoodofzero.

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
ygTheisoperimetricprofileofEuclideanspace
R
n
is
I
R
n
(
v
)=
n
(
ω
n
)
1
/
n
v
(
n
−
1
)
/
n
,where
ω
n
isthevolumeoftheunit
ballin
R
n
.Inacurvedmanifold,
I
M
(
v
)
∼
n
(
ω
n
)
1
/
n
v
(
n
−
1
)
/
n
as
v
tendsto0.

olopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
..nc=n1−n]∙∙∙+)nB(loVnr[∙∙∙+)1−nS(aerA1−n)a(Ilimsup
n
−
1
≤
c
n
.
0→ana

rSketchoftheproof

0Proposition

→ehtebIteL

rM
.Then

pDemonstration:
Fixapoint
p
∈M
.

ulimsup
I
(
a
)
≤
limsup
Area
(
∂
B
(
p
,
r
(
a
)))
a
→
0
a
nn
−
1
a
→
0
Vol
(
B
(
p
,
r
(
a
)))
nn
−
1

swith
r
(
a
)
suchthat
Vol
(
B
(
p
,
r
(
a
)))=
a
.Changingvariablesin
thelimits,wefind

mAaer

iln1−n))r,p(B(loV))r,p(B∂(aerA0→rpusmil=n1−n)))a(r,p(B(loV)))a(r,p(B∂(0→apusmilfoelfiorpcirtemireposiygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoyticitylanaimeS
itylanaimeSWehaveonlyapartialanswer.

ForacompactanalyticRiemanniann-manifold,isI
M
(
v
)
an
analyticfunctionofv
1
/
n
on
[
0
,
)
?

Question

Secondquestion

ygolopot-α,2CnissentcapmoCsemulov)llamsylirassecenton(yrartibrarofselbbub-oduesP3meroehTfofoorP2meroehTfofoorPyrtemoegcitylanalaernistluseR.1meroehTfofoorPstluserehTmelborpehTnoitcudortnIusnaPerreiPdna,illudraNonafetS,idlamirGataneRselfiorpcirtemireposifoytic