Renata Grimaldi Stefano
58 pages
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Description

Semianalyticity of isoperimetric profiles Renata Grimaldi, Stefano Nardulli, and Pierre Pansu Introduction The problem The results Proof of Theorem 1. Results in real analytic geometry Proof of Theorem 2 Proof of Theorem 3 Pseudo- bubbles for arbitrary (not necessarily small) volumes Compactness in C2,?- topology Semianalyticity of isoperimetric profiles Renata Grimaldi, Stefano Nardulli, and Pierre Pansu 10 septembre 2009

  • such currents

  • real analytic

  • round disks

  • there remains

  • riemannian manifold

  • compact real analytic

  • pi ?


Sujets

Informations

Publié par
Publié le 01 septembre 2009
Nombre de lectures 70
Langue English

Extrait

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

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