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Publié par | profil-urra-2012 |
Nombre de lectures | 34 |
Langue | English |
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REGULARITYOFOPTIMALTRANSPORTINCURVED
GEOMETRY:THENONFOCALCASE
G.LOEPERANDC.VILLANI
Abstract.
Weexploresomegeometricandanalyticconsequencesofacurvature
conditionintroducedbyMa,TrudingerandWanginrelationtothesmoothnessof
optimaltransportincurvedgeometry.Wediscussaconjectureaccordingtowhich
astrictversionoftheMa–Trudinger–Wangconditionissufficienttoproveregu-
larityofoptimaltransportonaRiemannianmanifold.Weprovethisconjecture
underasomewhatrestrictiveadditionalassumptionofnonfocality;atthesame
time,weestablishthestrikinggeometricpropertythatthetangentcutlocusis
theboundaryofaconvexset.Partialextensionsarepresentedtothecasewhen
thereisno“purefocalization”onthetangentcutlocus.
Contents
1.Introduction
2.VariousformsoftheMa–Trudinger–Wangcondition
3.MetricconsequencesoftheMa–Trudinger–Wangcondition
4.Uniformregularity
5.Convexityofinjectivitydomains
6.From
c
-convexityto
C
1
regularity
7.Stay-awayproperty
8.Ho¨ldercontinuityofoptimaltransport
9.Finalcommentsandopenproblems
AppendixA.Uniformconvexity
AppendixB.Semiconvexity
AppendixC.Differentialstructureofthetangentcutlocus
AppendixD.Acounterexample
References
1
29115191033363930424342535
2
G.LOEPERANDC.VILLANI
1.
Introduction
Thispaperhastwosides:ontheonehand,itisaworkonthesmoothnessof
optimaltransport;ontheotherhand,itisaworkonthestructureofthecutlocus.
Thelattercouldbediscussedindependentlyoftheformer,butsincetheinitial
motivationwasinoptimaltransporttheory,andsincebothfeaturesareintimately
entangled,weshallpresentbothproblematicstogether.Ourintroductionisreduced
totheminimumthatthereadershouldknowtounderstandthepaper;butmuch
moreinformationcanbefoundinthebooks[29,30];especially[30,Chapter12]is
alongandself-containedintroductiontotheregularityofoptimaltransport.
1.1.
Regularityofoptimaltransport:backgroundandmainresult.
After
Caffarelli[2,3,4]andUrbas[28]studiedthesmoothnessofoptimaltransportmaps
forthequadraticcostfunctionin
R
n
,theproblemnaturallyarosetoextendthese
resultstomoregeneralcostfunctions[29,Section4.3].Inthispaper,weshallonly
considertheimportantcasewhenthecostisthesquaredgeodesicdistanceona
Riemannianmanifold
M
;thiscostfunction,firststudiedbyMcCann[24],hasmany
applicationsinRiemanniangeometry[30,PartII].
Therewasalmostnoprogressonthesmoothnessissuebeforetheintroductionof
theMa–Trudinger–Wangtensor[22].Let
M
beaRiemannianmanifold,whichas
intherestofthispaperwillimplicitlybeassumedtobesmooth,connectedand
complete.Let
TM
=
∪
(
{
x
}×
T
x
M
)standforthetangentbundleover
M
,and
letcut(
M
)=
∪
(
{
x
}×
cut(
x
))denotethecutlocusof
M
.TheMa–Trudinger–
Wang(MTW)tensor
S
canbedefinedon
T
(
M
×
M
\
cut(
M
))asfollows[30,
Definition12.26].Let(
x,y
)
∈
M
×
M
\
cut(
M
),takecoordinatesystems(
x
i
)
1
≤
i
≤
n
,
(
y
j
)
1
≤
j
≤
n
around
x
and
y
respectively;set
c
(
x
′
,y
′
)=
d
(
x
′
,y
′
)
2
/
2,where
d
isthe
geodesicdistanceon
M
,andnotethat
c
is
C
∞
around(
x,y
).Write
c
i
(resp.
c
,j
)
forthepartialderivativewithrespectto
x
i
(resp.
y
j
),evaluatedat(
x,y
);
c
i,j
for
themixedsecondderivativewithrespectto
x
i
and
y
j
,etc.;andwrite(
c
i,j
)forthe
componentsoftheinverseof(
c
i,j
),alwaysevaluatedat(
x,y
).Thenforany
ξ
∈
T
x
M
,
η
∈
T
y
M
,
3X(1.1)
S
(
x,y
)
(
ξ,η
):=
c
ij,r
c
r,s
c
s,kℓ
−
c
ij,kℓ
ξ
i
ξ
j
η
k
η
ℓ
.
2
ijkℓrs
AccordingtoLoeper[20],KimandMcCann[16],thisformuladefinesacovariant
tensor.Moreover,asnotedin[20],if
ξ
and
η
areorthogonalunitvectorsin
T
x
M
,
then
S
(
x,x
)
(
ξ,η
)coincideswiththesectionalcurvatureat
x
alongtheplane
generatedby
ξ
and
η
[30,ParticularCase12.29].
REGULARITYOFOPTIMALTRANSPORT
3
Themainassumptionusedin[22,26,27]isthat
X(1.2)
S
(
x,y
)
≥
K
|
ξ
|
2
|
η
|
2
whenever
c
i,j
ξ
i
η
j
=0
,
jiwhere
K
isapositiveconstant(strongMTWcondition)or
K
=0(weakMTW
condition).Condition(1.2)impliesthatthesectionalcurvatureof
M
isbounded
belowby
K
.Loeper[21]showedthattheroundsphere
S
n
satisfies(1.2)forsome
K>
0(seealso[32]).
Thereisbynowplentyofevidencethattheseconditions,complicatedastheyseem,
arenaturalassumptionstodeveloptheregularitytheoryofoptimaltransport.In
particular,Loeper[20]showedhowtoconstructcounterexamplestotheregularityif
theweakMTWconditionisnotsatisfied.Thefollowingprecisestatementisproven
in[30,Theorem12.39];volstandsfortheRiemannianvolumemeasure.
Theorem1.1
(Necessaryconditionfortheregularityofoptimaltransport)
.
Let
M
beaRiemannianmanifoldsuchthat
S
(
x,y
)
(
ξ,η
)
<
0
forsome
x,y,ξ,η
.Thenthere
are
C
∞
positiveprobabilitydensities
f
and
g
on
M
suchthattheoptimaltransport
mapfrom
(
dx
)=
f
(
x
)vol(
dx
)
to
ν
(
dy
)=
g
(
y
)vol(
dy
)
,withcostfunction
c
=
d
2
,
isdiscontinuous.
(Forthesakeofpresentation,thistheoremisstatedin[30]underacompactness
assumption,buttheproofgoesthrougheasilytononcompactmanifolds.)
Conversely,smoothnessresultshavebeenobtainedundervarioussetsofassump-
tionsincludingeithertheweakorthestrongMTWcondition[10,16,19,20,22,27];
suchresultsarereviewedin[30,Chapter12].Forinstance,[22]furnishesinteriora
prioriregularityestimates(say
C
1
)ontheoptimaltransportmap,providedthatthe
optimaltransportplanissupportedinaset
D
⊂
M
×
M
suchthat(a)
c
isuniformly
smooth(say
C
4
)in
D
;(b)allsets(exp
x
)
−
1
(
D
x
)and(exp
y
)
−
1
(
D
y
)areconvex(in
T
x
M
and
T
y
M
respectively),where
D
x
=
{
y
;(
x,y
)
∈
D
}
,
D
y
=
{
x
;(
x,y
)
∈
D
}
,
andexpstandsfortheRiemannianexponential.(Themeaningofthenotation
(exp
x
)
−
1
willberecalledafterDefinition1.2.)Butsofar(a)and(b)havebeen
provenonlyinparticularcasessuchasthesphere
S
n
,oritsquotientslikethereal
projectivespace
RP
n
=
S
n
/
{±
Id
}
[16,20].ThereisalsoapartialresultbyDelanoe¨
andGe[6]workingonperturbationsofthesphereandassumingcertainrestrictions
onthesizeofthedata.
Inthispaperwesuggestthata(possiblyslightlymodified)strictformoftheMTW
condition
alone
isanaturalsufficientconditionforregularity.Weshallprovethis
conjectureonlyunderasimplifyingnonfocalityassumptionwhichwenowexplain.
Tobeginwith,letusintroducesomenotation:
4
G.LOEPERANDC.VILLANI
Definition1.2
(injectivitydomain,tangentcutandfocalloci)
.
Let
M
beaRie-
mannianmanifoldand
x
∈
M
.Forany
ξ
∈
T
x
M
,
|
ξ
|
=1,let
t
C
(
ξ
)bethefirsttime
′t
suchthat(exp
x
(
sξ
))
0
≤
s
≤
t
′
isnotminimizingfor
t>t
;andlet
t
F
(
ξ
)
≥
t
C
(
ξ
)be
thefirsttime
t
suchthat
d
tξ
exp
x
(thedifferentialofexp
x
at
tξ
)isnotone-to-one.
Wedefine
no
I(
x
)=
tξ
;0
≤
t<t
C
(
ξ
)=injectivitydomainat
x
;
onTCL(
x
)=
tξ
;
t
=
t
C
(
ξ
)=
∂
I(
x
)=tangentcutlocusat
x
;
onTFL(
x
)=
tξ
;
t
=
t
F
(
ξ
)=(first)tangentfocallocus
.
L