REGULARITY OF OPTIMAL TRANSPORT IN CURVED GEOMETRY: THE NONFOCAL CASE

Description

REGULARITY OF OPTIMAL TRANSPORT IN CURVED GEOMETRY: THE NONFOCAL CASE G. LOEPER AND C. VILLANI Abstract. We explore some geometric and analytic consequences of a curvature condition introduced by Ma, Trudinger and Wang in relation to the smoothness of optimal transport in curved geometry. We discuss a conjecture according to which a strict version of the Ma–Trudinger–Wang condition is sufficient to prove regu- larity of optimal transport on a Riemannian manifold. We prove this conjecture under a somewhat restrictive additional assumption of nonfocality; at the same time, we establish the striking geometric property that the tangent cut locus is the boundary of a convex set. Partial extensions are presented to the case when there is no “pure focalization” on the tangent cut locus. Contents 1. Introduction 2 2. Various forms of the Ma–Trudinger–Wang condition 9 3. Metric consequences of the Ma–Trudinger–Wang condition 11 4. Uniform regularity 15 5. Convexity of injectivity domains 19 6. From c-convexity to C1 regularity 30 7. Stay-away property 33 8. Holder continuity of optimal transport 36 9. Final comments and open problems 39 Appendix A. Uniform convexity 40 Appendix B. Semiconvexity 42 Appendix C. Differential structure of the tangent cut locus 43 Appendix D. A counterexample 52 References 53 1

• riemannian manifold


• transport


• any ? ?


• question nontrivial


• collects various sufficient


• optimal transport


• mtw condition


• topology —


• various forms

Sujets

• Riemannian manifold
• Transport
• Transportation theory

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 34
• Langue: English

Extrait

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