Metric Diffusion Along Foliations

63 Pages·2017·1.252 MB·English

by  Szymon M. Walczak (auth.)

SPRINGER BRIEFS IN MATHEMATICS
Szymon M. Walczak
Metric Diff usion
Along Foliations
123
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Szymon M. Walczak
Metric Diffusion Along
Foliations
123
SzymonM.Walczak
NationalScienceCenter
Kraków,Poland
FacultyofMathematics
andComputerScience
UniversityofŁódz´
Łódz´,Poland
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-3-319-57516-2 ISBN978-3-319-57517-9 (eBook)
DOI10.1007/978-3-319-57517-9
LibraryofCongressControlNumber:2017939353
MathematicsSubjectClassification:53C12,53C23
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Contents
1 WassersteinDistance ......................................................... 1
1.1 OptimalTransportationProblem........................................ 1
1.2 WassersteinDistance..................................................... 5
2 FoliationsandHeatDiffusion................................................ 11
2.1 BasicFacts ............................................................... 11
2.2 Holonomy ................................................................ 14
2.3 HarmonicMeasuresandHeatDiffusion................................ 16
3 CompactFoliations ........................................................... 21
3.1 Examples ................................................................. 21
3.2 TopologyoftheLeafSpace ............................................. 24
4 MetricDiffusion............................................................... 31
4.1 MetricDiffusion ......................................................... 31
4.2 MetricDiffusionAlongCompactFoliations ........................... 35
5 MetricDiffusionforNon-compactFoliations:Remarks ................. 49
References.......................................................................... 53
Index................................................................................ 55
vii
Introduction
Inthelastfewyears,theauthorofthisbookwaslookingforakindofadeformation
ofametriconafoliatedRiemannianmanifold,whichwillpreserve,insomesense,
the set of compact leaves. More precisely, given a foliation F on a compact
Riemannianmanifold.M;g/,wearelookingforadeformationD ofthestructureg
t
orinducedRiemannianmetricd suchthatthelimitofthesetC ofcompactleaves
under,forexample,theGromov–HausdorffconvergenceishomeomorphictoC=F.
Thefirstapproachtothisproblembywarpedfoliations[22,23]wasn’tsuccess-
ful.Brieflyspeaking,foracompactfoliatedRiemannianmanifold.M;F;g/,anda
smoothfunctionf WM !.0;1/constantalongtheleavesofF themetricinduced
bytheRiemannianstructureg definedby
f
g .v;w/Df2g.v;w/ forv;wtangenttoF;
f
g .v;w/Dg.v;w/ ifatleastoneofv;wisperpendiculartoF
f
iscalledthewarpedmetric.f iscalledthewarpingfunction,whilethemetricspace
.M;d /,whered isametricinducedbyg ,thewarpedfoliation.
f f f
Let f be a sequence of warping functions converging uniformly to zero on a
n
ReebfoliationR ofanannulusADfx2R2 W1(cid:2)kxk(cid:2)2g.In[23],thefollowing
wasproved:
TheoremI.1 The limit of warped Reeb foliation limGn!H1.M;dfn/n2N under the
Gromov–Hausdorffconvergenceisasingleton.
The above example shows that the two boundary compact leaves, which are
linked by a non-compact leaf, collapse, while warping, to the same point of the
limit. The same can be observed for compact foliations. In [22] it was shown that
the non-empty bad set of the compact foliation described by Epstein and Vogt in
[10]collapses,inGromov–Hausdorfftopology,tothesingleton.
The approach presented here uses more advanced tools. First, observe that the
Wassersteindistanced (see[19])oftwoDiracmeasuresonaPolishmetricspace
W
.X;d/ concentrated in x;y 2 X is equal to d.x;y/. Having d , with foliated heat
W
ix
x Introduction
diffusionoperatorsD (introducedbyL.Garnettin[11])wedefine,forgiventime
t
t (cid:3) 0, the metric Dd diffused along a foliation on a Riemannian manifold M
t
equippedwithafoliationF astheWassersteindistanceofDiracmeasuresdiffused
at time t. It occurs that Dd defines the same topology on M as the initial one
t
(Theorem4.1).
In further considerations, we concentrate our attention on compact foliations.
Thereasonistopological,namely,thattheleaveshavenoends,sothatthenatureof
theheatkerneliswellknown.Noticethatinthecaseofthemanifoldswithendsin
generalthereisnoknowledgeontheheatkernelbehaviour.Ontheotherhand,the
existenceofanon-emptybadsetcanproduceanumberofproblemsonconvergence
ofthefamily.M;Dd/.
t
The main purpose for this work is to try to answer the question, whether the
family.M;Dd/convergesornotind toaclosedsubsetofP.M/.Thiswillbe
t WH
themainsubjectofthestudiespresentedhere.
There is one more problem to settle. In the case of warped foliations, the
Gromov–Hausdorffconvergencewasused.Inthecaseofmetricdiffusionsomething
elseismoreappropriate.
LetusdenotebyP.M/thesetofallBorelprobabilitymeasuresonM.Thereis
anaturalisometricembeddingof.M;Dd/into.P.M/;d /definedby
t W
(cid:2) WM 3x7!Dı 2P.M/:
t t x
Hence,fort;s(cid:3)0wecanconsider.M;Dd/and.M;D d/astheclosedsubsetsof
t s
P.M/ (which is compact if M is so), and we shall use the Hausdorff distance of
(cid:2).M/and(cid:2) .M/in.P.M/;d /.Withtheaboveinmind,wewillwrite.M;Dd/or
t s W t
simplyM insteadof.(cid:2).M/;d /.
t t W
The book was planned to provide all necessary facts needed to understand
the metric diffusion along compact foliations, that is some basic facts from the
optimal transportation theory and the theory of foliations. Chapter 1 is devoted to
theWassersteindistance,KantorovichDualityTheorem,andthemetrizationofthe
weak-* topology by the Wasserstein distance. Moreover, we prove some technical
lemmasusedinfurtherconsiderations.InChapter2,wepresentsomebasicsabout
foliations, holonomy, and heat diffusion. They are necessary to understand the
notion of the metric diffusion. The compact foliations are discussed in Chapter 3
wherewerecallthesefactswhichareessentialforfurtherconsiderations.
The main results are presented in Chapter 4. We define the metric diffusion
Dd and study the topology of the metric space .M;Dd/. The remaining pages
t t
are devoted to the limits of diffused metrics along compact foliations. We prove
the necessary conditions for Wasserstein–Hausdorff convergence of the metric
diffusedalongcompactfoliationwithnon-emptyEpsteinhierarchy.Thefirstresult
(Theorem4.6)providesaninformationaboutthegeometryofthecompactfoliation,
that is it describes the leaf volume growth near connected components of the bad
sets. The second (Theorem 4.7) is rather measure-theoretic one. Enhancement of
the necessary conditions presented in Theorem 4.6 and Theorem 4.7 allows us to
formulatethesufficientconditionofWasserstein–Hausdorffconvergenceofmetrics
diffusedalongcompactfoliationsofdimensiononewithfiniteEpsteinhierarchy.
Introduction xi
Asakindofsupplement,wepresentsomefactsaboutthemetricdiffusionalong
non-compact foliations. We provide the full description of the limit for metrics
diffused along foliation with at least one compact leaf on the two-dimensional
torusT2.
I would like to express my thanks to Prof. Jesus A. Alvarez-Lopez from the
University of Santiago de Compostela for the initial idea of metric diffusion and
a number of fruitful discussions. Thanks are also due to Prof. PawełWalczak and
my colleagues from the University of Łódz´, namely Wojciech Kozłowski, Kamil
Niedziałomski, and Krzysztof Andrzejewski for helpfuldiscussions and important
remarks, and to Andrzej Komisarski for some probabilistic explanations. I’m also
grateful to Prof. Takashi Tsuboi from the University of Tokyo, who gave me the
opportunitytospendsometimeinTokyo,wherethisbookhasgotthecurrentshape.
Last but not least, I would like to acknowledge my gratitude to the University of
Łódz´,Prof.RyszardPawlak,theDeanoftheFacultyofMathematicsandComputer
Science,andtotheNationalCenterofScience(NCN,grant#6065/B/H03/2011/40)
forfinancialsupportduringtheresearchonmetricdiffusion.
Chapter 1
Wasserstein Distance
TheWassersteindistanceofBorelprobabilitymeasuresplaysaveryimportantrole
in metric diffusion along foliations. In this chapter we present some foundations
of the Optimal Transportation Theory, that is, the Kantorovich Duality Theorem
for optimal transportation problem and the definition of the Wasserstein distance,
togetherwiththeweak-(cid:4)topologymetrizationtheoremforthesetP.X/ofBorel
probabilitymeasuresonacompactmetricspaceX.Thechapterisclosedbysome
technicallemmasusedinthelaterconsiderations.
The full theory of the Optimal Transportation can be found in the excellent
monographs by C. Villani [18, 19]. We only present the main aspects, which are
needed to understand the theory of metric diffusion. Full description and precise
proofscanbefoundinthemonographsmentionedabove.
Ifnotmentioned,wewillalwaysassumethattopologicalspaces,metricspaces
ormanifoldsweconsiderarecompact.Forfullgenerality,referto[18]and[19].
1.1 OptimalTransportationProblem
The Wasserstein distance of measures on compact metric spaces comes from the
Monge–Kantorovich problem of existence of an optimal transportation plan. In
otherwords,imaginethatyouhaveapileofsomethingwhichshouldfillperfectly
a hole in the ground. Of course, both the pile and the hole have the same volume.
Therearemanywaysofdoingthis,buttheoneweareinterestedinisthatwiththe
lowestcost.Thecostcanbeunderstoodasaneffortneededtodothejob.
WenowrestrictourattentiontoBorelprobabilitymeasures(cid:3)and(cid:4) oncompact
spacesX andX0,respectively.ABorelprobabilitymeasure(cid:5) onX(cid:5)X0 iscalleda
couplingwithmarginals(cid:3)and(cid:4) (Figure1.1)ifitsatisfies
(cid:5).A(cid:5)X0/D(cid:3).A/and(cid:5).X(cid:5)B/D(cid:3).B/:
©TheAuthor(s)2017 1
S.M.Walczak,MetricDiffusionAlongFoliations,SpringerBriefsinMathematics,
DOI10.1007/978-3-319-57517-9_1