Second order analysis on (P_2(M),W_2)

173 Pages·2012·1.284 MB·English

by  Nicola Gigli

MEMOIRS
of the
American Mathematical Society
Number 1018
Second Order Analysis on
P
( (M), W )
2 2
Nicola Gigli
March 2012 • Volume 216 • Number 1018 (end of volume) • ISSN 0065-9266
American Mathematical Society
Number 1018
Second Order Analysis on
P
( (M), W )
2 2
Nicola Gigli
March2012 • Volume216 • Number1018(endofvolume) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Gigli,Nicola,1974-
Secondorderanalysison(P2(M),W2)/NicolaGigli.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 1018)
“Volume216,number1018(endofvolume).”
Includesbibliographicalreferences.
ISBN978-0-8218-5309-2(alk. paper)
1.Riemannianmanifolds. 2.Geometry,Differential. 3.Spacesofmeasures. I.Title.
QA649.G516 2011
516.3(cid:2).62—dc23 2011047060
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Contents
Introduction vii
Chapter 1. Preliminaries and notation 1
1.1. Riemannian manifolds 2
1.2. The distance W 5
2
1.3. Kantorovich’s dual problem 9
1.4. First order differentiable structure 15
Chapter 2. Regular curves 21
2.1. Cauchy Lipschitz theory on Riemannian manifolds 21
2.2. Definition and first properties of regular curves 24
2.3. On the regularity of geodesics 29
Chapter 3. Absolutely continuous vector fields 39
3.1. Definition and first properties 39
3.2. Approximation of absolutely continuous vector fields 44
Chapter 4. Parallel transport 49
4.1. The case of an embedded Riemannian manifold 49
4.2. Parallel transport along regular curves 52
4.3. Forward and backward parallel transport 57
4.4. On the question of stability and the continuity of μ(cid:2)→P 60
μ
Chapter 5. Covariant derivative 65
5.1. Levi-Civita connection 65
5.2. The tensor N 72
μ
5.3. Calculus of derivatives 77
5.4. Smoothness of time dependent operators 86
Chapter 6. Curvature 93
6.1. The curvature tensor 93
6.2. Related notions of curvature 98
Chapter 7. Differentiability of the exponential map 101
7.1. Introduction to the problem 101
7.2. Rigorous result 103
7.3. A pointwise result 107
Chapter 8. Jacobi fields 113
8.1. The Jacobi equation 113
8.2. Solutions of the Jacobi equation 115
8.3. Points before the first conjugate 119
iii
iv CONTENTS
Appendix A. Density of regular curves 121
Appendix B. C1 curves 131
Appendix C. On the definition of exponential map 139
Appendix D. A weak notion of absolute continuity of vector fields 147
Bibliography 153
Abstract
We develop a rigorous second order analysis on the space of probability mea-
sures on a Riemannian manifold endowed with the quadratic optimal transport
distance W . Our discussion comprehends: definition of covariant derivative, dis-
2
cussionoftheproblemofexistenceofparalleltransport,calculusoftheRiemannian
curvature tensor, differentiability of the exponential map and existence of Jacobi
fields. Thisapproachdoesnotrequireanysmoothnessassumptiononthemeasures
considered.
ReceivedbytheeditorMay19,2009and,inrevisedform,November13,2009.
ArticleelectronicallypublishedonJune21,2011;S0065-9266(2011)00619-2.
2000 MathematicsSubjectClassification. 53C15,49Q20.
Key wordsand phrases. Wessersteindistance,weakRiemannianstructure.
PartiallyfinancedbyKAMfaible,ANR-07-BLAN-0361.
(cid:2)c2011 American Mathematical Society
v

Introduction
The aim of this work is to build a solid theory of second order analysis in the
quadratic Wasserstein space over a Riemannian manifold. To our knowledge, this
topic has been investigated only by few authors: apart from the PhD thesis of the
author [10] and his paper with Ambrosio [1] (both of these works were concerned
withthecaseM =Rd),theonlyotherworkonthetopicofwhichweareawareisof
Lott [14] (who considered generic compact Riemannian manifolds). These works,
written independently at the same time, attack the problem from quite different
viewpoints: Lott was more concerned with the description of the second order
analysis, rather than with its construction, while the author and Ambrosio were
moreinterestedinprovingexistencetheoremsandinfindingtheminimalregularity
assumptions needed by the theory to work. This means that Lott assumed all of
the objects he was working with to be smooth enough to justify his calculations:
withthisapproachhewasabletofindouthowtheformulasforcovariantderivative
andcurvature tensor look like (he also describedthe Poisson structure of the space
P (M) when M has a Poisson structure itself - this is outside the scope of this
2
paper). On the other hand, the author and Ambrosio worked, inthe case M =Rd,
without regularity assumptions and were able to prove the existence of the parallel
transport along a certain dense class of curves; once this was done, they turned
to the definition of covariant derivative and curvature tensor and recovered the
formulas found also by Lott for the case M =Rd.
In this work we use the same approach used in [1, 10] to build the theory of
secondorderanalysisonP (M): ononehand,wereplicatemostoftheresultsvalid
2
in Rd to this more general case, and we recover Lott’s formulas in a more precise
contest which allow us to describe the minimal regularity assumptions needed by
theobjectstobewelldefined. Ontheotherhand,wepushtheinvestigationfurther,
showing,forinstance,thattheproblemofJacobifieldsiswellposed,andidentifying
the solutions of the Jacobi equation with the differential of the exponential map.
The theory we develop works without any regularity assumption on the mea-
suresinvolved: actually,wewillseethatwhatoftenmattersmore,issomeLipschitz
property of the vector fields involved.
Regarding the manifold, we assume that it is C∞, connected, complete and
without boundary.
The paper is structured as follows. In the first Chapter we recall the basic
facts of the theory. Although the material presented here is now standard among
specialists, we preferred to spend a bit of time in the introduction, mainly to fix
the notation and the terminology we will use in the work. In particular, we recall
the definition of tangent space Tan (P (M))⊂L2 (where L2 is the Hilbert space
μ 2 μ μ
vii
viii INTRODUCTION
of Borel tangent vector fields whose squared norm is μ-integrable):
(cid:2) (cid:3)
L2
Tan (P (M)):= ∇ϕ : ϕ∈C∞(M) μ,
μ 2 c
andthefactthatforagivenabsolutelycontinuouscurve(μ )in(P (M),W )there
t 2 2
exists a unique choice - up to a negligible set of times - of v ∈L2 such that
t μt
d
μ +∇·(v μ )=0 in the sense of distributions
dt t t t
and
W (μ ,μ )
(cid:7)v (cid:7) = lim 2 t+h t , a.e. t,
t μt h→0 |h|
where (cid:7)u(cid:7) is the norm of u∈L2. For such a choice it always holds
μ μ
v ∈Tan (P (M)), a.e. t,
t μt 2
and we will call this vector field the velocity vector field of the curve (μ ). These
t
properties of absolutely continuous curves on P (M) are proven for the case M =
2
Rd in [2], and can be generalized to the case of general Riemannian manifolds by,
for example, applying the Nash embedding theorem.
In the second Chapter we introduce the fundamental notion of regular curve
which will be the curves along which we are able to define and study the regularity
ofvectorfields. Inordertointroducethem,wefirstrecallthedefinitionofLipschitz
constant of a vector field v ∈L2:
μ
L(v):=inf lim sup (cid:7)∇ξn(x)(cid:7) ,
op
n→∞x∈M
where the infimum is taken w.r.t. all sequences n (cid:2)→ ξn of smooth vector fields
converging to v in L2 and (cid:7)·(cid:7) is the operator norm. Then we say that a given
μ op
absolutely continuous curve (μ ) is regular if its velocity vector field (v ) satisfies
t t
(cid:4)
1
L(v )dt<∞,
t
(cid:4) 0
1
(cid:7)v (cid:7)2 dt<∞.
t μt
0
Underthisassumptions andusingtheCauchy-Lipschitztheoremitisimmediateto
verifythe existenceand uniqueness of maps T(t,s,x), which we will callflow maps
satisfying
T(t,t,x)=x, ∀t,x∈supp(μ )
t
(cid:5) (cid:6)
d
T(t,s,x)=v T(t,s,x) , ∀t,x∈supp(μ ) a.e. s.
ds s t
The importance of regular curves is twofold: both geometric and algebraic.
From an algebraic point of view, the existence of the flow maps allows the
translation ofvectorfieldsalongaregularcurve. Thismeansthefollowing. Suppose
we have u∈L2 for some t. Then we can define the translated vector field τs(u)∈
μt t
L2 in the following way:
μs (cid:7)
the parallel transport of u(T(s,t,x)) along the curve
τs(u)(x):=
t r (cid:2)→T(s,r,x) from r =t to r =s.
INTRODUCTION ix
Since the parallel transport is always an isometry, it is not hard to check that
τs : L2 → L2 is actually an isometry. Also, from the group property of the flow
mtapsμitteasilyμsfollows that τs◦τr =τs.
r t t
From a geometric point of view, the importance of regular curves is due to the
factthattheangle between tangent spaces variessmoothlyalongthesecurves. This
meansthatifwehaveu∈Tan (P (M))andwetranslateittoobtainτs(u)∈L2
μt 2 t μs
(in general the result of this translation is no more a tangent vector), then τs(u) is
t
‘almost’ tangent for s close to t, in the following quantitative sense:
(cid:8) (cid:8) (cid:9) (cid:10) (cid:10) (cid:11)
(cid:8) (cid:5) (cid:6)(cid:8) (cid:10)(cid:2) (cid:10)
(0.1) (cid:8)τts(u)−Pμs τts(u) (cid:8)μs ≤(cid:7)u(cid:7)μt e tsL(vr)dr −1 ,
where P : L2 → Tan (P (M)) is the orthogonal projection. As we will see, this
μ μ μ 2
fact (explained and proven in Theorem 2.14), will be the key enabler for the proof
of existence of parallel transport.
We conclude the Chapter with the study of the problem of regularity of geodesics.
Here we will spend some time to improve the known result due to Fathi ([8]), that
if (v ) is the velocity vector field of a geodesic between measures with compact
t
support, then L(v )≤ C , for some C.
t min{t,1−t}
What we prove, is that under the same assumptions, there exist functions φ
t
whichare both− C -concave and− C -convex such that v =∇φ on
min{t,1−t} min{t,1−t} t t
supp(μ ) (as we will discuss, this fact is not a direct consequence of Fathi’s result
t
coupled with the tangency of the v ’s).
t
InthethirdChapter,wedefinethenotionofabsolutelycontinuousvectorfields
and study their first properties. The idea is the following: given a vector field (u )
t
along (μ ), i.e. a map t (cid:2)→ u ∈ L2 , we can say that it is absolutely continuous
t t μt
whenever the map
t(cid:2)→τt0(u )∈L2 ,
t t μt0
is absolutely continuous for any t . Observe the key role played by the translation
0
maps: usingthemweareabletocarrytheproblemoftimeregularityofavectorfield
definedondifferentL2 spacesfordifferenttimes,intoaquestionontheregularityof
a curve with values in the fixed Hilbert space L2 . Also, by the group property of
μt0
thetranslationmaps, itisimmediatetocheckthatt(cid:2)→τt0(u )∈L2 isabsolutely
t t μt0
continuous for any t if and only if it is for some t .
0 0
The definitionof(total)derivativeofanabsolutelycontinuous vectorfieldnow
comes pretty naturally: it is sufficient to set
d τt (u )−u
u := lim t+h t+h t
dt t h→(cid:9)0 h(cid:11)
d
=τt τ0(u ) , ∀t ,
0 dt t t 0
where the limit is intended in L2 . It is then immediate to verify that the total
μt
derivative of an absolutely continuous vector field exists for almost every time.
Among other properties of such derivation, an important one is the Leibniz rule:
(cid:14) (cid:15) (cid:14) (cid:15)
(cid:12) (cid:13)
d d d
u1,u2 = u1,u2 + u1, u2 , a.e. t,
dt t t μt dt t t t dt t
μt μt
where (cid:12)·,·(cid:13) is the scalar product in L2. The same idea of translating the vector
μ μ
field, works also for higher order of regularity: for instance, we can say that (u ) is
t