An intrinsic parallel transport in Wasserstein space

0.18 MB·

by  John Lott

AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE
JOHN LOTT
7
1
0
2 Abstract. If M is a smooth compact connected Riemannian manifold, let P(M) denote
theWassersteinspaceofprobabilitymeasuresonM. We describeageometricconstruction
n
a of parallel transport of some tangent cones along geodesics in P(M). We show that when
J everything is smooth, the geometric parallel transport agrees with earlier formal calcula-

9 tions.

]
G
D
. 1. Introduction
h
t
a
Let M be a smooth compact connected Riemannian manifold without boundary. The
m
space P(M) of probability measures of M carries a natural metric, the Wasserstein metric,
[
and acquires the structure of a length space. There is a close relation between minimizing

1
geodesics in P(M) and optimal transport between measures. For more information on this
v
7 relation, we refer to Villani’s book [13].
9
2 Otto discovered a formal Riemannian structure on P(M), underlying the Wasserstein
2 metric [10]. One can do formal geometric calculations for this Riemannian structure [6]. It
0
is an interesting problem to make these formal considerations into rigorous results in metric
.
1
geometry.
0
7
If M has nonnegative sectional curvature then P(M) is a compact length space with
1
: nonnegative curvature in the sense of Alexandrov [8, Theorem A.8], [12, Proposition 2.10].
v
Hence one can define the tangent cone T P(M) of P(M) at a measure µ ∈ P(M). If µ is
i µ
X
absolutelycontinuouswithrespect tothevolumeformdvol thenT P(M)isaHilbertspace
M µ
r
[8, Proposition A.33]. More generally, one can define tangent cones of P(M) without any
a
curvature assumption on M, using Ohta’s 2-uniform structure on P(M) [9]. Gigli showed
that T P(M) is a Hilbert space if and only if µ is a “regular” measure, meaning that it
µ
gives zero measure to any hypersurface which, locally, is the graph of the difference of two
convex functions [3, Corollary 6.6]. For examples of tangent cones at nonregular measures,
if S is an embedded submanifold of M, and µ is an absolutely continuous measure on S,
then T P(M) was computed in [7, Theorem 1.1].
µ
If γ : [0,1] → M is a smooth curve in a Riemannian manifold then one can define the
(reverse) parallel transport along γ as a linear isometry from T M to T M. If X is a
γ(1) γ(0)
finite-dimensional Alexandrov space then the replacement of a tangent space is a tangent
cone. If one wants to define a parallel transport along a curve c : [0,1] → X, as a map from
T X to T X, then there is the problem that the tangent cones along c may not look
c(1) c(0)
much alike. For example, the curve c may pass through various strata of X. One can deal
Date: January 6, 2017.
Research partially supported by NSF grant DMS-1207654and a Simons Fellowship.
1
2 JOHN LOTT
with this problem by assuming that c is in the interior of a minimizing geodesic. In this
case, Petrunin proved the tangent cones along c are mutually isometric, by constructing a
parallel transport map [11]. His construction of the parallel transport map was based on
passing to a subsequential limit in aniterative construction along c. It is not known whether
the ensuing parallel transport is uniquely defined, although this is irrelevant for Petrunin’s
result.
In the case of a smooth curve c : [0,1] → P∞(M) in the space of smooth probability
measures, one cando formalRiemannian geometry calculations onP∞(M) to writedown an
equation for parallel transport along c [6, Proposition 3]. It is a partial differential equation
in terms of a family of functions {η } . Ambrosio and Gigli noted that there is a weak
t t∈[0,1]
version of this partial differential equation [1, (5.9)]. By a slight extension, we will define
weak solutions to the formal parallel transport equation; see Definition 2.13.
Petrunin’s construction of parallel transport cannot work in full generality on P(M),
since Juillet showed that there is a minimizing Wasserstein geodesic c with the property
that the tangent cones at measures on the interior of c are not all mutually isometric [5].
However one can consider applying the construction on certain convex subsets of P(M).
We illustrate this in two cases. The first and easier case is when c is a Wasserstein geodesic
of δ-measures (Proposition 3.1). The second case is when c is a Wasserstein geodesic of
absolutely continuous measures, lying in the interior of a minimizing Wasserstein geodesic,
and satisfying a regularity condition. Suppose that ∇η ∈ T P(M) is an element of the
1 c(1)
tangent cone at the endpoint. Here ∇η ∈ L2(TM,dc(1)) is a square-integrable gradient
1
vector field on M and η is in the Sobolev space H1(M,dc(1)). For each sufficiently large
1
integer Q, we construct a triple
(1.1) (∇η ,∇η (0),∇η (1)) ∈ L2([0,1];L2(TM,dc(t)))⊕L2(TM,dc(0))⊕L2(TM,dc(1))
Q Q Q
with ∇η (1) = ∇η , which represents an approximate parallel transport along c.
Q 1
Theorem 1.2. Suppose that M has nonnegative sectional curvature. A subsequence of
{(∇η ,∇η (0),∇η (1))}∞ converges weakly to a weak solution (∇η ,∇η ,∇η ) of
Q Q Q Q=1 ∞ ∞,0 ∞,1
the parallel transport equation with ∇η = ∇η . If c is a smooth geodesic in P∞(M),
∞,1 1
η is smooth, and there is a smooth solution η to the parallel transport equation (2.6) with
1
η(1) = η , then lim (∇η ,∇η (0),∇η (1)) = (∇η,∇η(0),∇η(1)) in norm.
1 Q→∞ Q Q Q
Remark 1.3. In the setting of Theorem 1.2, we can say that ∇η is the parallel transport
∞,0
of ∇η along c to T P(M).
1 c(0)
Remark 1.4. We are assuming that M has nonnegative sectional curvature in order to
apply some geometric results from [11]. It is likely that this assumption could be removed.
Remark 1.5. A result related to Theorem 1.2 was proven by Ambrosio and Gigli when
M = Rn [1, Theorem 5.14], and extended to general M by Gigli [4, Theorem 4.9]. As
explained in [1, 4], the construction of parallel transport there can be considered to be
extrinsic, in that it is based on embedding the (linear) tangent cones into a Hilbert space
and applying projection operators to form the approximate parallel transports. Although
we instead use Petrunin’s intrinsic construction, there are some similarities between the two
constructions; see Remark 3.32. We use some techniques from [1], especially the idea of a
weak solution to the parallel transport equation.
AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE 3
Remark 1.6. Besides its inherent naturality, the intrinsic construction of parallel transport
givenhereislikelytoallowforextensions. Forexample, usingtheresultsof[7], itseemslikely
thatPetrunin’sconstructioncouldbeextendedtodefineparalleltransportalongWasserstein
geodesics of absolutely continuous measures on submanifolds of M. In the present paper we
have done this when the submanifolds have dimension zero or codimension zero.
The structure of this paper is as follows. In Section 2 we discuss weak solutions to the
parallel transport equation. In Section 3 we prove Theorem 1.2.
I thank Takumi Yokota and Nicola Gigli for references to the literature.
2. Weak solutions to the parallel transport equation
Let M be a compact connected Riemannian manifold without boundary. Put
∞ ∞
(2.1) P (M) = {ρ dvol : ρ ∈ C (M),ρ > 0, ρ dvol = 1}.
M M
Z
M
Given φ ∈ C∞(M), define a vector field V on P∞(M) by saying that for F ∈ C∞(P∞(M)),
φ
d
(2.2) (V F)(ρdvol ) = F ρdvol −ǫ∇i(ρ∇ φ)dvol .
φ M M i M
dǫ(cid:12)ǫ=0
(cid:0) (cid:1)
The map φ → Vφ passes to an isomorp(cid:12)(cid:12)hism C∞(M)/R → TρdvolMP∞(M). Otto’s Riemann-
ian metric on P∞(M) is given [10] by
(2.3) hV ,V i(ρdvol ) = h∇φ ,∇φ iρ dvol
φ1 φ2 M Z 1 2 M
M
= − φ ∇i(ρ∇ φ ) dvol .
1 i 2 M
Z
M
In view of (2.2), we write δ ρ = −∇i(ρ∇ φ). Then
Vφ i
(2.4) hV ,V i(ρdvol ) = φ δ ρ dvol = φ δ ρ dvol .
φ1 φ2 M Z 1 Vφ2 M Z 2 Vφ1 M
M M
To write the equation for parallel transport, let c : [0,1] → P∞(M) be a smooth curve.
We write c(t) = µ = ρ(t) dvol and define φ(t) ∈ C∞(M), up to a constant, by dc = V .
t M dt φ(t)
This is the same as saying
∂ρ
(2.5) +∇j(ρ∇ φ) = 0.
j
∂t
Let V be a vector field along c, with η(t) ∈ C∞(M). The equation for V to be parallel
η(t) η
along c [6, Proposition 3] is
∂η
(2.6) ∇ ρ ∇i + ∇ φ∇i∇jη = 0.
i j
(cid:18) (cid:18) ∂t (cid:19)(cid:19)
Lemma 2.7. [6, Lemma 5] If η,η are solutions of (2.6) then h∇η,∇ηi dµ is constant
M t
in t. R
Lemma 2.8. Given η ∈ C∞(M), there is at most one solution of (2.6) with η(1) = η , up
1 1
to time-dependent additive constants.
4 JOHN LOTT
Proof. By linearity, it suffices to consider the case when η = 0. FromLemma 2.7, ∇η(t) = 0
1
and so η(t) is spatially constant. (cid:3)
For consistency with later notation, we will write C∞([0,1];C∞(M)) for C∞([0,1]×M).
Lemma 2.9. (c.f. [1, (5.8)]) Given f ∈ C∞([0,1];C∞(M)), if η satisfies (2.6) then
d ∂f
(2.10) h∇f,∇ηidµ = h∇ ,∇ηidµ + Hess (∇η,∇φ)dµ .
t t f t
dt Z Z ∂t Z
M M M
Proof. We have
d d
(2.11) h∇f,∇ηidµ = h∇f,∇ηiρ dvol
t M
dt Z dt Z
M M
∂f ∂η
= h∇ ,∇ηiρ dvol + h∇f,∇ iρ dvol +
M M
Z ∂t Z ∂t
M M
∂ρ
h∇f,∇ηi dvol
M
Z ∂t
M
Then
d ∂f
(2.12) h∇f,∇ηidµ − h∇ ,∇ηidµ
t t
dt Z Z ∂t
M M
∂η
= (∇ f) ∇i ρ dvol − (∇ f)(∇iη)∇j(ρ∇ φ) dvol
i M i j M
Z (cid:18) ∂t(cid:19) Z
M M
∂η
= − f ∇ ρ∇i dvol − (∇ f)(∇iη)∇j(ρ∇ φ) dvol
i M i j M
Z (cid:18) ∂t(cid:19) Z
M M
= f∇ ρ(∇ φ)(∇i∇jη) dvol + ∇j((∇ f)(∇iη))(∇ φ)ρ dvol
i j M i j M
Z Z
M (cid:0) (cid:1) M
= − (∇ f)(∇ φ)(∇i∇jη)ρ dvol
i j M
Z
M
+ ∇j((∇ f)(∇iη))(∇ φ)ρ dvol
i j M
Z
M
= (∇j∇ f)(∇iη)(∇ φ)ρ dvol
i j M
Z
M
= Hess (∇η,∇φ)dµ .
f t
Z
M
(cid:3)
This proves the lemma.
Wenow weaken the regularity assumptions. Let Pac(M) denote the absolutely continuous
probabilitymeasuresonM withfullsupport. Supposethatc : [0,1] → Pac(M)isaLipschitz
curve whose derivative c′(t) ∈ T P(M) exists for almost all t. We can write c′(t) = V
c(t) φ(t)
with ∇φ(t) ∈ L2(TM,dc(t)). By the Lipschitz assumption, the essential supremum over
t ∈ [0,1] of k∇φ(t)k is finite. As before, we write c(t) = µ .
L2(TM,dc(t)) t
Definition 2.13. Let c : [0,1] → Pac(M) be a Lipschitz curve whose derivative c′(t) ∈
T P(M) exists for almost all t. Given ∇η ∈ L2(TM,dµ ), ∇η ∈ L2(TM,dµ ) and
c(t) 0 0 1 1
AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE 5
∇η ∈ L2([0,1];L2(TM,dµ )), we say that (∇η,∇η ,∇η ) is a weak solution of the parallel
t 0 1
transport equation if
(2.14) h∇f(1),∇η idµ − h∇f(0),∇η idµ =
1 1 0 0
Z Z
M M
1 ∂f
∇ ,∇η + Hess (∇η,∇φ) dµ dt
f t
Z Z (cid:18)(cid:28) ∂t (cid:29) (cid:19)
0 M
for all f ∈ C∞([0,1];C∞(M)).
Remark2.15. Inwhatfollows, therewouldbeanalogousresultsifwereplacedC∞([0,1];C∞(M))
everywhere by C0([0,1];C2(M))∩C1([0,1];C1(M)). We will stick with C∞([0,1];C∞(M))
for concreteness.
FromLemma2.9, ifcisasmoothcurveinP∞(M)andη ∈ C∞([0,1];C∞(M))isasolution
of (2.6) then (∇η,∇η(0),∇η(1)) is a weak solution of the parallel transport equation. We
now prove the converse.
Lemma 2.16. Suppose that c is a smooth curve in P∞(M). Given η ,η ∈ C∞(M) and η ∈
0 1
C∞([0,1];C∞(M)), if (∇η,∇η ,∇η ) is a weak solution of the parallel transport equation
0 1
then η satisfies (2.6), η(0) = η and η(1) = η (modulo constants).
0 1
Proof. In this case, equation (2.14) is equivalent to
(2.17) h∇f(1),∇η idµ − h∇f(0),∇η idµ =
1 1 0 0
Z Z
M M
h∇f(1),∇η(1)idµ − h∇f(0),∇η(0)idµ +
1 0
Z Z
M M
1 ∂η
f∇ ∇i +∇ φ∇i∇jη dµ dt.
i j t
Z Z (cid:18) ∂t (cid:19)
0 M
Taking f ∈ C∞([0,1];C∞(M)) with f(0) = f(1) = 0, it follows that (2.6) must hold. Then
taking all f ∈ C∞([0,1];C∞(M)), it follows that ∇η = ∇η(0) and ∇η = ∇η(1). Hence
0 1
η(0) = η and η(1) = η (modulo constants). (cid:3)
0 1
Lemma 2.18. Suppose that c is a smooth curve in P∞(M). Given ∇η ∈ L2(TM,dµ ),
0 0
∇η ∈ L2(TM,dµ ), ∇η ∈ L2([0,1];L2(TM,dµ )) and f ∈ C∞([0,1];C∞(M)), suppose that
1 1 t
(1) (∇η,∇η ,∇η ) is a weak solution to the parallel transport equation,
0 1
(2) f satisfies (2.6),
(3) ∇f(1) = ∇η ,
1
(4)
(2.19) |∇η |2 dµ ≤ |∇η |2 dµ
0 0 1 1
Z Z
M M
and
(5)
1
(2.20) |∇η|2 dµ dt ≤ |∇η |2 dµ
t 1 1
Z Z Z
0 M M
6 JOHN LOTT
Then ∇f(0) = ∇η , and ∇f(t) = ∇η(t) for almost all t.
0
Proof. From (2.6) (applied to f) and (2.14), we have
(2.21) h∇f(0),∇η idµ = h∇f(1),∇η idµ = h∇η ,∇η idµ .
0 0 1 1 1 1 1
Z Z Z
M M M
From Lemma 2.7,
(2.22) h∇f(0),∇f(0)idµ = h∇f(1),∇f(1)idµ = h∇η ,∇η idµ .
0 1 1 1 1
Z Z Z
M M M
Then
(2.23) |∇(η −f(0))|2dµ = |∇η |2 dµ − |∇η |2 dµ ≤ 0.
0 0 0 0 1 1
Z Z Z
M M M
Thus ∇f(0) = ∇η in L2(TM,dµ ).
0 0
Next, replacing f by tf in (2.14) gives
1
(2.24) h∇f,∇ηidµ dt = h∇f(1),∇η idµ = h∇η ,∇η idµ .
t 1 1 1 1 1
Z Z Z Z
0 M M M
Then
1
(2.25) |∇f −∇η|2 dµ dt =
t
Z Z
0 M
1 1 1
|∇f|2 dµ dt − 2 h∇f,∇ηidµ dt + |∇η|2 dµ dt =
t t t
Z Z Z Z Z Z
0 M 0 M 0 M
1
|∇f(1)|2dµ − 2 |∇η |2 dµ + |∇η|2 dµ dt =
1 1 1 t
Z Z Z Z
M M 0 M
1
|∇η|2 dµ dt − |∇η |2 dµ ≤ 0.
t 1 1
Z Z Z
0 M M
Thus ∇f(t) = ∇η(t) in L2(TM,dµ ), for almost all t. (cid:3)
t
3. Parallel transport along Wasserstein geodesics
3.1. Parallel transport in a finite-dimensional Alexandrov space. We recall the
construction of parallel transport in a finite-dimensional Alexandrov space X.
Let c : [0,1] → X be a geodesic segment that lies in the interior of a minimizing geodesic.
Then T X isanisometric product ofRwiththe normalconeN X. Wewant toconstruct
c(t) c(t)
a parallel transport map from N X to N X.
c(1) c(0)
Given Q ∈ Z+ and 0 ≤ i ≤ Q−1, define c : [0,1] → X by c (u) = c i+u . We define
i i Q
(cid:16) (cid:17)
an approximate parallel transport P : N X → N X as follows. Given v ∈ N X,
i ci(1) ci(0) ci(1)
let γ : [0,ǫ] → X be a minimizing geodesic segment with γ(0) = c (1) and γ′(0) = v.
i
For each s ∈ (0,ǫ], let µ : [0,1] → X be a minimizing geodesic with µ (0) = c (0) and
s s i
µ (1) = γ(s). Let w ∈ N X be the normal projection of 1µ′(0) ∈ T X. After passing
s s ci(0) s s ci(0)
to a sequence s → 0, we can assume that lim w = w ∈ N X. Then P (v) = w. If
i i→∞ si ci(0) i
X has nonnegative Alexandrov curvature then |w| ≥ |v|.
AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE 7
In [11], the approximate parallel transport froman appropriate dense subset L ⊂ N X
Q c(1)
to N X was defined to be P ◦P ◦...◦P . It was shown that by taking Q → ∞ and
c(0) 0 1 Q−1
applying a diagonal argument, in the limit one obtains an isometry from a dense subset of
N X to N X. This extends by continuity to an isometry from N X to N X.
c(1) c(0) c(1) c(0)
If X is a smooth Riemannian manifold then P is independent of the choices and can be
i
described as follows. Given v ∈ N X, let j (u) be the Jacobi field along c with j (0) = 0
ci(1) v v
and j (1) = v. (It is unique since c is in the interior of a minimizing geodesic.) Then
v
P (v) = j′(0).
i v
3.2. Construction of parallel transport along a Wasserstein geodesic of delta
measures. Let M be a compact connected Riemannian manifold without boundary. Let
γ : [0,1] → M be a geodesic segment that lies in the interior of a minimizing geodesic. Let
Π : T M → T M be (reverse) parallel transport along γ. Put c(t) = δ ∈ P(M).
γ(1) γ(0) γ(t)
Then {c(t)} is a Wasserstein geodesic that lies in the interior of a minimizing geodesic.
t∈[0,1]
We apply Petrunin’s construction to define parallel transport directly from the tangent cone
T P(M) to the tangent cone T P(M) (instead of the normal cones). From [7, Theorem
c(1) c(0)
∼
1.1], we know that T P(M) = P (T M).
c(t) 2 γ(t)
Proposition3.1. The paralleltransportmap fromT P(M) ∼= P (T M) to T P(M) ∼=
c(1) 2 γ(1) c(0)
P (T M) is the map µ → Π µ.
2 γ(0) ∗
Proof. Given Q ∈ Z+ and 0 ≤ i ≤ Q − 1, define γ : [0,1] → M by γ (u) = γ i+u
i i Q
(cid:16) (cid:17)
and c : [0,1] → P(M) by c (u) = δ . We define an approximate parallel transport
i i γi(u)
P : T P(M) → T P(M) as follows.
i ci(1) ci(0)
Given s ∈ R+ anda real vector space V, let R : V → V bemultiplication by s. Let ν bea
s
compactly-supported element ofP(T M). Forsmall ǫ > 0, thereisaWasserstein geodesic
γi(1)
σ : [0,ǫ] → P(M), with σ(0) = c (1) and σ′(0) corresponding to ν ∈ T PM, given by
i ci(1)
σ(s) = (exp ◦R ) ν. Given s ∈ (0,ǫ], let µ : [0,1] → P(M) be a minimizing geodesic
γi(1) s ∗ s
with µ (0) = c (0) = δ and µ (1) = σ(s). There is a compactly-supported measure
s i γi(0) s
τ ∈ P (T M) = T P(M) so that for v ∈ [0,1], we have µ (v) = (exp ◦R ) τ .
s 2 γi(0) ci(0) s γi(0) v ∗ s
If Q is large and ǫ is small then all of the constructions take place well inside a totally
convex ball, so τ is unique and can be written as τ = exp−1 ◦exp ◦R ν. Then
s s (cid:16) γi(0) γi(1) s(cid:17)∗
lim 1(τ −τ ) exists and equals (dexp )−1ν. Thus P = (dexp )−1.
s→0 s s 0 γi(0) ∗ i γi(0) ∗
Now
(3.2) P ◦P ◦...◦P = (dexp )−1 ◦(dexp )−1 ◦...◦(dexp )−1 .
0 1 Q−1 (cid:16) γ0(0) γ1(0) γQ−1(0) (cid:17)∗
Taking Q → ∞, this approaches Π . (cid:3)
∗
3.3. Construction of parallel transport along a Wasserstein geodesic of absolutely
continuous measures. Let M be a compact connected boundaryless Riemannian mani-
fold with nonnegative sectional curvature. Then (P(M),W ) has nonnegative Alexandrov
2
curvature.
8 JOHN LOTT
Let c : [0,1] → Pac(M) be a geodesic segment that lies in the interior of a minimizing
geodesic. Write c′(t) = V . Since φ(t) is defined up to a constant, it will be convenient to
φ(t)
normalize it by φ(t)dµ = 0. We assume that
M t
R
(3.3) sup kφ(t)k < ∞.
C2(M)
t∈[0,1]
In particular, this is satisfied if c lies in P∞(M).
Let N P(M) denote the normal cone to c at c(t). We want to construct a parallel
c(t)
transport map from N P(M) to N P(M).
c(1) c(0)
Given Q ∈ Z+ and 0 ≤ i ≤ Q − 1, define c : [0,1] → P(M) by c (u) = c i+u .
i i Q
(cid:16) (cid:17)
Correspondingly, write µ = µ . We define an approximate parallel transport P :
i,u i+u i
Q
N P(M) → N P(M), using Jacobi fields, as follows.
ci(1) ci(0)
Let us write c′(u) = V , i.e. φ (u) = 1φ i+u . The curve c is given by c (u) =
i φi(u) i Q Q i i
(F ) c (0), where F (x) = exp (u∇ φ (0)). Tha(cid:16)t is,(cid:17)for any f ∈ C∞(M),
i,u ∗ i i,u x x i
(3.4) f dc (u) = f(F (x))dµ (x).
i i,u i,0
Z Z
M M
If σ is a variation of φ (0), i.e. δφ (0) = σ , then taking the variation of (3.4) gives
i i i i
(3.5) f dδc (u) = h∇f,dexp (u∇ σ )i dµ (x)
Z i Z u∇xφi(0) x i Fi,u(x) i,0
M M
= u h∇f,W (u)idµ .
Z σi i,u
M
Here
(3.6) (W (u)) = dexp (∇ σ ),
σi y u∇xφi(0) x i
with y = F (x). The corresponding tangent vector at c (u) is represented by L (u) =
i,u i σi
Π W (u), where Π is orthogonal projection on Im∇ ⊂ L2(TM,dµ ). We can think
ci(u) σi ci(u) i,u
of J (u) = uL (u) as a Jacobi field along c . If v = J (1) = L (1) = Π W (1) then
σi σi i σi σi ci(1) σi
its approximate parallel transport along c is represented by w = J′ (0) = L (0) = ∇σ ∈
i σi σi i
Im∇ ⊂ L2(TM,dµ ).
i,0
AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE 9
Next, using (3.6), for f ∈ C∞(M) we have
(3.7)
d d d
hV ,L idµ = hV ,W idµ = h∇f,dexp (∇ σ )i dµ (x)
du Z f σi i,u du Z f σi i,u du Z u∇xφi(0) x i Fi,u(x) i,0
M M M
= Hess (f) dexp (∇ φ (0)),dexp (∇ σ ) dµ (x)+
Z Fi,u(x) u∇xφi(0) x i u∇xφi(0) x i i,0
M (cid:0) (cid:1)
h∇f,D dexp (∇ σ )i dµ (x)
Z ∂u u∇xφi(0) x i Fi,u(x) i,0
M
= Hess(f)(∇φ (u),W (u)) dµ +
Z i σi i,u
M
h∇f,D W (u)idµ .
Z ∂u σi i,u
M
Here ∂ is the vector at F (x) given by
u i,u
d
(3.8) ∂ = F (x) = dexp (∇ φ (0)).
u du i,u u∇xφi(0) x i
If instead f ∈ C∞([0,1];C∞(M)) then
d ∂f
(3.9) hV ,L idµ = ∇ ,L dµ +
du Z f σi i,u Z (cid:28) ∂u σi(cid:29) i,u
M M
Hess(f)(∇φ (u),W (u)) dµ +
Z i σi i,u
M
h∇f,D W (u)idµ .
Z ∂u σi i,u
M
We will need to estimate |W (u)−L (u)|2 dµ .
M σi σi i,u
R
Lemma 3.10. For large Q, there is an estimate
(3.11) |W (u)−L (u)|2 dµ ≤
Z σi σi i,u
M
const.kHess(φ (·))k2 kL (0)k2 .
i L∞([0,1]×M) σi L2(TM,dµi,0)
Here, and hereafter, const. denotes a constant that can depend on the fixed Riemannian
manifold (M,g).
Proof. Since Π is projection onto Im(∇) ⊂ L2(TM,dµ ), and ∇(σ ◦F−1) ∈ Im(∇), we
ci(u) i,u i i,u
have
(3.12) |W (u)−L (u)|2dµ ≤ |W (u)−∇(σ ◦F−1)|2 dµ
Z σi σi i,u Z σi i i,u g i,u
M M
= |(dF )−1W (u)−∇σ |2 dµ .
Z i,u ∗ σi i Fi∗,ug i,0
M
(Compare with [1, Proposition 4.3].) Defining T : T M → T M by
i,t,x x x
(3.13) T (z) = (dF )−1 dexp (z) ,
i,t,x i,u ∗ u∇xφi(0)
(cid:0) (cid:1)
10 JOHN LOTT
we obtain
(3.14) |W (u)−L (u)|2 dµ ≤
Z σi σi i,u
M
supkdF∗ dF (x)k·kT −Ik2 kL (0)k2 .
(cid:18)x∈M i,u i,u i,u,x (cid:19) σi L2(TM,dµi,0)
Since sup k∇φ(t)k < ∞, if Q is large then k∇φ (0)k is much smaller than
t∈[0,1] C0(M) i C0(M)
the injectivity radius of M. In particular, the curve {F (x)} lies well within a normal
i,u u∈[0,1]
ball around x. Now T can be estimated in terms of Hess(φ ). In general, if a function h
i,t,x i
on a complete Riemannian manifold satisfies Hess(h) = 0 then the manifold isometrically
splits off an R-factor and the optimal transport path generated by ∇h is translation along
the R-factor. In such a case, the analog of T is the identity map. If Hess(h) 6= 0 then the
i,t,x
divergence of a short optimal transport path from being a translation can be estimated in
terms of Hess(h). Putting in the estimates gives (3.11). (cid:3)
Using Lemma 3.10, we have
(3.15)
Hess(f)(∇φ (u),W (u)) dµ − Hess(f)(∇φ (u),L (u)) dµ ≤
(cid:12)Z i σi i,u Z i σi i,u(cid:12)
(cid:12) M M (cid:12)
(cid:12)(cid:12)const.kHess(f)kC0(M)kHess(φi(·))kL∞([0,1]×M)k∇φi(u)kL2(TM,dµi,0)kLσi(0(cid:12)(cid:12))kL2(TM,dµi,0).
Next, given x ∈ M, consider the geodesic
(3.16) γ (u) = F (x).
i,x i,u
Put
(3.17) j (u) = u(W (u)) ∈ T M.
σi,x σi γi,x(u) γi,x(u)
Then j is a Jacobi field along γ , with j (0) = 0 and j′ (0) = ∇ σ . Jacobi field
σi,x i,x σi,x σi,x x i
estimates give
(3.18) kD W (u)k ≤ const.k∇σ k k∇φ (·)k2 ,
∂u σi L2(TM,dµi,u) i L2(TM,dµi,u) i L∞([0,1]×M)
again for Q large.
Lemma 3.19. Define A : Im(∇) ⊂ L2(TM,dµ ) → Im(∇) ⊂ L2(TM,dµ ) by
i i,0 i,1
(cid:16) (cid:17) (cid:16) (cid:17)
(3.20) A (∇σ ) = L (1).
i i σi
Then for large Q, the map A is invertible for all i ∈ {0,...,Q−1}.
i
Proof. Define B : Im(∇) ⊂ L2(TM,dµ ) → Im(∇) ⊂ L2(TM,dµ ) by
i i,1 i,0
(cid:16) (cid:17) (cid:16) (cid:17)
(3.21) B (∇f) = ∇(f ◦F ).
i i,1
Then whenever ∇f ∈ L2(TM,dµ ), we have
i,1
(3.22) (A B )(∇f) = A (∇(f ◦F )) = L (1),
i i i i,1 f◦Fi,1