Max-Planck-Institut
fur Mathematik
in den Naturwissenschaften
Leipzig
Polar factorization of maps on
Riemannian manifolds
by
Robert J. McCann
Preprint no.:
41
1999
POLAR FACTORIZATION OF MAPS ON
RIEMANNIAN MANIFOLDS
Robert J. McCann
Department of Mathematics
University of Toronto, Toronto Ontario Canada M5S 3G3
[email protected]
May 27, 1999
Abstract
Let ( ) be a connected compact manifold, 3 smooth and without
boundary, equipped with a Riemannian distance ( ). If : ;! is
merely Borel and never maps positive volume into zero volume, we show = factors uniquely a.e. into the composition of a map ( ) = expx ;r ( )] and
a volume-preserving map : ;! , where : ;! R is an inmal
convolution with ( ) = 2 ( ) 2. Like the factorization it generalizes from
Euclidean space, this non-linear decomposition can be linearized around the
identity to yield the Hodge decomposition of vector elds.
The results are obtained by solving a Riemannian version of the MongeKantorovich problem, which means minimizing the expected value of the cost
( ) for transporting one distribution 0 of mass in 1 ( ) onto another.
A companion article extends this solution to strictly convex or concave cost
functions ( ) 0 of the Riemannian distance on non-compact manifolds.
M g
C
d x y
s
M
M
s
t x
u
c x y
d
M
M
t u
x
M
x y =
c x y
f
L
M
c x y
The author gratefully acknowledges the support of an American Mathematical Society Centennial Fellowship and National Science Foundation grant DMS 9622997 held at Brown University,
where much of this work was performed. He is also grateful for the hospitality of the Institut des
Hautes Etudes Scientiques and the Max-Planck Institut fur Mathematik in den Naturwissenschaften
during various stages of the work. These results were rst announced in the July 1997 NSF-CBMS
c 1999 by the author.
workshop on the Monge-Amp
ere equation at Boca Raton, FL.
1
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
2
1 Introduction
The purpose of this note is is twofold: to announce a solution of the Monge-Kantorovich
transportation problem in curved geometries, and to derive from it a factorization of
maps which extends Brenier's polar decomposition theorem from Euclidean space to
Riemannian manifolds.
In its original form, Brenier's theorem 4] factored each s : ;! Rn in L1(
Rn)
uniquely (on a bounded smooth domain Rn) into the composition s = t u of a
volume preserving map u : ;! with the gradient t = r of a convex function
: Rn ;! R f+1g. Here s is assumed not to map positive volume into zero
volume. The hypotheses were subsequently relaxed 5]6]23], and the theorem inspired a whole line of subsequent work and some fascinating applications: the Hodge
decomposition of vector elds 4], existence of convex solutions to the Monge-Ampere
equation 4]7], and the Brunn-Minkowksi inequality 24]1]2] are among its consequences Otto's work 28] is also striking, and further developments are reviewed in
Evans 15]. However, it has remained unclear how to formulate such a theorem in
curved geometries, where convex functions (and their gradients) hardly make sense,
let alone their connection with mappings. In the meantime, the optimal transportation problem of Monge 27] and Kantorovich 21], central to Brenier's proof, has been
studied intensively see e.g. Rachev and Ruschendorf 29]. An understanding of its
geometry obtained with Gangbo 19] for a large class of costs on Euclidean space sets
the stage for exploring its Riemannian structure.
For simplicity, let (M g) be a Riemannian manifold without boundary, connected,
compact, and C 3 smooth | meaning the metric tensor components gij (x) are twice
continuously dierentiable functions of local coordinates. The geodesic distance between
q x and y 2 M is denoted by d(x y ), and the volume element by d vol(x)
(= detgij (x)] dnx in local coordinates). We set c(x y) = d2(x y)=2 throughout
a companion work explores what happens when d2 =2 is replaced by another strictly
convex (or strictly concave) increasing function of distance 25]. Given nite Borel
measures 0 on M with the same total mass M ] = M ], Monge's problem
27] is to nd the map t : M ;! M minimizing the transportation cost
Z
C (s) = M c(x s(x)) d(x)
(1)
among all Borel maps s 2 S ( ) which push forward to , meaning
V ] = s;1 (V )]
(2)
holds for each measurable V M . When is absolutely continuous with respect to
Riemannian volume, denoted << vol, a unique optimal map t shall be shown to
exist in S ( ) and be characterized geometrically:
Given a function : M ;! R f1g, its in mal convolution c with c is dened
by
c (y ) = inf c(x y ) ; (x):
(3)
x2M
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
3
This transformation acts as an involution cc := ( c)c = if and only if is itself an
inmal convolution of some function with c, in which case is said to be c-concave 29,
x3.3]. The unique t 2 S ( ) of the form
t(x) = expx ;r (x)]
(4)
with : M ;! R c-concave yields the optimal map. Thus the existence of a potential
= cc whose gradient species which direction | r (x) 2 T M x | and how far |
jr (x)jx | to move the mass geodesically along M from -a.e.] x to its destination
characterizes optimality. In the Euclidean case, one recovers t(x) = x ; r (x) from
(4), while for c(x y) = jx ; yj2=2 the condition = cc reduces to convexity of
jxj2=2 ; . Verily is t(x) the gradient of a convex function.
Now x a nite non-negative Borel measure << vol on M and a Borel map
s : M ;! M . To polar factorize s, de ne a second Borel measure := s# using
(2), called the push-forward of through s. Clearly M ] = M ]. The optimal
map t(x) = expx;r (x)] pushing forward to with = cc is the rst factor
decomposing s. To nd the second factor, assume << vol, meaning s collapses
no set with positive measure onto a set of zero volume s is non-degenerate in the
terminology of Brenier. It is easy to guess that the optimal map t 2 S ( ) must
be the inverse to t in the sense that t(t (y)) = y -a.e.]. Setting u = t s ensures
t u = s holds -a.e., while u# = t# (s# ) = t# = so the measure is perserved
under u. Volume is preserved if we started out with = vol. Apart from sets of
measure zero, there is only one map in S ( ) of the form (4) with = cc, implying
t and u are uniquely determined -a.e.
The main part of our work will be devoted to proving existence and uniqueness of
a map t 2 S ( ) minimizing Monge's transportation cost (1), and establishing (4)
with = cc. Since the measures and are both nite and non-negative with the
same total mass, it costs no generality to normalize them so that M ] = M ] = 1
i.e. to restrict our attention to (Borel) probability measures. As in the Euclidean
case, our departure point is a dual problem 29] of Kantorovich type:
(
where
sup J ( ) = s2Sinf()
)2Lipc
J ( ) =
Z
Z
M
(x) d(x) +
is a linear functional dened on a convex subset
Lipc := f : M
M
c(x s(x)) d(x)
Z
M
(y ) d (y )
;! R continuous j (x) + (y) c(x y)g
(5)
(6)
(7)
of continuous functions. In fact, the supremum of J ( ) turns out to be attained
29, x2.3.12] | by a function = cc (with = c) from which we shall construct
a map minimizing Monge's cost. Compactness of the manifold facilitates a direct
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
4
proof of the existence of ( ) and the duality relation (5), following an approach of
Gangbo 17] also developed in Caarelli 8]9], and Gangbo and McCann 18]. We give
this proof after some preliminaries on inmal convolutions with c. A key ingredient
is the change of variables formula for pushed-forward measures s# = :
Z
M
h d(s#) =
Z
M
h(s(x)) d(x)
(8)
holds for all Borel s : M ;! M and h : M ;! R f1g, as is readily veried
from denition (2) by approximating h using simple functions. Alternately, the Riesz
representation theorem allows one to take (8) as the denition of s#. Before entering into further details, a few remarks are necessary concerning the history of this
manuscript.
This paper has been in gestation for quite a long time. It is the author's pleasure
to recall that the question of how to polar factorize vector elds on Riemannian manifolds was rst put to him by Dennis Sullivan (at a time when he was ill-prepared
to solve it, especially since the Euclidean example initially misled us into trying to
factor vector elds rather than maps) and again a year later by Tudor Ratiu (at a
time when he found himself better equipped). He is pleased to acknowledge both
of them, along with Stephen Semmes, for providing stimulating conversations during
the course of the work. He is also grateful to Michael Cullen and Robert Douglas,
who included a statement of the result in their announcement of its rst application
11]: nding simplifying variables for the semigeostrophic model of atmospheric dynamics on a sphere see also 12]3]. Their setting is actually a non-compact manifold
| the Northern hemisphere | with a conformally round metric proportional to the
Coriolis force, which degenerates on the equator. Such manifolds are treated in 25],
but require the additional hypothesis that (unbroken) minimal geodesics exist between every x 2 spt and y 2 spt . Interestingly enough, near the equator where
this hypothesis fails, their model breaks down and weather patterns change drastically because the atmosphere has no preferred direction to swirl! Finally, it must be
noted that Dario Cordero-Erausquin has independently obtained a solution to the
transportation problem on the at torus M = Tn 10].
We now proceed to state a series of standard lemmas and introduce some nonsmooth analysis as a prelude to our rst substantial remarks: Proposition 6 and
Lemma 7. These are used to prove our main results: Theorems 8, 9 and 11. We close
by highlighting how the formal relationship between the polar factorization of maps
and the Hodge decomposition of vector elds extends to the Riemannian setting.
2 Preliminaries
The results of this section, though well-known to part of our readership, are included
for ease of reference and completeness.
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
5
Lemma 1 (Lipschitz Cost) Let (M d) be a metric space whose diameter jM j :=
supfd(x z) j x z 2 M g is nite. For each y 2 M , the function (x) = d2(x y)=2 is
Lipschitz continuous:
j (x) ; (z)j jM j d(x z):
(9)
Proof: The triangle inequality shows that (x) := d(x y) has Lipschitz constant one:
(x) ; (z ) = d(x y ) ; d(z y ) d(x z )
(10)
for all x z 2 M . Also, (x) = d(x y) jM j < 1 is bounded. The desired estimate
(9) then follows easily for (x) = 2(x)=2:
2j (x) ; (z)j = j(x)((x) ; (z)) + (z)((x) ; (z))j
jM j d(x z) + jM j d(x z):
2
Lemma 2 (Inmal Convolutions are Lipschitz) Fix a metric space (M d) having nite diameter. Any : M ;! R f1g given by an in mal convolution
= cc with c(x y) = d2(x y)=2 is either identically in nite = 1 or Lipschitz
continuous throughout M . Indeed, it satis es (9).
Proof: More generally, suppose = c for some : M ;! R f1g, meaning
(x) = yinf
c(x y ) ; (y ):
(11)
2M
Observe 0 c(x y) jM j2=2 is bounded. Either is unbounded above, in which
case (11) yields = ;1 and the lemma holds trivially, or else is bounded below.
Fix z 2 M , and note (z) = +1 in (11) occurs only if := ;1 everywhere, in
which case = +1 again holds trivially. Thus we may assume that is nite
everywhere. Given any > 0, there exists y 2 M such that (z) + c(z y) ; (y),
while (x) c(x y) ; (y) holds because of (11). Subtracting these two inequalities
yields
(x) ; (z) c(x y) ; c(z y) + jM jd(x z) + by Lemma 1. Since the last inequality holds for all > 0, the Lipschitz estimate (9)
has been proved.
2
Proposition 3 (Dual Potentials) Fix Borel probability measures and on a separable metric space (M d) having nite diameter jM j. Set c(x y) = d2 (x y )=2. The
supremum (5) is attained by some : M ;! R satisfying = cc and its in mal
convolution =
c
with c.
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
6
Proof: Claim #1: If (u v) 2 Lipc then (vc v) 2 Lipc and J (u v) J (vc v). Moreover, (vc vcc) 2 Lipc and J (u v) J (vc vcc).
Proof of claim: Fix (u v) 2 Lipc. Since u is continuous and M is compact, the
inmal convolution (3) dening uc is nite-valued, hence Lipschitz continuous by
Lemma 2. Now (u uc) 2 Lipc follows immediately from (3) and (7). Moreover,
v (y ) c(x y ) ; u(x)
so an inmum over x yields v(y) uc(y) and hence J (u v) J (u uc). The symmetry
u $ v shows the rst half of the claim is established. Applying what we just proved
to (vc v) yields (vc vcc) 2 Lipc and J (vc v) J (vc vcc), from which the remaining
claim follows immediately.
To complete the proposition, choose a sequence ( n n) 2 Lipc for which J ( n n)
tends to its maximum value on Lipc. According to claim #1, the sequence (cn ccn)
also maximizes J on Lipc . Fix z 2 M and dene a new sequence (un vn) :=
(cn ; n ccn + n) in Lipc, again maximizing J since M ] = M ] the normalization constants n := cn(z) are selected to make un(z) = 0. Now Lemma 2 provides a uniform bound jM j for the Lipschitz constants of un and vn. Moreover, this
equicontinuous family of functions is uniformly bounded throughout M :
jun(x)j = jun(x) ; un(z)j jM jd(x z) jM j2 (12)
while jvn(y)j 3jM j2=2 follows from (3) since vn = ucn. The Ascoli-Arzela argument
extracts a pointwise convergent subsequence, also denoted (un vn), with Lipschitz
continuous limit (u v) 2 Lipc . Since and are nite measures, the dominated
convergence theorem yields J (u v) = limn J (un vn). The supremum (5) is attained
at (u v) and hence (by Claim #1) at ( ) := (vc vcc). Finally, vccc = vc 29, x3.3.5],
so cc = to complete the proof.
2
Lemma 4 (Rademacher's Theorem) Any function : M ;! R locally Lipschitz
with respect to the geodesic distance d(x y) on a connected, C 1 smooth Riemannian
manifold must be dierentiable outside a set Z M of zero volume. Its gradient
gives a Borel map from dom r := M n Z into the tangent bundle T M .
Proof: Fix z 2 M and normal coordinates : U ;! Rn centered at z = ;1(0) so
that the metric is diagonalized there: gij (z) = ij . Since the coecients gij (x) of the
quadratic form gh ix depend continuously on these coordinates, there is a smaller
neighbourhood of z, also denoted by U M , on which its largest eigenvalue does not
vary by more than a nite factor:
g h v vix k2
n
X
(vi)2
i=1
(13)
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
7
for all x 2 U and v 2 T M x. Choose > 0 small enough so that Bn(0 ) (U )
and replace U by the preimage ;1(Bn(0 )) of this ball. We shall show to be
dierentiable vol-a.e. on U . Since a connected Riemannian manifold is locally compact
and second countable from Kobayashi and Nomizu 22, Appendix 2], it is -compact,
hence covered by countably many such neighbourhoods U M . The dierentiability
of will therefore follow vol-a.e. on M .
The geodesic distance between x y 2 U is bounded by the length of the path
( ) := ;1((1 ; ) (x) + (y )) through U from x to y . Computing its arclength in
local coordinates yields
d(x y )
Z 1q
0
g h _ _ i d
k j (y) ; (x)j
(14)
after combining _ i( ) = i(y) ; i(x) with (13). Since was assumed Lipschitz
with respect to the Riemannian distance, it follows from (14) that f := ;1 is
Lipschitz with respect to the Euclidean metric on Bn(0 ). Rademacher's theorem
n
then asserts dierentiability of f Lebesgue a.e. on
q B (0 ). But Riemannian volume
can be expressed locally by a bounded multiple det gij (x)] of Lebesgue measure, so
dierentiability of has been established vol-a.e. on U to complete the rst claim.
To establish the remaining claim, it is necessary to recall the proof of Rademacher's
theorem from Evans and Gariepy 16, x3.1.2]. After extending f = ;1 continuously
to all of Rn, they observe that its upper derivative
D f (x) = lim
sup f (x + tv) ; f (x)
v
k!1 jtj2(01=k)\Q
t
in direction v 2 Rn is expressed as a limit of suprema of continuous functions of x,
hence Borel. The lower derivative Dv f dened by the analogous limit inmum is also
Borel, so the directional derivative Dv f is a Borel function on the set of full measure
where Dv f = Dv f . In particular, the n partial derivatives @xj f are Borel, as is
Fv = fx 2 Bn (0 ) j Dv = Dv f =
X j
v@
xj f g:
As Evans and Gariepy show, f is dierentiable on any countable intersection T Fvi of
these sets over a dense set of directions fvig 2 @ Bn (0 1). Outside
Fvi dierentiability
fails, so rf must be Borel. Clearly gkj @xj f is also Borel on T Fvi | and gives the
coordinates of r on U n Z . We conclude that both r and Z M are Borel. 2
The preceding proof used only that the metric tensor gij (x) was bounded (13)
and Borel | but not necessarily continuous. Indeed, with the denitions of De Cecco
and Palmieri 13], the lemma can be extended immediately to Lipschitz Riemannian
manifolds.
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
8
3 Method and Results
The chief technical complications arising in the Riemannian setting stem from nonuniqueness of minimal geodesics, and hence a lack of smoothness in the cost and
the distance function. However, all singularities come in the form of upward pointing
creases (and conical points) along the cut locus. Remarkably, this one-sidedness turns
out to permit the problem to be nessed by introducing appropriate notions from nonsmooth analysis. Indeed, on smooth manifolds, one may expect the cost c(x y) =
d2 (x y )=2 to have a semi-concavity property 19], though for present purposes it
suces to show it is superdierentiable in the following sense:
Fix x 2 M . A function : M ;! R is said to be superdierentiable at x with
supergradient p 2 T M x if
(expx v) (x) + g h p vix + o(jvjx)
(15)
holds for small v 2 T M x, where o(
)=
tends to zero with . Such ( supergradient,
point ) pairs (p x) form a subset @ T M of the tangent bundle we also express
their relationship (15) by writing p 2 @x. If the opposite inequality (expx v) (x) + g h q vix + o(jvjx) holds, is said to be subdierentiable with subgradient
q 2 @x T M x. When both inequalities hold, then is dierentiable at x and
its super- and sub- gradients coincide: p = q = r(x). This observation, which
proves crucial later, motivates a chain rule for supergradients used to establish the
connection between minimal geodesics and superdierentiability of the cost.
Lemma 5 (Chain Rule) Let : M ;! R and h : R ;! R have supergradients
p 2 @x and 2 @h(x) at some x 2 M . If h is non-decreasing then p 2 @ (h )x.
Proof: Applied to h, Denition (15) yields
h((x) + ) h((x)) + + o():
Since h is non-decreasing, setting = gh p vix + o(jvjx) and invoking (15) yields
h (expx v) h (x) + g h p vix + o(jvjx)
h((x)) + gh p vix + o(jvjx)
to complete the proof.
2
Proposition 6 (Superdierentiability of Geodesic Distance Squared)
Let (M g ) be a C 3 -smooth Riemannian manifold, possibly with boundary. Suppose
: 0 1] ;! M has minimal length among piecewise C 1 curves joining y = (0)
to x = (1) 2= @M , parameterized with constant speed. Then ( ) = d2 ( y)=2 has
supergradient _ (1) 2 @ x at x.
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
9
Proof: Since x lies in the interior of M , there is some > 0 and neighbourhood
X M of x such that: at each z 2 X , the exponential map expz maps the ball
B(0 ) T M z dieomorphically onto some open set Uz X , as in Milnor 26,
x10.3]. The proposition will rst be established when y = (0) 2 X , in which case
is actually dierentiable at expy _ (0) = x. We compute its derivative by linearizing
expx v 2 X around the origin and expy around _ (0):
(expx v) =
=
=
=
=
d2 y expy (exp;y 1 expx v) = 2
j exp;y 1(expx v) j2y = 2
2 = 2
1
_ (0) + D(exp;
)
D
(exp
)
v
+
o
(
j
v
j
)
x
0
x
x
y
y
2
;
1
j_ (0)jy =2 + gh _ (0) (D expy )_ (0) I viy + o(jvjx)
d2 (x y )=2 + g h _ (1) vix + o(jvjx)
so that r (x) = _ (1). Here the last equation follows from _ (1) = D(expy )_ (0) _ (0)
and Gauss' lemma, see do Carmo 14, x3.3.5] or Milnor 26, x10.5]. (Note that
C 3 smoothness of (M g ) ensures that the coecients ;kij in the geodesic equation
are continuously dierentiable the Picard theorem for ODEs set forth in Graves 20,
xIX.3.13 and xV.2.16] then guarantees dierentiability of the exponential map and
the equality
q of mixed partials 26, x8.7] needed to establish Gauss' lemma.) Since
d(x y ) = 2 (x) the chain rule yields rx d(x y ) = _ (1)=j_ (1)jx as long as x 6= y .
Now we return to the possibility that y 62 X . In this case, take z 2 X to
be any point lying on the geodesic near the endpoint x = (1). Applying the
foregoing argument to z 6= x instead of y yields rx d(x z) = _ (1)=j_ (1)jx. The
triangle inequality then gives
d(y expx v)
q
d(y z) + d(z x) + gh _ (1) vix = j_ (1)jx + o(jvjx)
= d(y x) + gh _ (1)=j_ (1)jx vix + o(jvjx)
so d(y ) = 2 ( ) is superdientiable
at x. Applying the one-sided chain rule,
p
2
Lemma 5, with h(d) = d =2 and = 2 yields d(y x) _ (1)=j_ (1)jx = _ (1) 2 @ x
to complete the proof.
2
The next lemma establishes a Young-like inequality together with conditions for
equality. These conditions determine the form of the optimal map, and play a critical
role in a uniqueness argument based on the original idea of Brenier 4]5].
Lemma 7 (Tangency) Let (M g) be a connected, compact Riemannian manifold,
C 3 -smooth and without boundary. Suppose = cc, meaning : M ;! R is an
in mal convolution (3) with c(x y ) = d2(x y )=2. Then
c(x y ) ; (x) ; c (y ) 0
(16)
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
10
for all x y 2 M . If a point x 2 M is selected where happens to be dierentiable,
then equality holds in (16) if and only if y = expx ;r (x)].
Proof: The inequality (16) follows immediately from the denition (3) of c it
appears also in Rachev and Ruschendorf 29, x3.3.3]. Now select a point x 2 M
where happens to be dierentiable. To establish the only if statement, assume
some y 2 M can be found for which (16) becomes an equality. Then for all z 2 M
one has
c(z y ) ; (z ) ; c (y )
0
= c(x y) ; (x) ; c(y):
Dening (z) := c(z y) and z := expx v yields
(expx v) = c(z y )
c(x y) ; (x) + (z)
= (x) ; (x) + (x) + gh r (x) vix + o(jvjx)
so (z) has subgradient r (x) 2 @x at x. On the other hand, the Hopf-Rinow
theorem 26, x10.19] 22, xIV.4.1{4] assures us of the existence of a minimal geodesic
: 0 1] ;! M from y to x given by (1 ; ) = expx ; _ (1)]. Like the manifold,
this geodesic will be C 3-smooth, whence Proposition 6 yields _ (1) 2 @x. Thus is both super- and subdierentiable at x, so it is dierentiable and its super- and
subgradients coincide: _ (1) = r (x), implying y = (0) = exp;r (x)] as desired.
For our xed x 2 M , we have now established that at most one point y 2 M
produces equality in (16). We have not yet shown that the equality must be achieved
by some y 2 M , nor used the condition = cc (except to know c(y) is nite).
We now do both. Indeed, since = cc is real-valued, Lemma 2 yields Lipschitz
continuity of c : M ;! R. Also, the manifold is compact, so the inmum dening
cc(x) :=
inf c(x y) ; c(y)
y 2M
is attained at some y 2 M . The same point produces equality in (16). But then
our previous argument shows y = expx;r (x)], so the if statement of the lemma is
satised.
2
Theorem 8 (Unique Optimal Maps) Fix a connected, compact Riemannian manifold, C 3 -smooth and without boundary, and a Borel probability measure << vol on
(M g). If : M ;! R is an in mal convolution = cc with c(x y) = d2(x y)=2,
then t(x) = expx;r (x)] minimizes (1) on S ( t#). Any other map s 2 S ( t#)
minimizing C (s) must coincide with t(x) -almost everywhere.
Proof: Taking = cc and t as above, Lemmas 2 and 4 show that the potential
is Lipschitz and dierentiable -a.e., while its gradient (and hence t) are Borel. Now
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
11
t 2 S ( t#) and (0 0) 2 Lipc so both domains are non-empty. For any (u v ) 2 Lipc
and s 2 S ( t#) one has
J (u v ) =
Z
ZM
M
u(x) d(x) +
Z
M
c(x s(x)) d(x)
= C (s )
v (s(x)) d(x)
(17)
(18)
(19)
from (6{8) and a change of variables applied to s#. Since both sides are nite, this
shows
Z
sup J (u v) s2Sinf
c(x s(x)) d(x)
(20)
(t )
(uv)2Lipc
#
M
for some 2 R. However Lemma 7 shows ( c) 2 Lipc satises (x) + c(t(x)) =
c(x t(x)) -a.e.]. Choosing (u v ) = ( c ) and s = t therefore leads to equality in
(18), and hence (20). Moreover J ( c) = = C (t) proves optimality of the map t.
Conversely, any map s 2 S ( t#) which achieves optimality must also satisfy
C (s) = = J ( c), so equality continues to hold in (18). From (16) we see (x) +
c (s(x)) = c(x s(x)) must hold pointwise -a.e. On the set of full measure where
is dierentiable, we conclude s(x) = expx;r (x)] = t(x) from Lemma 7.
2
Note that the Kantorovich duality (5) was established for = t# in the preceding
proof. It must therefore continue to hold under the hypotheses of Theorem 9.
Theorem 9 (Existence and Characterization of Optimal Maps) Let (M g) be
a connected, compact Riemannian manifold, C 3 -smooth and without boundary. Fix
c(x y ) = d2 (x y )=2 and two Borel probability measures << vol and arbitrary
on M . Then for some potential : M ;! R satisfying = cc the map t(x) =
expx;r (x)] pushes forward to . Modulo discrepancies on sets of -measure
zero, only one t 2 S ( ) can arise in this way.
Proof: To begin choose ( ) = ( cc c) which maximize (6) on Lipc. These exist
by Proposition 3. Both functions are Lipschitz according to Lemma 2, so r : M ;!
T M and hence t : M ;! M are Borel maps dened -a.e. in view of Lemma 4. One
way to prove that the map t(x) = expx;r (x)] pushes forward to is to integrate
each continuous function h 2 C (M ) against the measures and t# and show the two
integrals coincide. We do this following Gangbo 17], Gangbo and McCann 18] and
Caarelli 9]. For x y 2 M and jj < 1 dene perturbations (y) := c(y) + h(y)
and := ()c by
c(x y ) ; (y ) ; h(y ):
(21)
(x) := yinf
2M
Fix a point x where 0 is dierentiable. Continuity and compactness of M ensure
the inmum (21) is attained. For = 0 it is attained uniquely at y = t(x) in view of
12
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
Lemma 7, so for small , it must be attained at some nearby point y = t(x) + o(1).
Thus
c(x t(x)) ; (t(x)) ; h(y) (x) c(x y ) ; (y ) ; h(y )
for all y 2 M . Choosing y = t(x) yields (x) = 0 (x) ; h(t(x)) + o(), where the
error term satises a bound ;1 o() 2khk1 uniform in x in addition to vanishing
as ! 0. Because J ( ) attains its maximum at = 0, we deduce
Z
Z
J ( ) ; J ( 0 0 )
(x) ; 0 (x)
lim
= lim
d(x) + h(y ) d (y )
!0
=
M
Z
;h(t(x)) d(x) + h(y) d (y)
Z!0 M
MZ
= ;
= 0
M
h d(t#) +
M
Z
M
h d
where the dominated convergence theorem was used in the second equality and the
change of variables formula (8) in the third. But now the existence claim is established: t# = by the Riesz-Markov theorem since h 2 C (M ) was arbitrary.
To prove that we have characterized t uniquely in S ( ), let u = (uc)c denote
any other potential on M for which the map s(x) := expx;ru(x)] pushes forward
to . According to Theorem 8, both s and t minimize (1) on S ( ), hence they
coincide -almost everyhwere.
2
Corollary 10 (Invertibility) Take M , g, , , c = d2=2,
= cc and t from
Theorem 9. If << vol also, then the inverse map t (y) := expy ;r c (y)] belongs
to S ( ) and satis es t (t(x)) = x -a.e.] and t(t (y )) = y -a.e.].
Proof: We shall exploit the symmetry $ and ( c) $ ( c), where := c
is the inmal convolution of with c(x y), so that c = cc = and cc = c = .
Denote the set where is dierentiable by dom r M . Now dom r] = 1 by
Lemmas 2 and 4, so t# = implies U := dom r \ t;1 (dom r) is a Borel set of
full measure U ] = 1. For each x 2 U Lemma 7 yields
0 = c(x t(x)) ; c(t(x)) ; (x)
= c(t(x) x) ; (t(x)) ; c(x):
Since t(x) 2 dom r, we conclude x = expt(x) ;r(t(x))] from the same lemma
applied to . Thus x = t (t(x)) on U and hence -a.e., which we'll use to prove
t 2 S ( ).
Given any continuous function h 2 C (M ) we have
Z
M
h d(t ) =
#
Z
M
h(t (y )) d (y ) =
Z
M
h(t (t(x))) d(x) =
Z
M
h(x) d(x)
13
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
from the change of variables formula (8) applied to t# and = t#, and the fact
t (t(x)) = x -a.e.] established above. Since h 2 C (M ) was arbitrary this shows
t# = as desired.
Now it follows by symmetry that V := dom r \ (t );1(dom r ) is a Borel set
with V ] = 1, while t(t(y)) = y holds for each y 2 V to conclude the corollary. 2
Theorem 11 (Polar Factorization of Maps) Let (M g) be a connected, compact
Riemannian manifold, C 3 -smooth and without boundary. Fix c(x y) = d2 (x y)=2, a
Borel map s : M ;! M and a Radon measure << vol on M . If s# << vol,
meaning s maps no set with positive measure to a set of zero volume, then s = t u
-a.e.] for some map t(x) = expx;r (x)] with = cc and a measure preserving
map u# = . The two factoring maps u t : M ;! M are unique -a.e. and Borel.
Proof: Use (2) to dene := s# as the push-forward of the measure through the
map s : M ;! M . Compactness of M combines with the local niteness of Radon
measures to provide a normalization constant ;1 := M ] = M ] < 1 which
makes and probability measures, both absolutely continuous with respect to
the volume by hypothesis. Theorem 9 provides a potential = cc on M for which the
Borel map t(x) = expx;r (x)] pushes forward to , or equivalently t 2 S ( ).
Its corollary provides a Borel inverse t 2 S ( ) such that t(t(y)) = y holds on a
Borel set V M of full -measure: V ] = 1. Setting u := t s, one immediately
veries s(x) = t(u(x)) holds on s;1 (V ), hence -a.e. The existence proof is completed
by noting that t s 2 S ( ) follows from s 2 S ( ) and t 2 S ( ): three changes
of variables (8) yield
Z
M
h d(u#) =
Z
M
h(t (s(x))) d(x) =
Z
M
h(t (y )) d (y ) =
Z
M
h d
for each h 2 C (M ), proving u measure-preserving, u# = .
To establish the uniqueness of this decomposition, suppose s = t0 u0 holds on
a subset U 0 M of full -measure, where t0 (x) = expx;r(x)] with = cc and
u0 2 S ( ). Clearly t0 2 S ( s#), so t0 = t holds on a set U M of full measure by Theorem 9. For x 2 s;1(V ) \ u0;1(U ) \ U 0 , which is to say -a.e., one has
s(x) = t0 (u0 (x)) = t(u0 (x)) hence u(x) := t (s(x)) = u0 (x), to complete the uniqueness
proof.
2
4 Polar Factorization as Nonlinear Hodge Theorem
As a nal remark, in the Riemannian setting we expose the formal link between the
polar factorization of maps and the Hodge / Helmholtz decomposition of vector elds
uncovered in the Euclidean context by Brenier 4]. On a compact manifold, this
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
14
decomposition asserts that any smooth vector eld v : M ;! T M decomposes as
v = w + r where w is divergence free and : M ;! R. Just as a vector eld
generates (and linearly approximates) a one-parameter family of dieomorphisms, the
Hodge / Helmholtz decomposition arises from the linear approximation of the polar
factors in Theorem 11. Indeed, let s(x ) 2 M be the ow along the vector eld v
obtained by integrating
s_ (x ) = v(s(x ))
(22)
from the initial condition s(x 0) = x the dot denotes dierentiation with respect to .
At each instant , Theorem 11 factorizes the dieomorphism s(x ) = t(u(x ) ) as
the composition of a volume-preserving map u( ) with t( ) := exp;r ( )],
where (x ) is a d2=2-concave function of x for each . As we shortly derive, the
Hodge theorem follows formally from the relation
@
s_ (z 0) =
t(u(x ) )
@
(z0)
(23)
and the identications w = @u=@ and = ;@ =@ evaluated at = 0.
Since the derivation is formal, we simply assume the map u(x ) and potential
(x ) provided by Theorem 11 are C 2 smooth in both space and time. Now the
unique polar factors of s(x 0) = x are u(x 0) = x = t(x 0). As long as is small,
our smoothness assumption ensures all three maps stay close to the identity, so we
may work in a small neighbourhood of any (0 z) 2 T M in the tangent bundle,
where we have normal coordinates (p1 : : : pn x1 : : : xn) vanishing at (0 z). Letting
(p x) := expx p so that s(x ) = (;r (u(x ) ) u(x )), we compute
@ i @ i @ jk
@
@si j
= j u_ (z 0) + j ; g (u(x )) @xk (u(x ) )
@ (z0)
@x (0z)
@p (0z) @
(z0)
"
#
ik 2
2
= u_ i ; @g m @ k u_ m ; gik (z) @m k u_ m + @ k
@x z @x
@x @x
@@x (z0)
"
#
@_
= u_ i ; gik (z) @x
(24)
k
(z0)
with summation on like indices. Here the expressions i(0 x) = xi and i (p z) =
pi in normal coordinates were used to obtain the second equality, while (x 0) =
const throughout M (since t(x 0) = x) has been invoked to eliminate purely spatial
derivatives of (x 0) from the third equality. Identifying w(z) = u_ (z 0) and (z) =
; _ (z 0), we recover the Hodge / Helmholtz decomposition v = w + r globally from
(22-24) here w is divergence free since the C 2 dieomorphism u(z ) of M preserves
volumes at each .
RJMc/Polar Factorization of Maps on Manifolds/May 27, 1999
15
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