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OPTIMAL TRANSPORT AND CURVATURE

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OPTIMAL TRANSPORT AND CURVATURE ALESSIO FIGALLI AND CEDRIC VILLANI Introduction These notes record the six lectures for the CIME Summer Course held by the second author in Cetraro during the week of June 23-28, 2008, with minor modifications. Their goal is to describe some recent developements in the theory of optimal transport, and their applications to differential geometry. We will focus on two main themes: (a) Stability of lower bounds on Ricci curvature under measured Gromov–Hausdorff conver- gence. (b) Smoothness of optimal transport in curved geometry. The main reference for all the material covered by these notes (and much more) is the recent book of the second author [45]. These notes are organized as follows: • In Section 1 we recall some classical facts of metric and differential geometry; then in Section 2 we study the optimal transport problem on Riemannian manifolds. These sections introduce the basic objects and the notation. • In Section 3 we reformulate lower bounds on Ricci curvature in terms of the “displacement convexity” of certain functionals, and deduce the stability. Then in Section 4 we address the question of the smoothness of the optimal transport on Riemannian manifold. These two sec- tions, focusing on Problems (a) and (b) respectively, constitute the heart of these notes, and can be read independently of each other.

  • differential geometry

  • inf ?

  • riemannian manifold

  • speed minimizing

  • when ?x ?

  • such curves

  • riemannian metric

  • geometry only


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OPTIMAL TRANSPORT AND CURVATURE
´ALESSIO FIGALLI AND CEDRIC VILLANI
Introduction
These notes record the six lectures for the CIME Summer Course held by the second author
in Cetraro during the week of June 23-28, 2008, with minor modifications. Their goal is to
describe some recent developements in the theory of optimal transport, and their applications
to differential geometry. We will focus on two main themes:
(a) Stability of lower boundson Ricci curvature under measured Gromov–Hausdorff conver-
gence.
(b) Smoothness of optimal transport in curved geometry.
The main reference for all the material covered by these notes (and much more) is the recent
book of the second author [45].
These notes are organized as follows:
• In Section 1 we recall some classical facts of metric and differential geometry; then in Section
2 we study the optimal transport problem on Riemannian manifolds. These sections introduce
the basic objects and the notation.
• In Section 3 we reformulate lower bounds on Ricci curvature in terms of the “displacement
convexity” of certain functionals, and deduce the stability. Then in Section 4 we address the
question of the smoothness of the optimal transport on Riemannian manifold. These two sec-
tions, focusing on Problems (a) and (b) respectively, constitute the heart of these notes, and
can be read independently of each other.
• Section 5 is devoted to a recap and the discussion of a few open problems; finally Section 6
gives a selection of the most relevant references.
1. Bits of metric geometry
The apparent redundancy in the title of this section is intended to stress the fact that we
shall be concerned with geometry only from the metric point of view (rather than from the
topological, or differential point of view), be it either in some possibly nonsmooth metric space,
or in a smooth Riemannian manifold.
1.1. Length. Let (X,d) be a complete separable metric space. Given a Lipschitz curve γ :
[0,T]→X, we define its length by
( )
N X
L(γ) := sup d(γ(t ),γ(t )) 0 =t ≤t ≤...≤t =T .i i+1 0 1 N+1
i=0
It is easily checked that the length of a curve is invariant by reparameterization.
1´2 ALESSIO FIGALLI AND CEDRIC VILLANI
In an abstract metric space the velocity γ˙(t) of a Lipschitz curve does not make sense; still
it is possible to give a meaning to the “modulus of the velocity”, or metric derivative of γ, or
speed:
d(γ(t+h),γ(t))
|γ˙(t)| := limsup
|h|h→0
For almost all t, the above limsup is a true limit [1, Theorem 4.1.6], and the following formula
holds: Z
T
L(γ) = |γ˙(t)|dt. (1.1)
0
1.2. Length spaces. In the previous subsection we have seen how to write the length of curves
in terms of the metricd. But once the length is defined, we can introduce a new distance onX:
n o
′d(x,y) := inf L(γ) γ∈ Lip([0,1],X), γ(0) =x, γ(1) =y .
′ ′By triangle inequality, d ≥ d. If d = d we say that (X,d) is a length space. It is worth
′recording that (X,d) defined as above is automatically a length space.
1 2 2Example 1.1. TakeX =S ⊂R , andd(x,y) =|x−y| the standard Euclidean distance inR ;
|x−y|′then d(x,y) = 2arcsin , so (X,d) is not a length space. More generally, if X is a closed
2
nsubset ofR then (X,d) is a length space if and only if X is convex.
1.3. Geodesics. A curve γ : [0,1] → X which minimizes the length, among all curves with
γ(0) =x andγ(1) =y, is called a geodesic, or more properly a minimizing geodesic.
The property of being a minimizing geodesic is stable by restriction: if γ : [0,1] → X is a
geodesic, then for all a<b∈ [0,1], γ| : [a,b]→X is a geodesic from γ(a) to γ(b).[a,b]
A length space such that any two points are joined by a minimizing geodesic is called a
geodesic space.
Example 1.2. By the (generalized) Hopf–Rinow theorem, any locally compact complete length
space is a geodesic space [4].
ItisageneralfactthataLipschitzcurveγ canbereparameterized sothat|γ˙(t)|isconstant[1,
Theorem 4.2.1]. Thus, any geodesic γ :[0,1]→X can be reparameterized so that|γ˙(t)| =L(γ)
for almostallt∈ [0,1]. Inthiscase,γ iscalled aconstant-speedminimizing geodesic. Such
curves are minimizers of the action functional
Z 11 2A(γ) := |γ˙(t)| dt.
2 0
More precisely, we have:
Proposition 1.3. Let (X,d) be a length space. Then
s
Z 1
2d(x,y) = inf |γ˙(t)| dt ∀x,y∈X.
γ(0)=x,γ(1)=y 0
Moreover, if (X,d) is a geodesic space, then minimizers of the above functional are precisely
constant-speed minimizing geodesics.OPTIMAL TRANSPORT AND CURVATURE 3
Sketch of the proof. By (1.1) we know that
Z 1
d(x,y) = inf |γ˙(t)|dt ∀x,y∈X.
γ(0)=x,γ(1)=y 0
By Jensen’s inequality,
s
Z Z1 1
2|γ˙(t)|dt≤ |γ˙(t)| dt,
0 0
with equality if and only if |γ˙(t)| is constant for almost all t. The conclusion follows easily.

∞1.4. Riemannianmanifolds. Givenann-dimensionalC differentiablemanifoldsM,foreach
x∈M we denote by T M the tangent space to M at x, and by TM :=∪ ({x}×T M) thex x∈M x
whole tangent bundle of M. On each tangent spaceT M, we assume that is given a symmetricx
positivedefinitequadraticformg :T M×T M →Rwhichdependssmoothlyonx;g = (g )x x x x x∈M
is called a Riemannian metric, and (M,g) is a Riemannian manifold.
A Riemannian metric defines a scalar product and a norm on each tangent space: for each
v,w∈T Mx p
hv,wi :=g (v,w), |v| := g (v,v).x x x x
n 1 nLetU bean open subsetofR and Φ :U → Φ(U)=V ⊂M a chart. Givenx = Φ(x ,...,x )∈
∂ ∂Φ 1 nV, the vectors := (x ,...,x ), i= 1,...,n, constitute a basis ofT M: anyv∈T M cani x x∂x ∂xiP
n ∂ibe written as v = v . We can use this chart to write our metricg in coordinates insideii=1 ∂x
V: n nX X∂ ∂ i j i jg (v,v) = g , v v = g (x)v v ,x x iji j∂x ∂x
i,j=1 i,j=1

∂ ∂ ijwhere by definition g (x) :=g , . We also denote by g the coordinates of the inverseij x i j∂x ∂xP
ij −1 ij i iof g: g = (g ) ; more precisely, g g =δ , where δ denotes Kronecker’s delta:ij jkj k k

1 if i =k,iδ =k 0 if i =k.
In the sequel we will use these coordinates to perform many computations. Einstein’s con-P
k kvention of summation over repeated indices will be used systematically: a b = a b ,k kkP
i j i jg v v = g v v , etc.ij iji,j
1.5. Riemannian distance and volume. The notion of “norm of a tangent vector” leads to
the definition of a distance on a Riemannian manifold (M,g), called Riemannian distance:
Z q1
d(x,y) = inf g γ˙(t),γ˙(t) dtγ(t)
γ(0)=x,γ(1)=y 0
s
Z 1
= inf g γ˙(t),γ˙(t) dt ∀x,y∈X.γ(t)
γ(0)=x,γ(1)=y 0
This definition makes (M,d) a length space.
6´4 ALESSIO FIGALLI AND CEDRIC VILLANI
A Riemannian manifold (M,g) is also equipped with a natural reference measure, the Rie-
mannian volume:
q
1 nvol(dx) =n-dimensional Hausdorff measure on (M,d) = det(g )dx ...dx .ij
nThis definition of the volume allows to write a change of variables formula, exactly as inR (see
for instance [45, Chapter 1]).
1.6. Differentialandgradients. Given asmoothmapϕ:M →R, its differentialdϕ :TM →
R is defined as

)