Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf ·...
Transcript of Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf ·...
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Curvature of Metric Spaces
John LottUniversity of Michigan
http://www.math.lsa.umich.edu/˜lott
May 23, 2007
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Special Session on Metric Differential Geometry
Thursday, Friday, Saturday morning
Joint work with Cedric Villani.
Related work was done independently by K.-T. Sturm.
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Special Session on Metric Differential Geometry
Thursday, Friday, Saturday morning
Joint work with Cedric Villani.
Related work was done independently by K.-T. Sturm.
![Page 4: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/4.jpg)
Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
![Page 5: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/5.jpg)
Introduction
Goal : We have notions of curvature from differential geometry.
Do they make sense for metric spaces?
For example, does it make sense to say that a metric space is“positively curved”?
Motivations :1. Intrinsic interest2. Understanding3. Applications to smooth geometry
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Introduction
Goal : We have notions of curvature from differential geometry.
Do they make sense for metric spaces?
For example, does it make sense to say that a metric space is“positively curved”?
Motivations :1. Intrinsic interest2. Understanding3. Applications to smooth geometry
![Page 7: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/7.jpg)
Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
![Page 8: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/8.jpg)
Length structure of a Riemannian manifold
Say M is a smooth n-dimensional manifold. For each m ∈ M,the tangent space TmM is an n-dimensional vector space.
A Riemannian metric g on M assigns, to each m ∈ M, an innerproduct on TmM.
Example : M is a submanifold of RN
If v ∈ TmM, let g(v, v) be the square of the length of v.
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Length structure of a Riemannian manifold
Say M is a smooth n-dimensional manifold. For each m ∈ M,the tangent space TmM is an n-dimensional vector space.
A Riemannian metric g on M assigns, to each m ∈ M, an innerproduct on TmM.
Example : M is a submanifold of RN
If v ∈ TmM, let g(v, v) be the square of the length of v.
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Length structure of a Riemannian manifold
The length of a smooth curve γ : [0, 1] → M is
L(γ) =
∫ 1
0
√g(γ, γ) dt .
The distance between m0, m1 ∈ M is the infimal length ofcurves joining m0 to m1.
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Length structure of a Riemannian manifold
d(m0, m1) = inf{L(γ) : γ(0) = m0, γ(1) = m1}.
Fact : this defines a metric on the set M.
Any length-minimizing curve from m0 to m1 is a geodesic.
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Riemannian volume
Any Riemannian manifold M comes equipped with a smoothpositive measure dvolM .
In local coordinates, if g =∑n
i,j=1 gij dx i dx j then
dvolM =√
det (gij) dx1dx2 . . . dxn.
The volume of a nice subset A ⊂ M is
vol(A) =
∫A
dvolM .
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Sectional curvature
To each m ∈ M and each 2-plane P ⊂ TmM in the tangentspace at m, one assigns a number K (P), its sectionalcurvature.
Example : If M is two-dimensional then P is all of TmM andK (P) is the Gaussian curvature at m.
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Sectional curvature
To each m ∈ M and each 2-plane P ⊂ TmM in the tangentspace at m, one assigns a number K (P), its sectionalcurvature.
Example : If M is two-dimensional then P is all of TmM andK (P) is the Gaussian curvature at m.
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Ricci curvature
Ricci curvature is an averaging of sectional curvature.
Fix a unit-length vector v ∈ TmM.
Definition
Ric(v, v) = (n − 1) · (the average sectional curvature
of the 2-planes P containing v).
Example : S2 × S2 ⊂ (R3 × R3 = R6) has nonnegativesectional curvatures but has positive Ricci curvatures.
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Ricci curvature
Ricci curvature is an averaging of sectional curvature.
Fix a unit-length vector v ∈ TmM.
Definition
Ric(v, v) = (n − 1) · (the average sectional curvature
of the 2-planes P containing v).
Example : S2 × S2 ⊂ (R3 × R3 = R6) has nonnegativesectional curvatures but has positive Ricci curvatures.
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Ricci curvature
What does Ricci curvature control?
1. Volume growth
Bishop-Gromov inequality :If M has nonnegative Ricci curvature then balls in M grow nofaster than in Euclidean space.
That is, for any m ∈ M, r−n vol(Br (m)) is nonincreasing in r .
2. The universe
Einstein says that a matter-free spacetime has vanishing Riccicurvature.
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Ricci curvature
What does Ricci curvature control?
1. Volume growth
Bishop-Gromov inequality :If M has nonnegative Ricci curvature then balls in M grow nofaster than in Euclidean space.
That is, for any m ∈ M, r−n vol(Br (m)) is nonincreasing in r .
2. The universe
Einstein says that a matter-free spacetime has vanishing Riccicurvature.
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Summary
Any Riemannian manifold gets
1. Lengths of curves2. A metric space structure3. A measure4. Sectional curvatures5. Ricci curvatures
Question :To what extent can we recover the sectional curvatures fromjust the metric space structure?
To what extent can we recover the Ricci curvatures from justthe metric space structure and the measure?
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Summary
Any Riemannian manifold gets
1. Lengths of curves2. A metric space structure3. A measure4. Sectional curvatures5. Ricci curvatures
Question :To what extent can we recover the sectional curvatures fromjust the metric space structure?
To what extent can we recover the Ricci curvatures from justthe metric space structure and the measure?
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Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
![Page 22: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/22.jpg)
Length spaces
Say (X , d) is a compact metric space and γ : [0, 1] → X is acontinuous map.
The length of γ is
L(γ) = supJ
sup0=t0≤t1≤...≤tJ=1
J∑j=1
d(γ(tj−1), γ(tj)).
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Length spaces
Say (X , d) is a compact metric space and γ : [0, 1] → X is acontinuous map.
The length of γ is
L(γ) = supJ
sup0=t0≤t1≤...≤tJ=1
J∑j=1
d(γ(tj−1), γ(tj)).
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Length spaces
(X , d) is a length space if the distance between two pointsx0, x1 ∈ X equals the infimum of the lengths of curves joiningthem, i.e.
d(x0, x1) = inf{L(γ) : γ(0) = x0, γ(1) = x1}.
A length-minimizing curve is called a geodesic.
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Length spaces
(X , d) is a length space if the distance between two pointsx0, x1 ∈ X equals the infimum of the lengths of curves joiningthem, i.e.
d(x0, x1) = inf{L(γ) : γ(0) = x0, γ(1) = x1}.
A length-minimizing curve is called a geodesic.
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Length spaces
Examples of length spaces :1. The underlying metric space of any Riemannian manifold.2.
Nonexamples :1. A finite metric space with more than one point.2. A circle with the chordal metric.
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Length spaces
Examples of length spaces :1. The underlying metric space of any Riemannian manifold.2.
Nonexamples :1. A finite metric space with more than one point.2. A circle with the chordal metric.
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Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
![Page 29: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/29.jpg)
Alexandrov curvature
DefinitionA compact length space (X , d) has nonnegative Alexandrovcurvature if geodesic triangles in X are at least as “fat” ascorresponding triangles in R2.
The comparison triangle in R2 has the same sidelengthsFatness :(Length of bisector from x) ≥ (Length of bisector from x0)
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Alexandrov curvature
DefinitionA compact length space (X , d) has nonnegative Alexandrovcurvature if geodesic triangles in X are at least as “fat” ascorresponding triangles in R2.
The comparison triangle in R2 has the same sidelengthsFatness :(Length of bisector from x) ≥ (Length of bisector from x0)
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Alexandrov curvature
Example : the boundary of a convex region in RN hasnonnegative Alexandrov curvature.
1. If (M, g) is a Riemannian manifold then its underlying metricspace has nonnegative Alexandrov curvature if and only if Mhas nonnegative sectional curvatures.
2. If {(Xi , di)}∞i=1 have nonnegative Alexandrov curvature andlimi→∞(Xi , di) = (X , d) in the Gromov-Hausdorff topology then(X , d) has nonnegative Alexandrov curvature.
3. Applications to Riemannian geometry
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Alexandrov curvature
Example : the boundary of a convex region in RN hasnonnegative Alexandrov curvature.
1. If (M, g) is a Riemannian manifold then its underlying metricspace has nonnegative Alexandrov curvature if and only if Mhas nonnegative sectional curvatures.
2. If {(Xi , di)}∞i=1 have nonnegative Alexandrov curvature andlimi→∞(Xi , di) = (X , d) in the Gromov-Hausdorff topology then(X , d) has nonnegative Alexandrov curvature.
3. Applications to Riemannian geometry
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Gromov-Hausdorff topology
A topology on the set of all compact metric spaces (moduloisometry).
(X1, d1) and (X2, d2) are close in the Gromov-Hausdorfftopology if somebody with bad vision has trouble telling themapart.
Example : a cylinder with a small cross-section isGromov-Hausdorff close to a line segment.
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Gromov-Hausdorff topology
A topology on the set of all compact metric spaces (moduloisometry).
(X1, d1) and (X2, d2) are close in the Gromov-Hausdorfftopology if somebody with bad vision has trouble telling themapart.
Example : a cylinder with a small cross-section isGromov-Hausdorff close to a line segment.
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The question
Can one extend Alexandrov’s work from sectional curvature toRicci curvature?
Motivation : Gromov’s precompactness theorem
TheoremGiven N ∈ Z+ and D > 0,
{(M, g) : dim(M) = N, diam(M) ≤ D, RicM ≥ 0}
is precompact in the Gromov-Hausdorff topology on{compact metric spaces}/isometry.
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Gromov-Hausdorff space
Each point is a compact metric space.Each interior point is a Riemannian manifold (M, g) withdim(M) = N, diam(M) ≤ D and RicM ≥ 0.
The boundary points are compact metric spaces (X , d) withdimH X ≤ N. They are generally not manifolds.(Example : X = M/G.)
In some moral sense, the boundary points are metric spaceswith “nonnegative Ricci curvature”.
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Gromov-Hausdorff space
Each point is a compact metric space.Each interior point is a Riemannian manifold (M, g) withdim(M) = N, diam(M) ≤ D and RicM ≥ 0.
The boundary points are compact metric spaces (X , d) withdimH X ≤ N. They are generally not manifolds.(Example : X = M/G.)
In some moral sense, the boundary points are metric spaceswith “nonnegative Ricci curvature”.
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Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
![Page 39: Curvature of Metric Spaces - University of California, Berkeleymath.berkeley.edu/~lott/zlott.pdf · Special Session on Metric Differential Geometry Thursday, Friday, Saturday morning](https://reader033.fdocuments.us/reader033/viewer/2022060217/5f0657647e708231d417825b/html5/thumbnails/39.jpg)
Dirtmoving
Given a before and an after dirtpile, what is the most efficientway to move the dirt from one place to the other?
Let’s say that the cost to move a gram of dirt from x to y isd(x , y)2.
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Gaspard Monge
Memoire sur la theorie des deblais et des remblais (1781)
Memoir on the theory of excavations and fillings (1781)
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Gaspard Monge
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Wasserstein space
Let (X , d) be a compact metric space.
NotationP(X ) is the set of Borel probability measures on X.
That is, µ ∈ P(X ) iff µ is a nonnegative Borel measure on Xwith µ(X ) = 1.
DefinitionGiven µ0, µ1 ∈ P(X ), the Wasserstein distance W2(µ0, µ1) isthe square root of the minimal cost to transport µ0 to µ1.
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Wasserstein space
W2(µ0, µ1)2 = inf
{∫X×X
d(x , y)2 dπ(x , y)
},
where
π ∈ P(X × X ), (p0)∗π = µ0, (p1)∗π = µ1.
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Wasserstein space
Fact :(P(X ), W2) is a metric space, called the Wasserstein space.
The metric topology is the weak-∗ topology, i.e. limi→∞ µi = µ ifand only if for all f ∈ C(X ), limi→∞
∫X f dµi =
∫X f dµ.
PropositionIf X is a length space then so is the Wasserstein space P(X ).
Hence we can talk about its (minimizing) geodesics {µt}t∈[0,1],called Wasserstein geodesics.
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Wasserstein space
Fact :(P(X ), W2) is a metric space, called the Wasserstein space.
The metric topology is the weak-∗ topology, i.e. limi→∞ µi = µ ifand only if for all f ∈ C(X ), limi→∞
∫X f dµi =
∫X f dµ.
PropositionIf X is a length space then so is the Wasserstein space P(X ).
Hence we can talk about its (minimizing) geodesics {µt}t∈[0,1],called Wasserstein geodesics.
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Optimal transport
What is the optimal transport scheme between µ0, µ1 ∈ P(X )?
X = Rn, µ0 and µ1 absolutely continuous :Rachev-Ruschendorf, Brenier (1990)
X a Riemannian manifold, µ0 and µ1 absolutely continuous :McCann (2001)
Empirical fact : The Ricci curvature of the Riemannian manifoldaffects the optimal transport in a quantitative way.Otto-Villani (2000),Cordero-Erausquin-McCann-Schmuckenschlager (2001)
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Metric-measure spaces
DefinitionA metric-measure space is a metric space (X , d) equipped witha given probability measure ν ∈ P(X ).
A smooth metric-measure space is a Riemannian manifold(M, g) with a smooth probability measure dν = e−Ψ dvolM .
Idea : Use optimal transport on X to define what it means for(X , ν) to have “nonnegative Ricci curvature”.
(X , d) −→ (P(X ), W2)To one compact length space we have assigned another.Use the properties of the Wasserstein space (P(X ), W2) to saysomething about the geometry of (X , d).
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Metric-measure spaces
DefinitionA metric-measure space is a metric space (X , d) equipped witha given probability measure ν ∈ P(X ).
A smooth metric-measure space is a Riemannian manifold(M, g) with a smooth probability measure dν = e−Ψ dvolM .
Idea : Use optimal transport on X to define what it means for(X , ν) to have “nonnegative Ricci curvature”.
(X , d) −→ (P(X ), W2)To one compact length space we have assigned another.Use the properties of the Wasserstein space (P(X ), W2) to saysomething about the geometry of (X , d).
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Measured Gromov-Hausdorff limits
An easy consequence of Gromov precompactness :{(M, g,
dvolMvol(M)
): dim(M) = N, diam(M) ≤ D, RicM ≥ 0
}is precompact in the measured Gromov-Hausdorff topology on{compact metric-measure spaces}/isometry.
What can we say about the limit points? (Work ofCheeger-Colding)
What are the smooth limit points?
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Measured Gromov-Hausdorff limits
An easy consequence of Gromov precompactness :{(M, g,
dvolMvol(M)
): dim(M) = N, diam(M) ≤ D, RicM ≥ 0
}is precompact in the measured Gromov-Hausdorff topology on{compact metric-measure spaces}/isometry.
What can we say about the limit points? (Work ofCheeger-Colding)
What are the smooth limit points?
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Measured Gromov-Hausdorff (MGH) topology
Definitionlimi→∞(Xi , di , νi) = (X , d , ν) if there are Borel mapsfi : Xi → X and a sequence εi → 0 such that1. (Almost isometry) For all xi , x ′i ∈ Xi ,
|dX (fi(xi), fi(x′i ))− dXi
(xi , x ′i )| ≤ εi .
2. (Almost surjective) For all x ∈ X and all i , there is somexi ∈ Xi such that
dX (fi(xi), x) ≤ εi .
3. limi→∞(fi)∗νi = ν in the weak-∗ topology.
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Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
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Notation
X a compact Hausdorff space.
P(X ) = Borel probability measures on X , with weak-∗ topology.
U : [0,∞) → R a continuous convex function with U(0) = 0.
Fix a background measure ν ∈ P(X ).
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Entropy
The “negative entropy” of µ with respect to ν is
Uν(µ) =
∫X
U(ρ(x)) dν(x) + U ′(∞) µs(X ).
Hereµ = ρ ν + µs
is the Lebesgue decomposition of µ with respect to ν and
U ′(∞) = limr→∞
U(r)r
.
Uν(µ) measures the nonuniformity of µ w.r.t. ν. It is minimizedwhen µ = ν.
We get a function Uν : P(X ) → R ∪∞.
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Entropy
The “negative entropy” of µ with respect to ν is
Uν(µ) =
∫X
U(ρ(x)) dν(x) + U ′(∞) µs(X ).
Hereµ = ρ ν + µs
is the Lebesgue decomposition of µ with respect to ν and
U ′(∞) = limr→∞
U(r)r
.
Uν(µ) measures the nonuniformity of µ w.r.t. ν. It is minimizedwhen µ = ν.
We get a function Uν : P(X ) → R ∪∞.
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Effective dimension
N ∈ [1,∞] a new parameter (possibly infinite).
It turns out that there’s not a single notion of “nonnegative Riccicurvature”, but rather a 1-parameter family. That is, for each N,there’s a notion of a space having “nonnegative N-Riccicurvature”.
Here N is an effective dimension of the space, and must beinputted.
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Displacement convexity classes
Definition(McCann) If N < ∞ then DCN is the set of such convexfunctions U so that the function
λ → λN U(λ−N)
is convex on (0,∞).
DefinitionDC∞ is the set of such convex functions U so that the function
λ → eλ U(e−λ)
is convex on (−∞,∞).
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Displacement convexity classes
Example
UN(r) =
{Nr(1− r−1/N) if 1 < N < ∞,
r log r if N = ∞.
If U = U∞ then the corresponding functional is
Uν(µ) =
{∫X ρ log ρ dν if µ is a.c. w.r.t. ν,
∞ otherwise,
where µ = ρ ν.
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Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
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Convexity on Wasserstein space
(X , d) is a compact length space.
ν is a fixed probability measure on X .
We want to ask whether the negative entropy function Uν is aconvex function on P(X ).
That is, given µ0, µ1 ∈ P(X ), whether Uν restricts to a convexfunction along a Wasserstein geodesic {µt}t∈[0,1] from µ0 to µ1.
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Convexity on Wasserstein space
(X , d) is a compact length space.
ν is a fixed probability measure on X .
We want to ask whether the negative entropy function Uν is aconvex function on P(X ).
That is, given µ0, µ1 ∈ P(X ), whether Uν restricts to a convexfunction along a Wasserstein geodesic {µt}t∈[0,1] from µ0 to µ1.
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Nonnegative N-Ricci curvature
DefinitionGiven N ∈ [1,∞], we say that a compact measured lengthspace (X , d , ν) has nonnegative N-Ricci curvature if :
For all µ0, µ1 ∈ P(X ) with supp(µ0) ⊂ supp(ν) andsupp(µ1) ⊂ supp(ν), there is some Wasserstein geodesic{µt}t∈[0,1] from µ0 to µ1 so that for all U ∈ DCN and all t ∈ [0, 1],
Uν(µt) ≤ t Uν(µ1) + (1− t) Uν(µ0).
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Nonnegative N-Ricci curvature
Note : We only require convexity along some geodesic from µ0
to µ1, not all geodesics.
But the same geodesic has to work for all U ∈ DCN .
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What does this have to do with curvature?
Look at optimal transport on the 2-sphere.ν = normalized Riemannian density.Take µ0, µ1 two disjoint congruent blobs. Uν(µ0) = Uν(µ1).
Optimal transport from µ0 to µ1 goes along longitudes.Positive curvature gives focusing of geodesics.
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What does this have to do with curvature?
Look at optimal transport on the 2-sphere.ν = normalized Riemannian density.Take µ0, µ1 two disjoint congruent blobs. Uν(µ0) = Uν(µ1).
Optimal transport from µ0 to µ1 goes along longitudes.Positive curvature gives focusing of geodesics.
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Take a snapshot at time t .
The intermediate-time blob µt is more spread out, so it’s moreuniform with respect to ν.The more uniform the measure, the higher its entropy.So the entropy is a concave function of t , i.e. the negativeentropy is a convex function of t .
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Take a snapshot at time t .
The intermediate-time blob µt is more spread out, so it’s moreuniform with respect to ν.The more uniform the measure, the higher its entropy.
So the entropy is a concave function of t , i.e. the negativeentropy is a convex function of t .
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Take a snapshot at time t .
The intermediate-time blob µt is more spread out, so it’s moreuniform with respect to ν.The more uniform the measure, the higher its entropy.So the entropy is a concave function of t , i.e. the negativeentropy is a convex function of t .
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Main result
TheoremLet {(Xi , di , νi)}∞i=1 be a sequence of compact measured lengthspaces with
limi→∞
(Xi , di , νi) = (X , d , ν)
in the measured Gromov-Hausdorff topology.
For any N ∈ [1,∞], if each (Xi , di , νi) has nonnegative N-Riccicurvature then (X , d , ν) has nonnegative N-Ricci curvature.
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What does all this have to do with Ricci curvature?
Let (M, g) be a compact connected n-dimensional Riemannianmanifold.We could take the Riemannian measure, but let’s be moregeneral.
Say Ψ ∈ C∞(M) has ∫M
e−Ψ dvolM = 1.
Put ν = e−Ψ dvolM .
Any smooth positive probability measure on M can be written inthis way.
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What does all this have to do with Ricci curvature?
Let (M, g) be a compact connected n-dimensional Riemannianmanifold.We could take the Riemannian measure, but let’s be moregeneral.
Say Ψ ∈ C∞(M) has ∫M
e−Ψ dvolM = 1.
Put ν = e−Ψ dvolM .
Any smooth positive probability measure on M can be written inthis way.
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What does all this have to do with Ricci curvature?
For N ∈ [1,∞], define the N-Ricci tensor RicN of (Mn, g, ν) byRic + Hess(Ψ) if N = ∞,
Ric + Hess(Ψ) − 1N−n dΨ⊗ dΨ if n < N < ∞,
Ric + Hess(Ψ) − ∞ (dΨ⊗ dΨ) if N = n,
−∞ if N < n,
where by convention ∞ · 0 = 0.
RicN is a symmetric covariant 2-tensor field on M that dependson g and Ψ.
(If N = n then RicN is −∞ except where dΨ = 0. There,RicN = Ric.)
Ric∞ = Bakry-Emery tensor = right-hand side of Perelman’smodified Ricci flow equation.
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What does all this have to do with Ricci curvature?
For N ∈ [1,∞], define the N-Ricci tensor RicN of (Mn, g, ν) byRic + Hess(Ψ) if N = ∞,
Ric + Hess(Ψ) − 1N−n dΨ⊗ dΨ if n < N < ∞,
Ric + Hess(Ψ) − ∞ (dΨ⊗ dΨ) if N = n,
−∞ if N < n,
where by convention ∞ · 0 = 0.
RicN is a symmetric covariant 2-tensor field on M that dependson g and Ψ.
(If N = n then RicN is −∞ except where dΨ = 0. There,RicN = Ric.)
Ric∞ = Bakry-Emery tensor = right-hand side of Perelman’smodified Ricci flow equation.
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Abstract Ricci recovers classical Ricci
Recall that ν = e−Ψ dvolM .
TheoremFor N ∈ [1,∞], the measured length space (M, g, ν) hasnonnegative N-Ricci curvature if and only if RicN ≥ 0.
Classical case : Ψ constant, so ν = dvolvol(M) .
Then (Mn, g, ν) has abstract nonnegative N-Ricci curvature ifand only if it has classical nonnegative Ricci curvature, as soonas N ≥ n.
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Abstract Ricci recovers classical Ricci
Recall that ν = e−Ψ dvolM .
TheoremFor N ∈ [1,∞], the measured length space (M, g, ν) hasnonnegative N-Ricci curvature if and only if RicN ≥ 0.
Classical case : Ψ constant, so ν = dvolvol(M) .
Then (Mn, g, ν) has abstract nonnegative N-Ricci curvature ifand only if it has classical nonnegative Ricci curvature, as soonas N ≥ n.
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Curvature of Metric Spaces
Introduction
Five minute summary of differential geometry
Metric geometry
Alexandrov curvature
Optimal transport
Entropy functionals
Abstract Ricci curvature
Applications
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Smooth limit spaces
Had Gromov precompactness theorem. What are the limitspaces (X , d , ν)? Suppose that the limit space is a smoothmeasured length space, i.e.
(X , d , ν) = (B, gB, e−Ψ dvolB)
for some n-dimensional smooth Riemannian manifold (B, gB)and some Ψ ∈ C∞(B).
TheoremIf (B, gB, e−Ψ dvolB) is a measured Gromov-Hausdorff limit ofRiemannian manifolds with nonnegative Ricci curvature anddimension at most N then RicN(B) ≥ 0.
Note : the dimension can drop on taking limits.
The converse is true if N ≥ n + 2.
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Smooth limit spaces
Had Gromov precompactness theorem. What are the limitspaces (X , d , ν)? Suppose that the limit space is a smoothmeasured length space, i.e.
(X , d , ν) = (B, gB, e−Ψ dvolB)
for some n-dimensional smooth Riemannian manifold (B, gB)and some Ψ ∈ C∞(B).
TheoremIf (B, gB, e−Ψ dvolB) is a measured Gromov-Hausdorff limit ofRiemannian manifolds with nonnegative Ricci curvature anddimension at most N then RicN(B) ≥ 0.
Note : the dimension can drop on taking limits.
The converse is true if N ≥ n + 2.
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Smooth limit spaces
Had Gromov precompactness theorem. What are the limitspaces (X , d , ν)? Suppose that the limit space is a smoothmeasured length space, i.e.
(X , d , ν) = (B, gB, e−Ψ dvolB)
for some n-dimensional smooth Riemannian manifold (B, gB)and some Ψ ∈ C∞(B).
TheoremIf (B, gB, e−Ψ dvolB) is a measured Gromov-Hausdorff limit ofRiemannian manifolds with nonnegative Ricci curvature anddimension at most N then RicN(B) ≥ 0.
Note : the dimension can drop on taking limits.
The converse is true if N ≥ n + 2.
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Bishop-Gromov-type inequality
TheoremIf (X , d , ν) has nonnegative N-Ricci curvature and x ∈ supp(ν)then r−N ν(Br (x)) is nonincreasing in r .
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Lichnerowicz inequality
If (M, g) is a compact Riemannian manifold, let λ1 be thesmallest positive eigenvalue of the Laplacian −∇2.
TheoremLichnerowicz (1964)If dim(M) = n and M has Ricci curvatures bounded below byK > 0 then
λ1 ≥n
n − 1K .
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Sharp global Poincare inequality
TheoremIf (X , d , ν) has N-Ricci curvature bounded below by K > 0 andf is a Lipschitz function on X with
∫X f dν = 0 then∫
Xf 2 dν ≤ N − 1
N1K
∫X|∇f |2 dν.
Here
|∇f |(x) = lim supy→x
|f (y)− f (x)|d(y , x)
.
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Sharp global Poincare inequality
TheoremIf (X , d , ν) has N-Ricci curvature bounded below by K > 0 andf is a Lipschitz function on X with
∫X f dν = 0 then∫
Xf 2 dν ≤ N − 1
N1K
∫X|∇f |2 dν.
Here
|∇f |(x) = lim supy→x
|f (y)− f (x)|d(y , x)
.
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Open questions
1. Take any result that you know about Riemannian manifoldswith nonnegative Ricci curvature.
Does it extend to measured length spaces (X , d , ν) withnonnegative N-Ricci curvature?(Yes for Bishop-Gromov, no for splitting theorem.)
2. Take an interesting measured length space (X , d , ν). Does ithave nonnegative N-Ricci curvature?
This almost always boils down to understanding the optimaltransport on X .