\(A,B \subseteq \mathbb{R}^n\)  , \(0 \le \lambda \le 1\)  .

\[(1-\lambda) A + \lambda B : = \left\{(1-\lambda)a + \lambda b \, \mid \, a \in A, \, b \in B\right\}.\]

Theorem (Brunn-Minkowski) :

\(A,B \subseteq \mathbb{R}^n\)   Borel, nonempty, \(\qquad 0 \le \lambda \le 1\)  , \[\mathrm{Vol}((1-\lambda) A + \lambda B)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}(A)^{1/n} + \lambda\cdot\mathrm{Vol}(B)^{1/n}.\]

Theorem (Brunn-Minkowski) :

\(A,B \subseteq (M,g)\)   Borel, nonempty, \(\qquad 0 \le \lambda \le 1\)  , \[\mathrm{Vol}((1-\lambda) A + \lambda B)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}(A)^{1/n} + \lambda\cdot\mathrm{Vol}(B)^{1/n}.\]

\[ A_\lambda : = \left\{\gamma(\lambda T) \, \bigg\vert \, \begin{array}{c} \gamma:[0,T] \to M\,\,\text{unit-speed minimizing geodesic,}\\ \gamma(0) \in A_0,\,\, \gamma(T) \in A_1\end{array}\right\}. \]

Theorem (Cordero-Erausquin, McCann & Schmuckenschläger '01, Sturm '06): \((M,g)\)  complete \(n\)-dim Riemannian manifold, \[\mathrm{Ric}_g \ge 0\quad \implies \quad \forall A_0,A_1\,\, \text{Borel,} \,\,\mathrm{Vol}_g(A_0)\mathrm{Vol}_g(A_1) >0\] \[\mathrm{Vol}_g(A_\lambda)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}_g(A_0)^{1/n} + \lambda\cdot\mathrm{Vol}_g(A_1)^{1/n}.\]

\(\mathrm{Ric}_g \ge k \implies\) distorted Brunn-Minkowski.

In fact \(\iff\)  (Magnabosco, Portinale, Rossi '22).

Metric measure spaces (Sturm '06, Cavalletti-Mondino '15), Finsler (Ohta '09), Sub-Riemannian (Balogh, Kristály & Sipos '16, Barilari, Rizzi & Mondino '22), Lorentzian (McCann '20, Cavalletti-Mondino '20).

Theorem (Cordero-Erausquin, McCann & Schmuckenschläger '01, Sturm '06):   \((M,g,\mu)\)  complete \(n\)-dim weighted Riemannian manifold, \(N \in [n,\infty]\). \[\mathrm{Ric}_{\mu,N} \ge 0\quad \implies \quad \forall A_0,A_1\,\, \text{Borel,} \,\,\mathrm{Vol}_g(A_0)\mathrm{Vol}_g(A_1) >0\] \[ \mu(A_\lambda) \ge \begin{cases} \left((1-\lambda)\cdot\mu(A_0)^{1/N} + \lambda\cdot\mu(A_1)^{1/N}\right)^N & N \lt \infty \\ \mu(A_0)^{1-\lambda}\cdot\mu(A_1)^\lambda & N = \infty. \end{cases} \]

\(\mathrm{Ric}_{\mu,N} \ge k \implies\) distorted Brunn-Minkowski.

In fact \(\iff\)  (Magnabosco, Portinale, Rossi '22).

Metric measure spaces (Sturm '06, Cavalletti-Mondino '15), Finsler (Ohta '09), Sub-Riemannian (Balogh, Kristály & Sipos '16, Barilari, Rizzi & Mondino '22), Lorentzian (McCann '20, Cavalletti-Mondino '20).

Theorem (A. '25): Let \(M\) be a smooth manifold, let \(L:TM\to\mathbb{R}\) be a Tonelli Lagrangian satisfying [assumptions]. Let \(\mu\) be a measure on \(M\) with a smooth density, and let \(N \in [n,\infty]\).

Let \(A_0,A_1 \subseteq M\)   be Borel sets of positive measure and let \(0 \le \lambda \le 1 \), \[ A_\lambda : = \left\{\gamma(\lambda \ell) \, \bigg\vert \, \begin{array}{c} \gamma:[0,\ell] \to M \, \, \text{ is a minimizing extremal,} \\ \gamma(0) \in A_0, \, \,\gamma(\ell) \in A_1\end{array}\right\}. \]

If   \(\mathrm{Ric}_{L,\mu,N} \ge 0\)   on \(E^{-1}(0),\) then \[ \mu(A_\lambda) \ge \begin{cases} \left((1-\lambda)\cdot\mu(A_0)^{1/N} + \lambda \cdot \mu(A_1)^{1/N}\right)^N & N \lt \infty\\ \mu(A_0)^{1-\lambda}\cdot\mu(A_1)^\lambda & N = \infty. \end{cases} \]

Let \(\mathbb{C}\mathbf{H}^d\)  denote the complex hyperbolic space of complex dimension \(d.\)
A horocycle is a unit-speed curve \(\gamma\)  satisfying \[ \nabla_{\dot\gamma}\dot\gamma = \mathbf{J}\dot\gamma, \] where \(\mathbf{J}\)  is the complex structure.

For every \(x,y \in \mathbb{C}\mathbf{H}^d\)  there exists a unique horocycle \(\gamma:[0,T]\to \mathbb{C}\mathbf{H}^d\)  satisfying \(\gamma(0) = x\)  and \(\gamma(T) = y\) ; it is contained in the unique complex geodesic (totally-geodesic copy of the hyperbolic plane) containing \(x\)  and \(y\) .

Theorem (A.-Klartag '22, A. '25):   Let \(A_0,A_1 \subseteq \mathbb{C}\mathbf{H}^d\)  be Borel sets of positive measure and let \(0\le\lambda\le 1\) . Denote by \(A_\lambda\)  the set of points of the form \(\gamma(\lambda \ell)\) , where \(\gamma:[0,\ell]\to \mathbb{C}\mathbf{H}^d\)  is a horocycle satisfying \(\gamma(0) \in A_0\)  and \(\gamma(\ell) \in A_1\) . \[\implies \qquad \mathrm{Vol}(A_\lambda)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}(A_0)^{1/n} + \lambda\cdot\mathrm{Vol}(A_1)^{1/n}, \] where \(\mathrm{Vol}\)  denotes the hyperbolic volume measure and \(n = 2d\) .

Let \(S^{2d+1} = \{z \in \mathbb{C}^{d+1}\, \mid \, |z|=1\}\)    and let \(s \in (-1,1).\)   A contact magnetic geodesic of strength \(s\) is a solution to the ODE \[\nabla_{\dot\gamma}\dot\gamma = 2si\left(\dot\gamma - \eta(\dot\gamma)\cdot i\gamma\right), \qquad |\dot\gamma| \equiv 1,\] where \(\eta\) is the contact one-form \[\eta(v) = \mathrm{Re}\left\langle iz,v\right\rangle, \qquad v \in T_zS^{2d+1}, \,\, z \in S^{2d+1}.\]

• Contact magnetic geodesics of strength \(0\) are great circles.
• Reeb trajectories are contact magnetic geodesics of strength \(s.\)
• Legendrian contact magnetic geodesics are sub-Riemannian geodesics.

Theorem (A. '25): Let $A_0,A_1 \subseteq S^{2d+1}$  be Borel sets of positive measure and let $0\le\lambda\le 1$ . Denote by $A_\lambda$ the set of points of the form $\gamma(\lambda \ell)$, where $\gamma:[0,\ell]\to S^{2d+1}$   is a unit-speed contact magnetic geodesic of strength $s\in(-1,1)$ satisfying $\gamma(0) \in A_0$ and $\gamma(\ell) \in A_1$. Then $$\mathrm{Vol}(A_\lambda)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}(A_0)^{1/n} + \lambda\cdot\mathrm{Vol}(A_1)^{1/n},$$ where $\mathrm{Vol}$   denotes the spherical volume measure and $n = 2d+1$.

A Tonelli Lagrangian \(L\) on a manifold \(M\) is a function on the tangent bundle \(TM\) which is smooth on \(TM\setminus\mathbf{0}\) and fiberwise strongly convex and superlinear.

The action of a curve \(\gamma:[0,T]\to M \)   is: \[\mathrm{A}(\gamma) : = \int_0^T L(\dot\gamma(t))dt.\]

A minimizing extremal of \(L\) is a curve which minimizes the action \(\mathrm{A}\) among all curves with the same endpoints. Extremals are solutions to the Euler-Lagrange equation.

• Riemannian metrics:  \(L = g/2.\)
• Mechanical Lagrangians:  \(L = g/2 - U,\)  where \(U\) is a potential depending on position.
• Magnetic:  \(L = g/2 + \eta,\)  where \(\eta\) is a one-form.
• Finsler metrics: fiberwise  quadratic 2-homogeneous.
• Isotropic:  \(L = \phi(F),\)  where \(F\) is Riemannian or Finslerian metric and \(\phi\) is convex (e.g. \(|\cdot|^p\)).

\(L = g/2 + \eta\) \(g\) — Riemannian metric,   \(\eta\) — one-form.

\[\text{Euler-Lagrange:}\qquad \nabla_{\dot\gamma}\dot\gamma = \mathrm{Y}\dot\gamma, \quad \text{where } \left\langle\mathrm{Y}\,\cdot\,, \,\cdot \, \right\rangle = d\eta.\]

The Hamiltonian is: \[H(p) : = \sup_{v \in T_xM}p(v) - L(v), \quad p\in T^*_xM,\,\,x \in M.\]

The supremum is achieved at \(\mathcal{L}p,\) where \(\mathcal{L} : T^*M \to TM\)   is the Legendre transform associated to \(L.\)

A curve is an extremal if and only if it is the projection to \(M\) of an integral curve of the Hamiltonian flow \(\Phi^H\)   on \(T^*M.\)

The energy is the function \(E : = H \circ \mathcal{L}^{-1}\)   on \(TM.\) If \(\gamma\) is a minimizing extremal then \(E(\dot\gamma)\equiv 0.\)

For \(u:M \to \mathbb{R},\) \[\nabla u : = \mathcal{L}du \qquad \text{ and } \qquad \mathbf{L} u : = \mathrm{div}_\mu(\nabla u)\] (typically nonlinear in \(u\)).

The Hamilton-Jacobi equation is \[H(du) = 0.\] If \(u\) is a \(C^2\) solution to the H-J equation, then the integral curves of \(\nabla u\) are zero-energy extremals.

The Ricci curvature of a Riemannian manifold is a fiberwise quadratic form which is a trace of the Riemann Curvature tensor: \[\mathrm{Ric}(v) = \mathrm{tr}\left(w \mapsto R(w,v)v \right), \qquad v \in TM.\]

If \(V \) is a vector field such that \[\nabla_VV = 0, \qquad V\vert_x = v \qquad \text{ and } \qquad \nabla V\vert_x = 0,\] then \(V\mathrm{div} V\vert_x = -\mathrm{Ric}(v).\)

If \(V = \nabla u\)   for a function \(u\) then \((d\Delta u)(\nabla u)\vert_x = -\mathrm{Ric}(v).\)

Geometric interpretation: If we take a small object and let it flow along "parallel" geodesics in the direction \(v,\) then the second (logarithmic) derivative of the object's volume will be roughly \(-\mathrm{Ric}(v).\)

• Grifone, Ann. Inst. Fourier, 1972.
• Foulon, Ann. Inst. H. Poincaré Phys. Théor., 1986.

• Agrachev, Gamkrelidze, J. Dyn. Control Syst., 1997.
• Agrachev, J. Dyn. Control Syst., 1998.
• Agrachev, Barilari, Rizzi, Mem. Amer. Math. Soc., 2018.

• Lee, J. Funct. Anal., 2013.
• Ohta, Anal. Geom. Metr. Spaces, 2014.

\(\Sigma:=\) Euler-Lagrange vector field on \(TM\): \[\Sigma = v^i\partial_{x^i} + \Sigma^i\partial_{v^i}, \qquad \Sigma^i = L^{v^iv^j}\!\left(L_{x^j} - v^k L_{v^jx^k}\right).\]

The associated nonlinear connection is a splitting \[TTM = \mathcal{V}TM \oplus \mathcal{H}TM.\]

\[ \mathrm{Ric}_L \;: =\; \varepsilon^i\!\left([\Sigma,E_i]\right) \]

Bochner-Weitzenböck formula: for any solution \(u\) of \(H(du) = 0,\) \[(d\Delta u)(\nabla u) + |\nabla^2 u|_{g_u}^2 + \mathrm{Ric}_L(\nabla u) = 0,\] where \(g_u : = \left(\partial_{v_iv_j}L\circ\nabla u\right)dx^idx^j.\)

\(\mu\) measure on \(M\) coming from a smooth density \(\omega\).

\[\pi^*\omega \;=\; e^{-\psi}\sqrt{\det g}\,\bigl|dx^1\wedge\cdots\wedge dx^n\bigr|, \qquad \psi: TM\setminus\mathbf{0} \to \mathbb{R},\] \[g_{ij} = \partial_{v^iv^j}L.\]

\(\Lambda := \tfrac12(\Sigma - [v^i\partial_{v^i},\Sigma]) = \Lambda^i\partial_{v^i}\quad\) (vanishes iff \(\Sigma\) is a spray), \[ \Lambda_\parallel := \frac{g_{ij}v^i\Lambda^j}{g_{ij}v^iv^j}, \qquad \Lambda_\perp^2 := \frac{g_{ij}(\Lambda^i-\Lambda_\parallel v^i)(\Lambda^j-\Lambda_\parallel v^j)}{g_{ij}v^iv^j}. \]

\[ \mathrm{Ric}_{L,\mu,N} \;:=\; \mathrm{Ric} + \Sigma^2\psi \;-\; \frac{(\Sigma\psi - \Lambda_\parallel)^2}{N-n} \;+\; 2\Lambda_\perp^2 + \Lambda_\parallel^2 \]

Weighted Bochner inequality: for any solution \(u\) of \(H(du) = 0,\) \[(d\mathbf{L}u)(\nabla u) + \frac{(\mathbf{L}u)^2}{N-1} + \mathrm{Ric}_{L,\mu,N}(\nabla u) \;\le\; 0\] where \[\mathbf{L}u := \mathrm{div}_\mu(\nabla u) = \Delta u - (\Sigma\psi)\circ\nabla u.\]

\[L = g/2 - U\] where \(g\) is a Riemannian metric and \(U\) is a smooth negative function. \[ \mathrm{Ric}_{L,\mathrm{Vol}_g,N} = \mathrm{Ric}_g + \Delta V\cdot g -\left(1+\frac{1}{N-n}\right)\cdot (dV)^2 \qquad \text{ on } E^{-1}(0),\] \[\text{where} \qquad V : = -\frac12\log(-U).\]

\[L = g/2 + 1/2 - \eta\] \[\mathrm{Ric}_{L,\mathrm{Vol}_g,N} = \mathrm{Ric}_g - \mathrm{div}_g\mathrm{Y} + \frac14|\mathrm{Y}|_g^2 + \frac12 \, g\circ \mathrm{Y} \quad \text{ on } \quad E^{-1}(0),\] where the magnetic field \(\mathrm{Y}\) is defined by \[\left\langle\mathrm{Y}v,w\right\rangle = d\eta(v,w), \qquad v,w \in T_xM, \, \, x \in M.\] If \((M,g,\omega)\) is a Kähler manifold of complex dim. \(d\) and \(d\eta = c\cdot \omega\) then \[\mathrm{Ric}_{L,\mathrm{Vol}_g,n} = \mathrm{Ric}_g + c^2\cdot \frac{d+1}{2}.\]

Gouda '97, Grognet '99, Wojtkowski '00, Bai-Adachi '13, Assenza '24.

\((\mathcal X_0,\mu_0),(\mathcal X_1,\mu_1)\)  - probability spaces.
\(\mathrm{c}: \mathcal{X}_0 \times \mathcal{X}_1 \to \mathbb{R}\)   - cost function.

The optimal transport (Monge-Kantorovich) problem is the problem of finding a map \(T:\mathcal{X}_0\to\mathcal{X}_1\)   minimizing \[ \int_{\mathcal X_0} \mathrm{c}(x_0,T(x_0)) d\mu_0(x_0)\] among all maps pushing forward \(\mu_0\) to \(\mu_1.\)

Such \(T\) is called an optimal transport map .

Given a Lagrangian \(L\) and a pair of probability measures \(\mu_0,\mu_1\) on \(M,\) we can set \(\mathcal X _0 = \mathcal X_1 = M\) and define a cost function \[\mathrm{c}(x_0,x_1) : = \inf_\gamma \mathrm{A}(\gamma) = \inf_\gamma\int L(\dot\gamma), \qquad x_0,x_1 \in M,\] where the infimum is over all curves \(\gamma\) joining \(x_0\) to \(x_1.\)

If \(L = (g+1)/2 \)   then \(\mathrm{c} = d_g\). In general, the function \(\mathrm{c}\) satisfies the triangle inequality, but may:

attain negative values

be asymmetric

not grow linearly along extremals

Theorem (........, Bernard-Buffoni '05, Fathi-Figalli '07): Under classical assumptions on the Lagrangian \(L,\) for every pair \(\mu_0,\mu_1\) of absolutely-continuous, compactly supported probability measures on \(M,\) there exists an optimal transport map \(T\) from \(\mu_0\) to \(\mu_1.\)

In fact, there exists a family \(\{T_\lambda\}_{0 \le \lambda \le 1}\)   of maps such that:

• \(T_0 = \mathrm{Id}\)   and \(T_1\) is an optimal transport map from \(\mu_0\) to \(\mu_1.\)
• Each \(T_\lambda\) is an optimal transport map from \(\mu_0\)   to \[\mu_\lambda : = (T_\lambda)_*\mu_0.\]
• \(\mu_0\)-a.e. curve \(\lambda\mapsto T_\lambda(x)\)   is a minimizing extremal.
• There exists a viscosity subsolution \(u\) to the Hamilton-Jacobi equation, and \(\ell : M \to [0,\infty)\)   such that \[T_\lambda = \pi\circ\Phi_{\lambda\ell}^H\circ du \qquad \text{\(\mu_0\) -a.e.}\]

The family \(\{\mu_\lambda\}_{0 \le \lambda \le 1}\)   of probability measures is called a displacement interpolation between \(\mu_0\) and \(\mu_1.\)

McCann '97, Cordero-Eruasquin-McCann-Schmuckenschläger '01, von Renesse-Sturm '05, Lott-Villani '07: \((M,g)\) complete Riemannian manifold.

\(\mathrm{Ric}_g \ge 0 \iff \) displacement convexity: for every displacement interpolation \(\mu_\lambda = f_\lambda\mathrm{Vol}_g\) with cost \(\tfrac12 d_g^2,\) \[\mathrm{Ent}[\mu_\lambda|\mathrm{Vol}_g]:=\int \log(f_\lambda) d\mu_\lambda, \quad \text{is convex in } \lambda.\]

• Arbitrary lower Ricci curvature and upper dimension bounds; arbitrary reference measures (weighted Ricci curvature) (C-M-S '01, von Renesse-Sturm '05, Lott-Villani '07).
• Synthetic curvature-dimension bounds for mms's via optimal transport (Sturm '06, Lott-Villani '07).

\(L^1\) localization (needle decomposition):

• Evans-Gangbo '00, Caffarelli-Feldman-McCann '01, Feldman-McCann '02 - The solution to the Monge-Kantorovich problem with cost d provides a disintegration of the manifold into disjoint geodesics ("transport rays").
• Klartag '14 - together with CD bounds, this can be used to prove geometric inequalities on weighted Riemannian manifolds (Gromov-Milman '87, Lovasz-Simonovits '93).
• Finsler (Ohta '15), mms (Cavalletti-Mondino '15), Sub-Riemannian (Milman '19), Lorentzian (Braun-McCann '23, Cavalletti-Mondino '24).

• \(M\) smooth manifold of dimension \(n \ge 2.\)
• \(L:TM \to \mathbb{R}\)   Lagrangian which is:
  • Smooth away from the zero section
  • Fiberwise strongly convex and superlinear
  • Supercritical (i.e. closed curves have positive action)
  and such that:

  • Every pair of points can be joined by a minimizing extremal.
  • For every compact set \(A\) there exists a compact set \(A'\) such that all minimizing extremals with endpoints in \(A\) are contained in \(A'.\)
• \(\mu\) measure on \(M\) with a smooth, positive density.

Theorem (A. '25): Let \(N \in [n,\infty]\) . The following are equivalent:

• \(\mathrm{Ric}_{L,\mu,N} \ge 0\)   on \(E^{-1}(0)\)  .
• For every local solution \(u\) to the H-J equation \(H(du) = 0\)  , \[ (d\mathbf{L}u)(\nabla u) + \frac{(\mathbf{L}u)^2}{N-1} \le 0. \]
• \(\forall \mu_0,\mu_1\in\mathcal{P}_1(L)\)   there exists a displacement interpolation \(\{\mu_\lambda\}_{0 \le \lambda \le 1}\)  between \(\mu_0\) and \(\mu_1\) such that \( \mathrm{S}_{N}[\mu_\lambda|\mu] \) is convex in \(\lambda.\)

Theorem (A. '25): Let \(N \in [n,\infty]\) and \(K \in \mathbb{R}.\) TFAE:

1. \(\mathrm{Ric} _{L,\mu,N} \ge {K}\) on \(SM.\)
2. Every local solution \(u\) to the Hamilton-Jacobi equation \(H(du) = 0\) satisfies \[ (d\mathbf{L} u)(\nabla u) + \frac{(\mathbf{L} u)^2}{N-1} + {K} \le 0. \]
3. For every \(\mu_0 = f_0\mu,\;\mu_1 = f_1\mu \in \mathcal{P}_1(L)\,\,\) there exists a displacement interpolation \(\mu_\lambda\) between \(\mu_0\) and \(\mu_1\) such that for every \(\lambda \in [0,1],\) \[ \small \mathrm{S}_{N}[\mu_\lambda|\mu] \le \begin{cases} \mathbb{E}\left[\tau_{1-\lambda}^{{K},{N}}\cdot \left(-f_0(\gamma(0))^{-1/{N}}\right) + \tau_{\lambda}^{{K},{N}}\cdot \left(-f_1(\gamma(T))^{-1/{N}}\right)\right] & {N} < \infty\\[9pt] (1-\lambda)\cdot\mathrm{S}_\infty[\mu_0|\mu] + \lambda\cdot\mathrm{S}_\infty[\mu_1|\mu] - \frac{K}{2}\cdot\lambda\cdot(1-\lambda)\cdot \mathbb{E} T^2 & {N} =\infty, \end{cases} \] where \(\gamma:[0,T] \to M\) is a random minimizing extremal such that \[\gamma(\lambda T)\sim \mu_\lambda \quad \text{ for all } \quad 0\le\lambda\le1.\]

• depends only on the dynamics on \(E^{-1}(0).\)
• Autonomous Lagrangian, unrestricted time \(\rightarrow\) extends \(L^1\) d.c. (Cavalletti & Milman '16, Cavalletti, Gigli & Santarcangelo '20).
• Consequences of curvature-dimension bounds:
  • Brunn-Minkowski
  • Prékopa–Leindler / Borell-Brascamp-Lieb
  • Bishop-Gromov
  • Bonnet-Myers
  • Isoperimetry
  • Poincaré / log-Sobolev
• cf. Lee '13, Ohta '14, Schachter '17, Yang '22.

Theorem (A. '25, cf. Klartag '14'): Let \(N \in (-\infty,\infty]\setminus[0,n)\)   and \(K \in \mathbb{R}\)   and suppose that \(\mathrm{Ric}_{L,\mu,N} \ge K.\) Let \(f : M \to \mathbb{R}\)   be a \(\mu\)-intergrable function satisfying \[\int_Mfd\mu = 0, \qquad \qquad {\exists x_0\in M \quad \int_M\left(|\mathrm{c}(x_0,\cdot)| + |\mathrm{c}(\cdot,x_0)|\right)fd\mu < \infty.}\]

\(\implies \, \exists\) Borel measures \(\{\mu_\alpha\}_{\alpha \in \mathscr{A}}\)   and a measure \(\nu\) on \(\mathscr{A}\) such that:

• For \(\nu\)-almost every \(\alpha \in \mathscr{A},\) either \(\mu_\alpha\) is a Dirac measure, or \(\mu_\alpha = \left(\gamma_\alpha\right)_*m_\alpha\)  where \(m_\alpha\) is a measure on an interval \(I_\alpha \subseteq \mathbb{R}\)   satisfying \(\mathrm{CD}({K},N)\) with respect to the Euclidean metric on \(\mathbb{R},\) and \(\gamma_\alpha : I_\alpha \to M\)   is a minimizing extremal.
• Disintegration of measure: The measure \(\mu\) disintegrates as \[\mu = \int_{\mathscr{A}}\mu_\alpha d\nu(\alpha).\]
• Mass Balance: For \(\nu\)-almost every \(\alpha \in \mathscr{A},\) \[ \int_Mfd\mu_\alpha = 0. \]

Part I: \(L^1\) optimal transport (Evans-Gangbo '99, Feldman-McCann '02, Caffarelli- Feldman-McCann '02. Also: Bernard-Buffoni '06, Figalli '07, Fathi-Figalli '10). Let \(u : M \to \mathbb{R}\) satisfy

\[ \int_M fu\, d\mu = \inf\left\{\int_M fv \, d\mu \quad \Big\vert \quad v : M \to \mathbb{R}, \,\, H(dv) \le 0\right\}. \]

A transport ray of \(u\) is a maximal curve \(\gamma : I \to \mathbb{R}\) with the properties

\[ \dot\gamma \equiv \nabla u \qquad \text{ and } \qquad E(\dot\gamma) \equiv 0. \]

Note that in this case

\[ H(du\vert_{\gamma(t)}) = E(\nabla u\vert_{\gamma(t)}) = E(\dot\gamma(t)) = 0 \qquad \text{ for all $t \in I$.} \]

If \(x\) is not contained in such a curve then we say that \(\{x\}\) is a (degenerate) transport ray.

For every Borel set \(A\subseteq M\) which is a union of transport rays,

\[ \int_A f \,d\mu = 0 \qquad \text{(Mass balance)}. \]

Let \(\{\gamma_\alpha:I_\alpha \to M\}_{\alpha\in\mathscr{A}}\) be the collection of transport rays. Make a change of variables:

\[ \begin{aligned} \mathscr{A}\times\mathbb{R} &\to M\\ (\alpha,t) &\mapsto \gamma_\alpha(t). \end{aligned} \]

For every smooth \(\phi : M \to \mathbb{R},\)

\[ \int_M\phi\,d\mu = \int_{\mathscr{A}}\int_{I_\alpha}\phi(\gamma_\alpha(t))\rho(\alpha,t)dt\,d\alpha. \]

How do we determine the ''Jacobian" \(\rho\)? since we have freedom in choosing the measure on \(\mathscr{A},\) we only need to determine \(\rho\) up to a multiplicative constant depending on \(\alpha.\) Thus it suffices to determine

\[ \partial_t\log\rho. \]

But in this coordinate chart \(\partial/\partial t = \dot\gamma_\alpha = \nabla u,\) so

\[ \partial_t\log\rho = \mathrm{div}_\mu(\partial / \partial t) = \mathrm{div}_\mu(\nabla u ) = \mathbf{L} u. \]

For every \(\alpha \in \mathscr{A}\) Define a measure \(m_\alpha\) on the interval \(I_\alpha\) by

\[ dm_\alpha(t) = e^{-\psi_\alpha(t)}dt, \qquad \text{ where } \qquad \frac{d\psi_\alpha}{dt} = -\mathbf{L} u \circ\gamma_\alpha. \]

Define a needle \(\mu_\alpha\) by

\[ \mu_\alpha = (\gamma_\alpha)_* m_\alpha. \]

Then for every smooth \(\phi : M \to \mathbb{R},\)

\[ \begin{aligned} \int_M\phi\,d\mu & = \int_{\mathscr{A}}\int_{I_\alpha}\phi(\gamma_\alpha(t))e^{-\psi_\alpha(t)}dt\,d\nu(\alpha)\\ & = \int_{\mathscr{A}}\int\phi d\mu_\alpha\,d\nu(\alpha). \end{aligned} \]

By mass balance, for \(\nu\)-almost every \(\alpha \in \mathscr{A},\)

\[ \int f\, d\mu_\alpha = 0. \]

Part II: It remains to show that \(\nu\)-a.e needle \(\mu_\alpha\) satisfies \(\mathrm{CD}(K,N).\) Assume for simplicity \(K=0,N=\infty,\mu = \mathrm{Vol}_g\,\,\) and recall

\[ \mu_\alpha = (\gamma_\alpha)_*m_\alpha, \qquad dm_\alpha(t) = e^{-\psi_\alpha(t)}dt, \qquad \dot\psi_\alpha = - \mathbf{L} u\circ\gamma_\alpha. \]

We need to prove that \(\ddot \psi_\alpha \ge 0.\)

Since \(H(du) = 0\) on nondegenerate transport rays, in some sense, \(u\) solves the Hamilton-Jacobi equation \(H(du) = 0\) on the set of nondegenerate transport rays.

Therefore, by the Bochner inequality, loosely speaking, on the set of nondegenerate transport rays

\[ (d\mathbf{L} u)(\nabla u) \le 0. \]

Hence

\[ \ddot\psi_\alpha = \frac{d}{dt}\left(-\mathbf{L} u\circ\gamma_\alpha\right) = -(d\mathbf{L} u)(\dot\gamma_\alpha) = -(d\mathbf{L} u)(\nabla u) \ge 0 \]

for \(\nu\)-almost \(\alpha\) such that \(\gamma_\alpha\) is nondegenerate.