Magnetic Brunn-Minkowski inequalities

Rotem Assouline (IMJ-PRG)

Séminaire d'Analyse
IMT

May 2026

The Brunn-Minkowski inequality

$A_0,A_1 \subseteq \mathbb{R}^n$ , $0 \le \lambda \le 1$ .

\[A_\lambda : =(1-\lambda) A_0 + \lambda A_1 = \left\{(1-\lambda)a_0 + \lambda a_1 \, \mid \, a_0 \in A_0, \, a_1 \in A_1\right\}.\]

The Brunn-Minkowski inequality

Theorem (Brunn-Minkowski) :

$A_0,A_1 \subseteq \mathbb{R}^n$ Borel, nonempty, $\qquad 0 \le \lambda \le 1$ , \[\mathrm{Vol}(A_\lambda)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}(A_0)^{1/n} + \lambda\cdot\mathrm{Vol}(A_1)^{1/n}.\]

The Brunn-Minkowski inequality

Theorem (Brunn-Minkowski) :

$A_0,A_1 \subseteq (M,g)$ Borel, nonempty, $\qquad 0 \le \lambda \le 1$ , \[\mathrm{Vol}(A_\lambda)^{1/n} \ge (1-\lambda)\cdot\mathrm{Vol}(A_0)^{1/n} + \lambda\cdot\mathrm{Vol}(A_1)^{1/n}.\]

The Brunn-Minkowski inequality

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

The Brunn-Minkowski inequality

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).

The Brunn-Minkowski inequality

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

$(M,g,\mu)$ weighted Riemannian manifold, $N \in [n,\infty]$, $\mathrm{Ric}_{\mu,N} \ge k \implies$ distorted Brunn-Minkowski for $\mu$ with exponent $1/N$.

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

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

Horocyclic Brunn-Minkowski inequality

Let $\mathbf{H}$ denote the hyperbolic plane.

A horocycle is a curve of signed geodesic curvature $\equiv 1$.
For every $x,y \in \mathbf{H}$ there exists a unique horocycle $\gamma:[0,T]\to \mathbf{H}$ satisfying $\gamma(0) = x$ and $\gamma(T) = y$ .

Horocyclic Brunn-Minkowski inequality

Theorem (A.-Klartag '22): Let $A_0,A_1 \subseteq \mathbf{H}$ 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 $\gamma:[0,\ell]\to \mathbf{H}$ is a horocycle satisfying $\gamma(0) \in A_0$ and $\gamma(\ell) \in A_1$ . \[\implies \qquad \mathrm{Vol}(A_\lambda)^{1/2} \ge (1-\lambda)\cdot\mathrm{Vol}(A_0)^{1/2} + \lambda\cdot\mathrm{Vol}(A_1)^{1/2}, \] where $\mathrm{Vol}$ denotes the hyperbolic volume measure.

Magnetic geodesics

$(M,g)$-Riemannian manifold, $\Omega$ closed two-form on $M.$

A magnetic geodesic is a smooth curve $\gamma$ satisfying \[ g\left(\nabla_{\dot\gamma}\dot\gamma,\cdot\right) = \Omega(\dot\gamma,\cdot). \]
We assume that $\Omega$ is exact: \[ \Omega = d\eta \] for some one-form $\eta$.
A minimizing magnetic geodesic is a curve $\gamma$ minimizing \[ \mathrm{Len}[\gamma] - \int_\gamma\eta \] among all piecewise-$C^1$ curves joining its endpoints.

$\Omega$ is the magnetic field and $\eta$ is the magnetic potential.

Magnetic geodesics: Examples

The Euclidean plane
$\Omega = \kappa\,dx\wedge dy, \quad \kappa\in\mathbb{R}$

Circular arcs of radius $1/\kappa.$

The Hyperbolic plane
$\Omega = d\mathrm{Vol}.$

Horocycles.

Magnetic geodesics

$(M,g)$ Riemannian manifold, $\eta$ one-form, $\Omega = d\eta$.

$A_0,A_1\subseteq M, \quad 0 \le \lambda \le 1$

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

$\Omega = 0.5\,dx\wedge dy$ $\Omega = 0$ $\Omega = 0.5\,dx\wedge dy$ $\Omega = dx\wedge dy$

Magnetic Ricci curvature

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 $-\mathrm{Ric}(v).$

Magnetic Ricci curvature

Let $\Omega$ be a closed 2-form on $M$ and let $SM = \{v\in TM : g(v,v)=1\}$ be the unit tangent bundle. The magnetic Ricci curvature is \[\mathrm{Ric}_\Omega(v) \,:=\, \mathrm{Ric}(v) \,-\, (\delta\Omega)(v) \,+\, \tfrac12|\iota_v\Omega|^2 \,+\, \tfrac14|\Omega|^2, \qquad v \in SM.\] cf. (Gouda '97, Grognet '99, Wojtkowski '00, Bai-Adachi '13, Assenza '24)

Here $\delta$ is the codifferential, $\iota_v\Omega = \Omega(v,\cdot)$, and $|\Omega|^2 = \sum_{i,j}\Omega(e_i,e_j)^2$ for any orthonormal frame $(e_j)$.

If $\tilde g = g - (d\varphi + A)^2 $ is the Kaluza-Klein metric on $M\times S^1$ then $\mathrm{Ric}_\Omega(v) = \mathrm{Ric}_{\tilde g}(\tilde v),$ where $\tilde v = v + (1-\eta(v))\partial_\varphi.$

Magnetic Ricci curvature

Proposition (Magnetic Bochner formula): Let $\sigma$ be a one-form on an open set $U\subseteq M$ satisfying \[d\sigma = \Omega \qquad \text{and}\qquad |\sigma|\equiv 1.\] Then \[\,-\langle\sigma,\,d\delta\sigma\rangle \,+\, |\Sigma|^2 \,+\, \mathrm{Ric}_\Omega(\sigma^\sharp) \,=\, 0\,\] where $\Sigma$ is the symmetric part of $\nabla\sigma\big|_{\ker\sigma}$, and $\sigma = \langle \sigma^\sharp,\cdot\rangle.$

In particular, \[-\langle\sigma,\,d\delta\sigma\rangle \,+\, \frac{(\delta\sigma)^2}{n-1} \,+\, \mathrm{Ric}_\Omega(\sigma^\sharp) \,\le\, 0,\] with equality iff $\Sigma$ is scalar.

When $\Omega = 0$, $\nabla\sigma$ is symmetric with $\nabla\sigma\vert_{\sigma^\sharp} = 0$, and locally $\sigma = du$ and $\Sigma = \nabla^2u.$

Magnetic Ricci curvature

Setting $V = \sigma^\sharp$, the unit vector field $V$ satisfies the magnetic geodesic equation \[|V|\equiv 1 \qquad \text{and}\qquad \langle\nabla_V V,\,\cdot\,\rangle = \Omega(V,\,\cdot),\] and the Bochner identity and inequality become \[V(\mathrm{div}\,V) \,+\, |\Sigma|^2 \,+\, \mathrm{Ric}_\Omega(V) \,=\, 0,\] \[V(\mathrm{div}\,V) \,+\, \frac{(\mathrm{div}\,V)^2}{n-1} \,+\, \mathrm{Ric}_\Omega(V) \,\le\, 0,\] where $\Sigma$ is the symmetric part of $\nabla V\big|_{V^\perp}$.

When $\Omega = 0$, $\nabla V$ is symmetric with $\nabla V\vert_{V} = 0$, and locally $V = \nabla u$ and $\Sigma = \nabla^2u.$

Magnetic Ricci curvature

Examples:

Riemannian surfaces with $\Omega = \kappa\,\Omega_g$: \[\mathrm{Ric}_\Omega \,=\, K + \kappa^2 + \star\, d\kappa, \qquad \mathrm{Ric}_\Omega \ge 0 \,\iff\, K + \kappa^2 \ge |d\kappa|.\]
Kähler manifolds $(M,g,\omega)$, $\dim_\mathbb{C} M = d$, with $\Omega = c\,\omega$: \[\mathrm{Ric}_\Omega \,=\, \mathrm{Ric} \,+\, c^2 \cdot \tfrac{d+1}{2}.\]
- The unit ball $B\subseteq\mathbb{C}^d$: $\mathrm{Ric}_\Omega \equiv c^2(d+1)/2 > 0, \quad c\in[-1,1].$
- Complex hyperbolic space $\mathbb{C}\mathbf{H}^d$ with $c = 1$: $\mathrm{Ric}_\Omega \equiv 0.$
Sasakian manifolds $(M,g,\eta)$ of dimension $2m+1$, with $\Omega = c\,d\eta$: \[\mathrm{Ric}_\Omega \,=\, \mathrm{Ric} \,+\, 2c^2(m+1) \,-\, 4cm\,\eta \,-\, 2c^2\,\eta^2.\]
- Odd-dimensional spheres $S^{2d+1}$ with $c\in(-1,1)$: $\mathrm{Ric}_\Omega \ge 0.$
- Heisenberg group $\mathbb{H}^m$ with $c=1$: $\mathrm{Ric}_\Omega = 2m(1-\eta)^2 \ge 0.$

Magnetic Brunn-Minkowski inequality

Global assumptions:

Every pair of points $x,y\in M$ can be joined by a minimizing magnetic geodesic.
For every closed piecewise-$C^1$ curve $\gamma$, \[\int_\gamma \eta < \mathrm{Len}[\gamma].\]
For every compact $A\subseteq M$ there exists a compact $\tilde A\supseteq A$ containing every minimizing magnetic geodesic with endpoints in $A$.

If $M$ is compact and $|\eta|{<}1$ then all three conditions hold.

Magnetic Brunn-Minkowski inequality

Theorem (A. '25): The following are equivalent:

$\mathrm{Ric}_\Omega \ge 0$ on $SM$.
For every Borel $A_0, A_1\subseteq M$ of positive measure and $\lambda\in[0,1]$, \[\mathrm{Vol}(A_\lambda)^{1/n} \,\ge\, (1-\lambda)\cdot\mathrm{Vol}(A_0)^{1/n} \,+\, \lambda\cdot\mathrm{Vol}(A_1)^{1/n},\] where $n = \dim M$.

Magnetic Brunn-Minkowski inequality

Theorem (A. '25): For $k\in\mathbb{R}$, the following are equivalent:

$\mathrm{Ric}_\Omega \ge k$ on $SM$.
For every Borel $A_0, A_1\subseteq M$ of positive measure and $\lambda\in[0,1]$, \[\mathrm{Vol}(A_\lambda)^{1/n} \,\ge\, \tau_{1-\lambda}^{k,n}(A_0,A_1)\cdot\mathrm{Vol}(A_0)^{1/n} \,+\, \tau_\lambda^{k,n}(A_0,A_1)\cdot\mathrm{Vol}(A_1)^{1/n},\] where $n = \dim M$.

Magnetic Brunn-Minkowski inequality

For $k\in\mathbb{R}$, $t\in[0,1]$ and $\ell \ge 0$, \[\tau_t^{k,n}(\ell) := \begin{cases} t^{1/n}\!\left(\dfrac{\sin\!\left(t\ell\sqrt{k/(n-1)}\right)}{\sin\!\left(\ell\sqrt{k/(n-1)}\right)}\right)^{\!1-1/n}, & k>0,\ \ell\sqrt{k/(n-1)}<\pi,\\[6pt] t, & k=0,\\[4pt] t^{1/n}\!\left(\dfrac{\sinh\!\left(t\ell\sqrt{-k/(n-1)}\right)}{\sinh\!\left(\ell\sqrt{-k/(n-1)}\right)}\right)^{\!1-1/n}, & k<0. \end{cases}\]

For $A_0, A_1\subseteq M$, \[ \tau_t^{k,n}(A_0, A_1) \,:=\, \inf \left\{ \quad \tau_t^{k,n}(\ell)\quad\middle| \quad \begin{array}{c} \exists\text{ minimizing magnetic geodesic } \\ \gamma:[0,\ell]\to M\text{ joining }A_0\text{ to }A_1 \end{array} \right\}. \]

Example: horocyclic Brunn-Minkowski inequality

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. '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 $n = 2d.$

If $A_0$ and $A_1$ are concentric balls, then equality holds.

Example: Contact magnetic Brunn-Minkowski on $S^{2d+1}$

Let $S^{2d+1} = \{z \in \mathbb{C}^{d+1}\, \mid \, |z|=1\}.$ A contact magnetic geodesic of strength $s \in (-1,1)$ is a solution to \[\nabla_{\dot\gamma}\dot\gamma = 2si\left(\dot\gamma - \eta(\dot\gamma)\cdot i\gamma\right), \qquad |\dot\gamma| \equiv 1,\] 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 $n = 2d+1$.

Example: Contact magnetic Brunn-Minkowski on $\mathbb{H}^m$

Let $\mathbb{H}^m \cong \mathbb{R}^{2m+1}$ be the Heisenberg group. A contact magnetic geodesic on $\mathbb{H}^m$ is a minimizing magnetic geodesic corresponding to the metric and one-form \[g = \tfrac14\left(|dx|^2+|dy|^2+(dz-\langle y, dx\rangle)^2\right)\quad\text{and}\quad \eta = \tfrac12\left(dz - \langle y, dx \rangle\right).\]

Reeb trajectories $t\mapsto (0,0,t)$ are contact magnetic geodesics.
Legendrian contact magnetic geodesics are sub-Riemannian geodesics.

Theorem (A. '26+): Let $A_0,A_1 \subseteq \mathbb{H}^m$ be Borel sets of positive measure and $0\le\lambda\le 1$. Denote by $A_\lambda$ the set of points of the form $\gamma(\lambda \ell)$, where $\gamma:[0,\ell]\to \mathbb{H}^m$ is a unit-speed minimizing contact magnetic geodesic 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 Riemannian volume measure and $n=2m+1$.

Example: Closed magnetic potentials

Suppose $\eta$ is closed: $\Omega = d\eta = 0$. Then:

magnetic geodesics are ordinary geodesics;
$\mathrm{Ric}_\Omega = \mathrm{Ric}$.
The class of minimizing magnetic geodesics depends on the cohomology class $[\eta]\in H^1(M)$.
Each cohomology class gives rise to a potentially different Minkowski average, and all of them satisfy the (undistorted) Brunn-Minkowski inequality whenever $(M,g)$ is geodesically convex and $\mathrm{Ric}\ge 0$.

Example: Closed magnetic potentials

Geodesic vs magnetic Minkowski average on the flat torus with closed potential eta = (dtheta_1 + dtheta_2)/2

On the proof

$L^1$ 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).
Tonelli Lagrangians (A '25).

On the proof

Theorem (A. '25, cf. Klartag '14'): Let $N \in (-\infty,\infty]\setminus[0,n)$ and $k \in \mathbb{R}$ and suppose that $\mathrm{Ric}_\Omega \ge k.$ Let $f : M \to \mathbb{R}$ be a $\mu$-integrable 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 magnetic geodesic.
Disintegration of measure: The measure $\mu$ disintegrates as \[\mu = \int_{\mathscr{A}}\mu_\alpha d\nu(\alpha).\]
Mass Balance: For $\nu$-almost every $\alpha \in \mathscr{A},$ \[ \int_Mfd\mu_\alpha = 0. \]

On the proof

Proof sketch:

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

\[ \int_M fu\, d\mathrm{Vol}_g = \inf\left\{\int_M fv \, d\mathrm{Vol}_g \quad \Big\vert \quad v : M \to \mathbb{R}, \,\, |dv + \eta| \le 1\right\}. \]

A transport ray of $u$ is a maximal curve $\gamma : I \to \mathbb{R}$ with the properties \[ \dot\gamma \equiv \nabla u + \eta^\sharp \qquad \text{ and } \qquad |\dot\gamma| \equiv 1. \]

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

On the proof

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 $\phi : M \to \mathbb{R},$ \[ \int_M\phi\,d\mathrm{Vol}_g = \int_{\mathscr{A}}\int_{I_\alpha}\phi(\gamma_\alpha(t))\rho(\alpha,t)dt\,d\alpha. \]

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 $\partial/\partial t = \dot\gamma_\alpha = \nabla u + \eta^\sharp$ so \[ \partial_t\log\rho = \mathrm{div}(\partial / \partial t) = \mathrm{div}(\nabla u + \eta^\sharp) = \mathrm{div}(\sigma^\sharp) = -\delta\sigma \qquad \text{where $\sigma = du + \eta$}. \]

On the proof

For every $\alpha \in \mathscr{A}$, define a measure $m_\alpha$ on the interval $I_\alpha$ by

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

Define a needle $\mu_\alpha$ by \[ \mu_\alpha = (\gamma_\alpha)_* m_\alpha. \]

Then for every smooth $\phi : M \to \mathbb{R},$ \[ \begin{aligned} \int_M\phi\,d\mathrm{Vol}_g & = \int_{\mathscr{A}}\int_{I_\alpha}\phi(\gamma_\alpha(t))e^{-\psi_\alpha(t)}dt\,d\nu(\alpha)\\ & = \int_{\mathscr{A}}\int\phi d\mu_\alpha\,d\nu(\alpha). \end{aligned} \]

By mass balance, for $\nu$-almost every $\alpha \in \mathscr{A},$ \[ \int f\, d\mu_\alpha = 0. \]

On the proof

Part II: It remains to show that $\nu$-a.e needle $\mu_\alpha$ satisfies $\mathrm{CD}(k,n),$ i.e.

\[ \ddot\psi_\alpha \,\ge\, k \,+\, \frac{\dot\psi_\alpha^{\,2}}{n-1}, \qquad \text{where}\qquad dm_\alpha(t) = e^{-\psi_\alpha(t)}dt, \quad \dot\psi_\alpha = \delta\sigma\circ\gamma_\alpha. \]

Loosely speaking, the one-form $\sigma = du + \eta$ satisfies $d\sigma = \Omega$ and $|\sigma|\equiv 1$ on the set of nondegenerate transport rays.

Therefore, by the Bochner inequality and the assumption $\mathrm{Ric}_\Omega \ge k$, on the set of nondegenerate transport rays \[ \langle d\delta\sigma,\sigma\rangle \,\ge\, \frac{(\delta\sigma)^2}{n-1} \,+\, \mathrm{Ric}_\Omega(\sigma^\sharp) \,\ge\, \frac{(\delta\sigma)^2}{n-1} \,+\, k. \]

Hence \[ \ddot\psi_\alpha = \frac{d}{dt}\left(\delta\sigma\circ\gamma_\alpha\right) = (d\delta\sigma)(\dot\gamma_\alpha) = \langle d\delta\sigma,\sigma\rangle \,\ge\, k \,+\, \frac{\dot\psi_\alpha^{\,2}}{n-1} \] for $\nu$-almost $\alpha$ such that $\gamma_\alpha$ is nondegenerate.