Optimal Transport and Rigidity Phenomena

• Optimal Transport and Rigidity Phenomena is defined as the study of efficient mass relocation between probability measures under cost functions, emphasizing deterministic maps in non-branching spaces.
• The framework leverages geometric and measure-theoretic conditions such as cyclic monotonicity and absolute continuity to guarantee the existence and uniqueness of optimal transport maps.
• Applications extend to ensuring canonical measures in RCD spaces and deriving functional inequalities through controlled interpolations along geodesics.

Optimal transport investigates the most efficient way to move mass between configurations in geometric or probabilistic spaces, under a prescribed cost function. Rigidity phenomena, in this context, refer to situations where the structure of optimizers (maps, plans, interpolations, or underpinning measure-theoretic data) is uniquely determined, exhibits strong stability, or is uniquely characterized by the geometric properties of the background space, cost, or participating measures. Recent advances elucidate the fundamental mechanisms leading to such rigidity, especially in metric measure spaces admitting geodesics, with curvature-dimension conditions, or which are essentially non-branching.

1. Existence and Structure of Optimal Transport Maps

A central objective is to determine conditions under which an optimal transport plan is induced by a deterministic map $T\colon M \to M$, meaning that the transport plan $\pi$ is concentrated on the graph of $T$: $\pi = (\text{id} \times T)_\# \mu$. In the context of geodesic metric measure spaces $(M, d, m)$ with $m$ a $\sigma$-finite Borel measure, uniqueness and existence of transport maps are inextricably linked to the regularity of the initial measure $\mu$ and the geometry of $(M, d, m)$. Specifically, if $\mu \ll m$ (absolutely continuous) and the space is essentially non-branching and satisfies a non-degeneracy property (either quantitative or qualitative), then for any $p$-Wasserstein optimal coupling between $\mu$ and another probability measure $\nu$, $\pi$ is induced by a measurable map $T$ with

$$\pi = (\text{id} \times T)_\# \mu = \text{unique optimizer}.$$

The necessity of $\mu \ll m$ is fundamental: Rademacher-type differentiability properties and the existence of intermediate absolutely continuous measures $\mu_t = (e_t)_\# \sigma \ll m$ (for $t \in (0,1)$) exclude multivalued or branching behavior at almost every point.

Under these conditions, transport plans satisfy $c_p$-cyclic monotonicity and the “selection dichotomy”: for $\mu$-almost every $x$, the fiber $\{y : (x,y) \in \operatorname{supp}\pi\}$ is a singleton. This rigidity is formalized via measurable selection and dichotomy theorems and is enforced by the underlying non-branching geometry.

2. Rigidity from Non-Branching and Non-Degeneracy

A metric measure space $(M, d, m)$ is $p$-essentially non-branching if, for any optimal $p$-Wasserstein dynamical coupling (considered as a measure on the space of geodesics) between absolutely continuous marginals, the coupling is concentrated on a set of geodesics that do not branch. Explicitly, if the coupling is disintegrated at any time $t \in (0,1)$, then for $\mu_t$-almost every point, the preimage consists of a single geodesic.

Non-branching is vital because it ensures that, under cyclic monotonicity, there is no possibility for the optimal plan to “split” mass in a non-unique way along the geodesics connecting starting and terminal points. The quantitative qualitative non-degeneracy condition further ensures that mass does not concentrate along negligible sets during interpolation, allowing for control over densities and intermediate measure regularity.

Prominent classes where these mechanisms hold include spaces with the measure contraction property $\mathrm{MCP}(K,N)$, curvature-dimension condition $\mathrm{CD}^*(K,N)$, or $\mathrm{RCD}^*(K,N)$, all of which are now known to be essentially non-branching. In such settings, uniqueness and existence of optimal transport maps are guaranteed for initial data that are absolutely continuous, and the structure of the interpolating measures and supports is strongly controlled.

3. Measure Rigidity in Metric Measure Spaces

A salient achievement is the measure rigidity phenomenon. If $(M, d, m_1)$ and $(M, d, m_2)$ are two essentially non-branching metric measure spaces, each with “good transport behavior” (all optimal couplings with absolutely continuous first marginal are induced by maps) and qualitative non-degeneracy, then $m_1$ and $m_2$ must be mutually absolutely continuous: $m_1 \ll m_2 \quad \text{and} \quad m_2 \ll m_1.$ This is shown by constructing absolutely continuous interpolation measures along optimal geodesics, establishing that any jump across mutually singular supports would contradict the propagation of absolute continuity.

This rigidity greatly impacts geometric analysis, especially in spaces endowed with synthetic finite-dimensional Ricci lower bounds such as $\mathrm{RCD}(K,N)$ spaces: any two reference measures satisfying the above properties (e.g., competing notions of normalized volume measure) are necessarily equivalent. Thus, analytic and geometric invariants defined using such measures are canonical up to equivalence, precluding the existence of “exotic” or “singular” alternative measures compatible with the same synthetic Ricci lower bound structure.

4. Technical and Structural Formulations

The mathematical characterization of these rigidity phenomena relies on several formulations:

• $p$–Wasserstein distance:

$$W_p(\mu_0, \mu_1) = \left( \inf_{\pi \in \Pi(\mu_0, \mu_1)} \int d(x,y)^p \, d\pi(x,y) \right)^{1/p}.$$

• Transport map induced optimizer: If $\pi = (\text{id} \times T)_\# \mu_0$, then for $\mu_0$-a.e. $x$, there exists a unique target $y = T(x)$ receiving mass from $x$.
• Dynamical coupling and interpolations: Any $p$-optimal coupling can be realized via a measure $\sigma$ on $\mathrm{Geo}_{[0,1]}(M,d)$ such that $\pi = (e_0, e_1)_\# \sigma$; the interpolated measures at time $t$ are $\mu_t = (e_t)_\# \sigma$.
• Selection dichotomy: For any Borel set $\Gamma \subseteq M \times M$ with full first marginal, for $\mu$-a.e. $x$, either $\#\{y:(x,y)\in \Gamma\}=1$ (map case), or there is a set of positive measure where multiple selections occur (branching).
• Non-degeneracy (measure contraction): For every $R>0$, $x_0 \in M$ and $A \subseteq B_R(x_0)$ measurable,

$$m(A_{t,x}) \geq f(t) m(A) \quad \forall x \in B_R(x_0), t \in (0,1),$$

where $A_{t,x}$ is the set of $t$-midpoints along geodesics to $x$.

• Density bounds along interpolation: If $\mu_t = f_t m$,

$\|f_t\|_\infty \leq g(t) \|f_0\|_\infty,$

where $g(t)$ depends on $f(t)$ from the non-degeneracy condition.

These formulations delineate the analytic and geometric constraints underlying the rigidity of optimal transport.

5. Applications and Broader Implications

Rigidity of transport maps and background measures has several geometric and analytic consequences:

• Unique optimal transport maps: In spaces such as $\mathrm{RCD}(K,N)$, every optimal transport from an absolutely continuous initial measure is deterministic; this underpins the regularity theory of optimal transport in non-smooth spaces.
• Measure uniqueness: Any two “natural” measures on a fixed metric space satisfying the essential non-branchings and non-degeneracy are equivalent. The geometric structure “selects” the measure and eliminates the emergence of alternative reference measures.
• Localization and functional inequalities: The uniqueness and regularity of interpolated measures aid in proving localization results, Poincaré inequalities, and promoting measure-theoretic rigidity to geometric and functional rigidity.
• Further research: Prospective directions include relaxation of the non-branching and non-degeneracy hypotheses, exploring rigidity for more general cost functions or other $p$, and the impact for measures not of finite entropy. There is also interest in how measure rigidity relates to the structure of the isometry group and group actions on non-smooth metric measure spaces.

6. Quantitative and Qualitative Aspects; Examples

Some examples highlighting the importance and sharpness of these phenomena include:

• $\mathrm{MCP}(K,N)$ and $\mathrm{RCD}(K,N)$ spaces: These settings are essentially non-branching and non-degenerate, so all the rigidity results above apply.
• Spaces failing non-branching: Branching geodesics allow for non-deterministic optimal couplings, so rigidity fails. Analyzing how “partial non-branching” or weaker conditions affect rigidity is an active area.
• Quantitative rigidity: In cases with explicit non-degeneracy bounds (e.g., $f(t) = (1-t)^N$), one obtains explicit density and regularity estimates along interpolations, reinforcing the deterministic and rigid structure of optimal transport.

7. Summary

The deep connection between (i) the uniqueness and regularity of optimal transport maps, (ii) essential non-branching of geodesics, and (iii) the rigidity of admissible background measures, is now recognized as a cornerstone of the optimal transport theory on metric measure spaces. In particular, any essentially non-branching, qualitatively non-degenerate metric measure space supports unique optimal transport from absolutely continuous initial data, and measure rigidity ensures mutual absolute continuity of all such admissible measures. This powerful confluence of measure-theoretic, geometric, and analytic rigidity is central to the modern theory of metric measure spaces with curvature-dimension conditions and has broad implications for the paper of non-smooth spaces in analysis and geometry (Kell, 2017).