on curvature-dimension geometry of metric measure spaces

Overview

Sturm’s two Acta Mathematica papers introduced one of the foundational formulations of lower Ricci curvature bounds for metric measure spaces (Sturm, 2006a; Sturm, 2006b). The idea is to replace the tensor inequality

$${\mathrm{Ric}}_g\ge Kg$$

on a smooth Riemannian manifold by convexity of entropy along optimal-transport geodesics in the Wasserstein space of probability measures.

This post explains the main definitions, the curvature-dimension condition $\mathrm{CD}(K,N)$, the comparison theorems recovered from it, and the proof mechanisms that make the theory stable under measured Gromov-Hausdorff convergence.

Metric Measure Geometry

A metric measure space is a triple

$$(X,d,\mathfrak{m})$$

where $(X,d)$ is a complete separable metric space and $\mathfrak{m}$ is a reference measure. The intended model is a weighted Riemannian manifold

$$(M,d_g,e^{-V}{\mathrm{vol}}_g),$$

but the definition allows singular limits, Alexandrov spaces, measured graphs after scaling, and spaces without a differentiable structure.

The geometry is encoded in probability measures. Let ${\mathcal{P}}_2(X)$ be the set of Borel probability measures $\mu$ with finite second moment. The quadratic Wasserstein distance is

$$W_2({\mu }_0,{\mu }_1{)}^2=\underset{\pi \in \Pi ({\mu }_0,{\mu }_1)}{\inf }{\int }_{X\times X}d(x,y{)}^{2}d\pi (x,y),$$

where $\Pi ({\mu }_0,{\mu }_1)$ is the set of couplings with marginals ${\mu }_0$ and ${\mu }_1$.

Wasserstein geodesics

A curve $({\mu }_t{)}_{0\le t\le 1}$ in ${\mathcal{P}}_2(X)$ is a constant-speed Wasserstein geodesic if

$$W_2({\mu }_s,{\mu }_t)=|s-t|W_2({\mu }_0,{\mu }_1).$$

In a smooth manifold, such a geodesic is induced by moving mass along minimizing geodesics in $M$. Thus convexity of functionals on ${\mathcal{P}}_2(M)$ can detect curvature of $M$.

The basic entropy functional is

$${\mathrm{Ent}}_{\mathfrak{m}}(\mu )={\int }_X\rho \log \rho d\mathfrak{m}$$

when $\mu =\rho \mathfrak{m}$, and $+\infty$ otherwise. On a smooth Riemannian manifold, the statement

$$\mathrm{Ric}\ge K$$

is equivalent, in the Otto-Villani formalism, to $K$-convexity of entropy along $W_2$-geodesics:

$${\mathrm{Ent}}_{\mathfrak{m}}({\mu }_t)\le (1-t){\mathrm{Ent}}_{\mathfrak{m}}({\mu }_0)+t{\mathrm{Ent}}_{\mathfrak{m}}({\mu }_1)-\frac{K}{2}t(1-t)W_2({\mu }_0,{\mu }_1{)}^2.$$

This is the infinite-dimensional curvature condition usually denoted $\mathrm{CD}(K,\infty )$.

Curvature-Dimension Conditions

The finite-dimensional condition $\mathrm{CD}(K,N)$ refines entropy convexity by inserting the distortion coefficients that appear in the Jacobian comparison theorem. The point is that an $N$-dimensional manifold with Ricci curvature at least $K$ does not merely make entropy convex; it controls how volumes distort along geodesics in the model space of constant curvature $K/(N-1)$.

For absolutely continuous measures ${\mu }_i={\rho }_i\mathfrak{m}$, one uses Rényi-type entropy functionals

$$S_N(\mu \mid \mathfrak{m})=-{\int }_X{\rho }^{1-1/N}d\mathfrak{m}.$$

The condition $\mathrm{CD}(K,N)$ says, roughly, that for every pair ${\mu }_0,{\mu }_1$ there is an optimal dynamical plan whose interpolating measures ${\mu }_t$ satisfy a convexity inequality with model distortion factors. In the flat case $K=0$, the core inequality has the schematic form

$$S_N({\mu }_t\mid \mathfrak{m})\le -(1-t)\int {\rho }_0(x{)}^{-1/N}d\pi (x,y)-t\int {\rho }_1(y{)}^{-1/N}d\pi (x,y),$$

with the appropriate signs following the convention for $S_N$. For $K\ne 0$, the coefficients are no longer linear in $t$; they are trigonometric or hyperbolic model coefficients.

Synthetic lower Ricci bound

A metric measure space satisfies $\mathrm{CD}(K,N)$ if entropy has the same displacement convexity behavior that holds on smooth $N$-dimensional manifolds with

$$\mathrm{Ric}\ge K.$$

The definition is intrinsic: it uses only $d$, $\mathfrak{m}$, and optimal transport.

The word synthetic is important. The definition is not a weak form of a tensor inequality; it is a replacement for that inequality in settings where there may be no tangent bundle. Its strength is tested by whether it recovers the theorems that lower Ricci curvature is supposed to imply.

Comparison Geometry

Sturm proves that $\mathrm{CD}(K,N)$ yields analogues of the classical comparison theorems.

The Bishop-Gromov principle compares volume growth of balls. If $K=0$, an $N$-dimensional nonnegatively Ricci curved space should satisfy monotonicity of

$$\frac{\mathfrak{m}(B_r(x))}{r^N}$$

as $r$ varies. For $K\ne 0$, $r^N$ is replaced by the volume of a ball in the $N$-dimensional model space with curvature $K/(N-1)$.

The Brunn-Minkowski inequality appears in a metric form. For sets $A_0,A_1\subset X$, define the set of $t$-intermediate points

$$A_t=\{{\gamma }_t:{\gamma }_0\in A_0,{\gamma }_1\in A_1,\gamma \text{ minimizing geodesic}\}.$$

Curvature-dimension convexity gives lower bounds for $\mathfrak{m}(A_t)$ in terms of $\mathfrak{m}(A_0)$ and $\mathfrak{m}(A_1)$, with the same model coefficients as in the entropy inequality.

Bonnet-Myers also survives. If $K>0$ and $N<\infty$, then spaces satisfying $\mathrm{CD}(K,N)$ have bounded diameter of the model order

$$\mathrm{diam}(X)\le \pi \sqrt{\frac{N-1}{K}}.$$

This is a striking test of the definition: a purely transport-theoretic convexity condition forces a global metric diameter bound.

Proof Mechanism

The central proof strategy is to express classical Jacobian comparison through mass transport. On a smooth manifold, if a map sends $x$ to $T(x)$ along minimizing geodesics, the interpolating map is

$$T_t(x)={\exp }_x(t\nabla \phi (x)).$$

The density ${\rho }_t$ of $(T_t{)}_{\#}{\mu }_0$ is controlled by the Jacobian determinant of $T_t$. Ricci lower bounds enter through a differential inequality for this determinant.

Sturm reverses this logic. Instead of differentiating a Jacobian, he makes the resulting displacement-convex inequality the definition. The smooth theorem tells us the definition is correct on manifolds; the metric formulation makes it meaningful after taking limits.

Proof: Stability Under Measured Convergence

Suppose

$$(X_i,d_i,{\mathfrak{m}}_i)\to (X,d,\mathfrak{m})$$

in a measured Gromov-Hausdorff sense and each $X_i$ satisfies $\mathrm{CD}(K,N)$. To prove the limit also satisfies $\mathrm{CD}(K,N)$, choose two absolutely continuous measures ${\mu }_0,{\mu }_1$ on $X$ and approximate them by measures ${\mu }_0^i,{\mu }_1^i$ on $X_i$.

For each $i$, the curvature-dimension condition gives an optimal interpolation ${\mu }_t^i$ satisfying the entropy inequality. Compactness of probability measures gives a subsequential limit ${\mu }_t$. Lower semicontinuity of entropy and convergence of the transport costs allow passage to the limit in the inequality. The limit curve is a Wasserstein geodesic because the metric costs converge. Therefore the same convexity inequality holds in $X$.

This argument is the structural reason the theory is useful: lower Ricci bounds become closed conditions under geometric limits.

Conceptual Consequences

The theory separates three roles that coincide on smooth manifolds.

First, the metric determines geodesics and transport costs. Second, the measure determines entropy and volume comparison. Third, the dimension parameter controls the strength of convexity. A singular space can therefore be studied by asking whether these three structures interact as they do on a manifold with lower Ricci curvature.

This is why Sturm’s construction became a base layer for later developments such as $\mathrm{RCD}(K,N)$ spaces, where one adds an infinitesimal Hilbertianity condition to rule out Finsler behavior. The original $\mathrm{CD}$ condition captures Ricci lower bounds but still permits normed spaces whose tangent behavior is not Euclidean.

Branching and infinitesimal structure

In branching spaces, optimal transport geodesics may split, and finite-dimensional curvature-dimension conditions can behave less rigidly than in smooth geometry. Later refinements impose non-branching or Hilbertian assumptions when one wants sharper differential calculus and splitting theorems.

Smooth Verification and Distortion Coefficients

The definition is easiest to trust after checking the smooth model. Let ${\mu }_0={\rho }_0{\mathrm{vol}}_g$ and let

$$T(x)={\exp }_x(-\nabla \phi (x))$$

be the optimal map to ${\mu }_1$. The interpolation is

$$T_t(x)={\exp }_x(-t\nabla \phi (x)),{\mu }_t=(T_t{)}_{\#}{\mu }_0.$$

The density is determined by the Jacobian identity

$${\rho }_t(T_t(x))\det D{T}_t(x)={\rho }_0(x).$$

Ricci lower bounds enter through comparison estimates for $\det D{T}_t$. In dimension $N$, these estimates are encoded by distortion coefficients, often denoted ${\tau }_{K,N}^{(t)}(\theta )$, where $\theta =d(x,T(x))$. They compare the actual infinitesimal expansion of geodesic transport with the expansion in the simply connected model space of constant curvature $K/(N-1)$.

For $K=0$, the coefficient is essentially linear in $t$, and one recovers the concavity of ${\rho }_t^{-1/N}$ along transport rays. For $K>0$, the coefficient involves sine functions; for $K<0$, hyperbolic sine functions appear. This is why the finite-dimensional condition is stronger than plain convexity of Boltzmann entropy: it remembers the volume distortion predicted by Jacobi-field comparison.

Stability of the definition

The smooth verification uses differentiability of $T_t$ and Jacobi fields. Sturm’s definition keeps only the final integral inequality, which makes sense without differentiability. Stability under measured convergence then follows from compactness and lower semicontinuity rather than from pointwise convergence of curvature tensors.

Proof Details Behind Comparison Theorems

Bishop-Gromov comparison follows by choosing one measure concentrated in a small ball around a base point and the other uniformly distributed in a larger annulus. The $\mathrm{CD}(K,N)$ inequality controls the measure of intermediate sets along geodesics. Letting the small ball shrink forces a comparison between the volumes of concentric balls and the corresponding model volumes.

The Bonnet-Myers estimate uses the same distortion coefficients in a limiting form. If $K>0$, the model coefficient becomes singular at distance

$$\pi \sqrt{\frac{N-1}{K}}.$$

A geodesic longer than this would force the curvature-dimension inequality to compare positive endpoint masses through a zero or negative model coefficient, which is impossible. Thus the transport inequality forbids excessive diameter.

This proof pattern is important: one does not build geodesic polar coordinates. Instead, one tests the synthetic inequality on carefully chosen probability measures. The geometry is extracted from how mass can be interpolated.

Transferable Mechanisms

The most portable idea is to prove geometric theorems through convexity in a space of probability measures. Lott and Villani developed an independent optimal-transport formulation of Ricci curvature lower bounds at the same time, confirming that the transport interpretation was not an artifact of Sturm’s presentation (Lott & Villani, 2009).

The later metric-measure calculus of Ambrosio, Gigli, and Savaré shows how this viewpoint feeds into analysis: weak gradients, heat flow, and Bochner-type inequalities can be constructed without a smooth tangent bundle (Ambrosio, Gigli & Savare, 2013). For other problems, this is the lesson: if a differential inequality can be rewritten as convexity or monotonicity of a variational functional, then it may survive singular limits where tensors no longer exist.

Links

• on convergence of graph Laplacian to manifold’s Laplacian --- both topics ask how analytic structures survive passage from discrete or approximate spaces to limiting geometric objects.
• on KLS localization lemma --- optimal transport and one-dimensional localization are parallel ways of reducing high-dimensional inequalities to controlled geodesic or needle families.
• on concentration of information and log-concave distributions --- entropy and displacement convexity are central mechanisms behind concentration inequalities in log-concave geometry.

References

🐻  Ambrosio, L., Gigli, N. & Savare, G. 2013. Calculus and Heat Flow in Metric Measure Spaces and Applications to Spaces with Ricci Bounds from Below. Inventiones Mathematicae 195(2), 289–391.

🐻  Lott, J. & Villani, C. 2009. Ricci Curvature for Metric-Measure Spaces via Optimal Transport. Annals of Mathematics 169(3), 903–991.

🐻  Sturm, K.-T. 2006a. On the Geometry of Metric Measure Spaces. Acta Mathematica 196(1), 65–131.

🐻  Sturm, K.-T. 2006b. On the Geometry of Metric Measure Spaces. II. Acta Mathematica 196(1), 133–177.