Rigidity of the Borell-Brascamp-Lieb Inequality on Weighted Riemannian Manifolds

Rigidity of the Borell-Brascamp-Lieb Inequality on Weighted Riemannian Manifolds

Introduction and Context

This paper addresses the rigidity phenomena associated with the Borell-Brascamp-Lieb (BBL) inequality in the context of weighted Riemannian manifolds with curvature-dimension conditions. The BBL inequality, a functional generalization of the Brunn-Minkowski inequality, is foundational in geometric analysis and optimal transport. It interpolates between the Prékopa-Leindler and Brunn-Minkowski inequalities and is essential for deriving regularity and comparison results in spaces with lower bounds on Ricci curvature and generalized dimension.

The author generalizes the known rigidity results for unweighted manifolds, specifically the curvature rigidity theorem of Balogh and Kristály, to the weighted measure setting. This broadens the scope to manifolds equipped with a smooth weight function, where the notion of $N$-Ricci curvature, $Ric_{m,N}$, replaces the traditional Ricci tensor.

Main Results

Weighted Borell-Brascamp-Lieb Inequality

The paper establishes a version of the BBL inequality for smooth weighted Riemannian manifolds $(M, g, m = e^{-\psi} \mathrm{vol}_g)$ satisfying the lower curvature-dimension condition $Ric_{m,N} \geq Kg$, with $N \in [n, \infty)$ and $K \in \mathbb{R}$. The proof leverages optimal transport theory, specifically the Monge-Ampère equation and regularity of optimal maps, to establish an interpolation inequality for density evolution under displacement interpolation.

The result recovers the standard Euclidean BBL inequality in the flat case and extends curvature distortion terms via the volume distortion coefficients tailored to the weighted and curved setting.

Rigidity Theorem

A central contribution is the precise rigidity statement: if equality holds in the BBL inequality for such weighted manifolds, then strong geometric and measure-theoretic conclusions must hold. Explicitly, for almost every point and for geodesics determined by optimal transport between extremizing densities, the following conditions are deduced:

• The $N$-Ricci curvature along the geodesic coincides identically with the lower curvature bound $K$.
• All sectional curvatures along these geodesics equal $\frac{K}{N-1}$.
• The weight function $e^{-\psi}$, when restricted to these geodesics, evolves as the $Ric_{m,N}$0-th power of a trigonometric, linear, or hyperbolic function of time parameter $Ric_{m,N}$1, according to the sign of $Ric_{m,N}$2.

This structure theorem generalizes classical results for unweighted manifolds, explicitly showing that equality in the BBL inequality constrains the local geometry and the measure in rigid ways dictated by the curvature-dimension data.

Implications for the Brunn-Minkowski Inequality

Since the BBL inequality implies the Brunn-Minkowski (BM) inequality, the paper also derives rigidity statements for the BM inequality in the weighted setting. In the case $Ric_{m,N}$3, sectional curvature vanishes along transport geodesics and the density of the measure evolves polynomially. For positive and negative curvature, the transport sets and measures behave analogously, but rigidity forces (in the positive curvature case) the transported sets to coincide up to null sets, and (in the negative curvature case) precludes equality for compact sets of positive measure.

The results further confirm the equivalence in such spaces between the BBL inequality, the BM inequality, and the curvature-dimension condition in the sense of optimal transport.

Technical Approach

The paper follows a classical line of argument via the analysis of the Jacobian of optimal transport maps along geodesics, extending the techniques of Cordero-Erausquin, McCann, and Schmuckenschläger, and of Balogh and Kristály, to incorporate measure-weights and the $Ric_{m,N}$4-Ricci tensor. Central to the argument is the careful analysis of the Riccati equation satisfied by the differential of the transport map, the concavity properties arising from the curvature-dimension condition, and the propagation of equality cases through unique solutions to the corresponding second-order ODEs.

A key lemma establishes that equality in the interpolation inequality at a single interior time parameter forces equality at all times, by invoking ODE uniqueness and the strong maximum principle for subsolutions.

Theoretical and Practical Implications

The rigidity results have significant theoretical implications. They offer precise geometric characterization of cases of equality in fundamental inequalities, which in turn indicate the uniqueness and stability properties of extremizers. These results facilitate the classification of manifolds and measures where the BBL inequality is saturated, thereby contributing to the structure theory of metric measure spaces with Ricci curvature bounds.

From a practical perspective, such rigidity theorems underpin uniqueness and stability in geometric variational problems, inform regularity in PDEs derived from functional inequalities, and motivate analogous questions in spaces beyond smooth Riemannian geometry (e.g., metric measure spaces with synthetic curvature).

Directions for Further Research

Several avenues are suggested:

• Quantitative stability versions of the BBL rigidity are open, with potential for optimal transport-inspired stability inequalities.
• Extensions of rigidity phenomena to Finsler manifolds or synthetic, non-smooth metric measure spaces ($Ric_{m,N}$5 or $Ric_{m,N}$6) are anticipated.
• Generalization to Lorentzian geometry and curvature-dimension conditions in spacetime is posited, with connections to recent progress in timelike optimal transport.
• The cases $Ric_{m,N}$7 and $Ric_{m,N}$8 are not yet treated and remain as natural open questions.

Conclusion

The paper provides a comprehensive and technically robust account of the rigidity of the Borell-Brascamp-Lieb inequality in the weighted Riemannian manifold context. The main theorem establishes that equality in the BBL inequality enforces stringent geometric and analytic symmetry: sectional curvatures, $Ric_{m,N}$9-Ricci curvature, and the weighted measure exhibit precise behaviors determined by the curvature-dimension data. These findings reveal strong links between optimal transport, functional inequalities, and Riemannian geometry, and set the stage for further developments in metric measure theory and geometric analysis (2604.00562).