[Paper Review] Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds

Young‐Heon Kim|ArXiv.org|Sep 11, 2007

Geometric Analysis and Curvature Flows6 references15 citations

TL;DR

This paper constructs explicit counterexamples to the conjecture that nonnegative or positive sectional curvature on a Riemannian manifold implies the A3w condition, a key requirement for continuity of optimal transport maps. By analyzing radially symmetric surfaces with shallow conical singularities and their smooth perturbations, the authors demonstrate that even in positively curved manifolds, optimal transport maps can be discontinuous for smooth measures, thereby disproving a long-standing open question in optimal transport and Riemannian geometry.

ABSTRACT

Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature.

Motivation & Objective

• To resolve Question 1.1: whether nonnegative or positive sectional curvature implies the A3w condition for optimal transport.
• To disprove the conjecture that A3w follows from nonnegative curvature, which had been suspected by Trudinger and supported by known results on spheres.
• To construct explicit examples of complete Riemannian manifolds with everywhere positive (or nonnegative) curvature that violate A3w.
• To demonstrate that discontinuous optimal transport maps can exist even for smooth source and target measures on such manifolds.
• To clarify the geometric meaning of A3w by showing it is not implied by curvature positivity, but rather a stronger, independent condition.

Proposed method

• Construct a 2D surface Σ_ϑ as a smooth, radially symmetric perturbation of a flat cone with conical angle 2π−2ϑ, where 0<ϑ≪1.
• Ensure the metric is flat (K≡0) outside a small ball B(O,1), and has small positive curvature (0<K<1/10000) inside B, preserving radial symmetry.
• Use the Riemannian distance squared cost c(x,ȳ)=dist²(x,ȳ)/2 as the transport cost function.
• Show that the cost function is differentiable in a large region, enabling the use of geodesic analysis and c-segments.
• Prove that the local DASM (degenerate A3w) condition fails at points near the origin by analyzing c-segments that bend inside the curved region.
• Extend the construction to higher dimensions via radial symmetry or Riemannian products, and perturb to achieve strict positive curvature while preserving the violation of A3w.

Experimental results

Research questions

• RQ1 Does every Riemannian manifold with nonnegative sectional curvature satisfy the A3w condition for optimal transport?
• RQ2 Can a manifold with positive sectional curvature fail to satisfy A3w, thereby allowing discontinuous optimal transport maps?
• RQ3 Is the A3w condition implied by curvature positivity, or is it a strictly stronger geometric condition?
• RQ4 Can explicit counterexamples be constructed in both open and compact settings?
• RQ5 What is the geometric role of cross-curvature in determining the regularity of optimal transport maps?

Key findings

• The paper constructs a 2-dimensional complete Riemannian manifold with everywhere nonnegative curvature that violates the A3w condition.
• The constructed manifold is a smooth, radially symmetric perturbation of a flat cone with small conical angle, resulting in small positive curvature in a bounded region.
• The violation of A3w is demonstrated via failure of the local DASM condition, shown by analyzing c-segments that deviate from straight-line paths in the curved region.
• The counterexample can be extended to higher dimensions by isometrically embedding the 2D example into higher-dimensional radial manifolds.
• By perturbing the nonnegatively curved example, a compact, positively curved manifold is constructed that still violates A3w, proving that positive curvature does not imply A3w.
• The result confirms Trudinger’s suspicion and shows that A3w is not a consequence of curvature positivity, but a distinct, stronger condition.