Extend the main results to 2-dimensional Cartan–Hadamard manifolds

Establish that the paper’s principal well-posedness and characterization results—specifically, existence and uniqueness of solutions to reflected backward stochastic differential equations with possibly nonzero generators, and existence and uniqueness of I-martingales with prescribed drifts and terminal values—remain valid when the state space D is a two-dimensional Cartan–Hadamard manifold (i.e., simply connected with non-positive sectional curvature) satisfying the regularity condition (R) of Assumption 1.1.

Background

The paper proves existence and uniqueness results for reflected BSDEs and for I-martingales with drifts in bounded, simply connected two-dimensional domains that are locally C2-diffeomorphic to convex sets, leveraging the CAT(0) property to obtain convexity of the squared geodesic distance. These results are formulated as Theorems 3.2, 4.1, 4.2, and 4.3.

Cartan–Hadamard manifolds are CAT(0) spaces, implying strong convexity properties for squared geodesic distance and well-behaved Fréchet means. The authors conjecture that the same geometric tools should extend their proofs from Euclidean domains with boundary to two-dimensional Cartan–Hadamard manifolds meeting Assumption 1.1, with adaptations such as replacing Euclidean lines by geodesics and using parallel transport for linearizations.

References

We conjecture that our main results (Theorems 3.2, 4.1, 4.2 and 4.3) remain valid in the setting where D is a 2-dimensional Cartan-Hadamard manifold (i.e., simply connected with a non-positive sectional curvature), satisfying the regular- ity condition of Assumption 1.1. The reason is that Cartan-Hadamard manifolds are CAT(0) spaces, implying that the square of their geodesic distance function is convex and smooth. In addition, it is well known that, in these manifolds, the Fréchet means of compactly supported probability measures exist, are unique and depend smoothly on the measures. To establish such an extension, in all the proofs herein, one would need to replace the Euclidean lines with geodesics and to perform linearizations via parallel translations along geodesics.

— Martingales On A Euclidean Manifold With A Boundary And Reflected BSDES In Non-Convex Domains  (2512.13200 - Arnaudon et al., 15 Dec 2025) in Section 1.1 (Motivation and main contributions)