Rigorous convergence of Ricci curvature

Establish a rigorous analysis of the convergence of the Ricci curvature defined through Zeitlin-type finite-dimensional approximations in the various fluid settings considered in the paper.

Background

The paper defines Ricci curvature on Hamiltonian diffeomorphisms of the two-dimensional torus as a large-N limit of normalized Ricci curvature tensors on SU(N), equipped with metrics induced by several discrete Laplacians. Numerical experiments suggest convergence for fixed Fourier modes, and an explicit limiting integral is conjectured for the adjoint Laplacian. However, the paper does not provide a rigorous convergence proof, either for the principal torus model or for the extensions involving Sobolev metrics, rectangular domains, and the Coriolis force.

References

Second, a rigorous analysis of the convergence of the Ricci curvature in various settings remains an important open question.

— Ricci curvature for fluid models on the torus via Zeitlin's quantization  (2609.01259 - Ishida et al., 1 Sep 2026) in Introduction, subsection “Future work”

We expect a similar result for the Ricci curvature with respect to the adjoint Laplacian $\Delta{\ad}$:

— Ricci curvature for fluid models on the torus via Zeitlin's quantization  (2609.01259 - Ishida et al., 1 Sep 2026) in Conjecture labeled “Ricci convergence,” Section 3, subsection “Asymptotics toward Ricci curvature on HDiff(T²)”