Strict asymptotic ordering of Ricci curvature across modes

Prove that, for the adjoint Laplacian, the limiting normalized Ricci curvature of the modes indexed by (0,k₂) is strictly decreasing with k₂>0, and that the limiting values for the lowest modes indexed by (0,1) and (1,1) coincide.

Background

The paper investigates whether Ricci curvature reflects the known distinction between Lyapunov-stable gravest Euler modes and unstable higher-frequency modes. Numerical computations suggest that the limiting normalized Ricci curvature becomes more negative as the mode number increases, while the two lowest-mode values appear to coincide. These observations are stated as conjectured asymptotics rather than proved results.

References

We speculate the following asymptotics:

— Ricci curvature for fluid models on the torus via Zeitlin's quantization  (2609.01259 - Ishida et al., 1 Sep 2026) in Conjecture in Section 4, subsection “Nonlinear stability of the Euler flows”