The Large-N Limit of Ricci curvature in the Zeitlin Model on the Torus
Abstract
The Euler equations of hydrodynamics describe the motion of an ideal incompressible fluid. Arnold observed that their solutions are geodesics on the group of volume-preserving diffeomorphisms of the fluid domain, equipped with the L2 metric, so that the curvature of this group governs how nearby fluid motions separate. Sectional curvature takes both signs, and Ricci curvature, its average over all directions, is the natural measure of which sign prevails, but in infinite dimensions the sum defining it diverges. When the fluid domain is the flat torus, we define Ricci curvature through Zeitlin's finite-dimensional models, which approximate the Euler equations by geodesic equations on the matrix groups SU(N). We compute the Ricci curvature of these models exactly and prove that, normalized by dimension, it converges as N → ∞ to an explicit integral. The limit is strictly negative in every Fourier mode and decreases logarithmically with the wavenumber, the behavior conjectured in \cite{lichtenfelz2026ricci} for the sphere. We also show that the diameter of SU(N) in the Zeitlin metric grows at most linearly in N.
Related
- Ricci curvature for fluid models on the torus via Zeitlin's quantization
- Torus and Positive Mass stability for metrics with Ricci curvature lower bound
- Under Ricci flow, a 3-torus goes flat
- Long-Time Behavior of the α-Ricci Flow on the 2-Torus
- Torus actions, almost non-negative curvature and fundamental groups
BibTeX
@misc{lichtenfelz2026largen,
title = {{The Large-$N$ Limit of Ricci curvature in the Zeitlin Model on the Torus}},
author = {Leandro Lichtenfelz and Isabelle Raad and Justin Valletta},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02323},
eprint = {2611.02323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02323},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The Large-N Limit of Ricci curvature in the Zeitlin Model on the Torus
AU - Leandro Lichtenfelz
AU - Isabelle Raad
AU - Justin Valletta
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02323
UR - https://arxiv.org/abs/2611.02323
AB - The Euler equations of hydrodynamics describe the motion of an ideal incompressible fluid. Arnold observed that their solutions are geodesics on the group of volume-preserving diffeomorphisms of the fluid domain, equipped with the L^2 metric, so that the curvature of this group governs how nearby fluid motions separate. Sectional curvature takes both signs, and Ricci curvature, its average over all directions, is the natural measure of which sign prevails, but in infinite dimensions the sum defining it diverges. When the fluid domain is the flat torus, we define Ricci curvature through Zeitlin's finite-dimensional models, which approximate the Euler equations by geodesic equations on the matrix groups SU(N). We compute the Ricci curvature of these models exactly and prove that, normalized by dimension, it converges as N → ∞ to an explicit integral. The limit is strictly negative in every Fourier mode and decreases logarithmically with the wavenumber, the behavior conjectured in \cite{lichtenfelz2026ricci} for the sphere. We also show that the diameter of SU(N) in the Zeitlin metric grows at most linearly in N.
ER -