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1. Pseudo-locality for a coupled Ricci flow

   Guo, Bin; Huang, Zhijie; Phong, Duong H. (July 2018, Communications in analysis and geometry)

   Let (M,g,\phi) be a solution to the Ricci flow coupled with the heat equation for a scalar field \phi. We show that a complete, \kappa-noncollapsed solution (M,g,\phi) to this coupled Ricci flow with a Type I singularity at time T<\infty will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base point is Type I singular. A key ingredient is a version of Perelman pseudo-locality for the coupled Ricci flow. 

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2. Ricci flow on Courant algebroids

   https://doi.org/10.1142/S0219199725500373  

   Streets, Jeffrey; Strickland-Constable, Charles; Valach, Fridrich (April 2025, Communications in Contemporary Mathematics)

   We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short-time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow. 

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3. Maxwell’s Demon: Controlling Entropy via Discrete Ricci Flow over Networks

   Sandhu, Romeil; Liu, Ji (January 2020, International Conference on Network Science)

   In this work, we propose to utilize discrete graph Ricci flow to alter network entropy through feedback control. Given such feedback input can “reverse” entropic changes, we adapt the moniker of Maxwell’s Demon to motivate our approach. In particular, it has been recently shown that Ricci curvature from geometry is intrinsically connected to Boltzmann entropy as well as functional robustness of networks or the ability to maintain functionality in the presence of random fluctuations. From this, the discrete Ricci flow provides a natural avenue to “rewire” a particular network’s underlying geometry to improve throughout and resilience. Due to the real-world setting for which one may be interested in imposing nonlinear constraints amongst particular agents to understand the network dynamic evolution, controlling discrete Ricci flow may be necessary (e.g., we may seek to understand the entropic dynamics and curvature “flow” between two networks as opposed to solely curvature shrinkage). In turn, this can be formulated as a natural control problem for which we employ feedback control towards discrete Ricci-based flow and show that under certain discretization, namely Ollivier-Ricci curvature, one can show stability via Lyapunov analysis. We conclude with preliminary results with remarks on potential applications that will be a subject of future work. 

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4. Optimal Transport and Generalized Ricci Flow

   https://doi.org/10.3842/SIGMA.2024.003  

   Kopfer, Eva; Streets, Jeffrey (January 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   We prove results relating the theory of optimal transport and generalized Ricci flow. We define an adapted cost functional for measures using a solution of the associated dilaton flow. This determines a formal notion of geodesics in the space of measures, and we show geodesic convexity of an associated entropy functional. Finally, we show monotonicity of the cost along the backwards heat flow, and use this to give a new proof of the monotonicity of the energy functional along generalized Ricci flow. 

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5. Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

   https://doi.org/10.1142/S1793525321500461  

   Brannan, Michael; Gao, Li; Junge, Marius (September 2023, Journal of Topology and Analysis)

   We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors. 

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