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1. Scalar Curvature, Entropy, and Generalized Ricci Flow

   https://doi.org/10.1093/imrn/rnad002  

   Streets, Jeffrey (February 2023, International Mathematics Research Notices)

   Abstract We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction–diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles. 

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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. Parabolic Frequency on Manifolds

   https://doi.org/10.1093/imrn/rnab052  

   Holck Colding, Tobias; Minicozzi II, William P (April 2021, International Mathematics Research Notices)

   Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness. 

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4. Uniqueness and stability of Ricci flow through singularities

   https://doi.org/DOI: https://dx.doi.org/10.4310/ACTA.2022.v228.n1.a1  

   Bamler, Richard; Kleiner, Bruce (July 2022, Acta mathematica)

   We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper. 

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5. Matrix Li–Yau–Hamilton estimates under Ricci flow and parabolic frequency

   https://doi.org/10.1007/s00526-024-02668-x  

   Li, Xiaolong; Zhang, Qi S. (February 2024, Calculus of Variations and Partial Differential Equations)

   Abstract We prove matrix Li–Yau–Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply these estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature. 

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