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1. 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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2. 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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3. A generalization of Geroch's conjecture

   https://doi.org/10.1002/cpa.22137  

   Brendle, Simon; Hirsch, Sven; Johne, Florian (January 2024, Communications on Pure and Applied Mathematics)

   Abstract The Theorem of Bonnet–Myers implies that manifolds with topology do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so‐calledm‐intermediate curvature), and use stable weighted slicings to show that for and the manifolds do not admit a metric of positivem‐intermediate curvature. 

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4. 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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5. Some aspects of Ricci flow on the 4-sphere

   https://doi.org/10.53733/152  

   Chang, Sun-Yung Alice; Chen, Eric (September 2021, New Zealand Journal of Mathematics)

   In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere. 

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