https://neckpinch.com/authors/marco-flaim/New arXiv papers by Marco Flaim, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.18942arxiv:2603.18942Thu, 19 Mar 2026 00:00:00 GMTMarco Flaim, Erik Hupp, Karl-Theodor Sturm — We present a synthetic notion of scalar curvature (and its integral) for Riemannian manifolds and metric measure spaces, defined in terms of the initial slope of a Gaussian (double) integral. We explicitly calculate the integral scalar curvature for Lipschitz gluings of smooth Riemannian manifolds and for cones. In dimension 2, the former coincides with the formula derived by Gauss-Bonnet, whereas the latter differs. The extension to the time-dependent case allows us to characterize Ricci flows as super Ricci flows with minimal integral curvature functional.Ricci flowScalar curvatureWeak & singular flowsLimit spaces & RCD https://arxiv.org/abs/2511.11477arxiv:2511.11477Fri, 14 Nov 2025 00:00:00 GMTMatthias Erbar, Marco Flaim, Eric Hupp, Zhenhao Li, Timo Schultz, Karl-Theodor Sturm — We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from optimal transport, and the asymptotic behaviour of volumes. Each notion equivalently characterises (weighted) Ricci flow for smooth families of weighted Riemannian manifolds. We discuss the features of the different notions on various examples.Ricci flowWeak & singular flowsLimit spaces & RCD https://arxiv.org/abs/2507.05032arxiv:2507.05032Mon, 07 Jul 2025 00:00:00 GMTMarco Flaim, Erik Hupp — Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. Some results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds and Ricci flows. However we observe that the monotonicity of Perelman's functionals (𝓕, 𝓦, reduced volume), which hold true for Ricci flows and Ricci nonnegative manifolds, cannot be strictly generalized to super Ricci flows. In 2010 Buzano introduced a condition which still generalizes Ricci flows and Ricci nonnegative manifolds, and on which Perelman's monotonicities do hold. We provide characterizations of this condition using optimal transport and understand it heuristically as Ricci nonnegativity of the space-time. This interpretation is consistent with its equivalence to Ricci nonnegativity on Perelman's infinite dimensional manifold. More precisely, we prove that for smooth evolutions of Riemannian manifolds, this condition is equivalent to a Bochner inequality (resembling Perelman's Harnack inequality but for the forward heat flow), a gradient estimate for the heat flow, a Wasserstein contraction along the adjoint heat flow, the convexity of a modified entropy along Wasserstein geodesics, and an Evolutionary Variational Inequality (EVI). The optimal transport statements use Perelman's L distance as cost, as first studied on Ricci flows by Topping and by Lott. We also consider dimensionally improved and weighted versions of these conditions. The dimensional Bochner inequality and all gradient estimates for the forward heat equation, along with the EVIs, appear to be new even for general Ricci flows, and are related to the Hamiltonian perspective on the L distance. Most of our proofs do not use tensor calculus or Jacobi fields, suggesting the possibility of future extensions to more singular settings.Ricci flowPerelman entropyReduced volumeHarnack inequalitiesWeak & singular flowsLimit spaces & RCD