https://neckpinch.com/authors/karl-theodor-sturm/New arXiv papers by Karl-Theodor Sturm, 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/2404.15755arxiv:2404.15755Wed, 24 Apr 2024 00:00:00 GMTKarl-Theodor Sturm — Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvaturedimension condition formulated in functional-analytic terms by Bakry-Èmery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, ii) second order calculus, upper Ricci bounds, and transformation formulas, iii) distribution-valued Ricci bounds which e.g. allow singular effects of non-convex boundaries to be taken into account.Ricci flowHarnack inequalitiesWeak & singular flowsLimit spaces & RCDStability & uniquenessGeometric inequalities