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1. Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

   https://doi.org/10.1007/s00454-023-00507-y  

   Hoorn, Pim van der; Lippner, Gabor; Trugenberger, Carlo; Krioukov, Dmitri (June 2023, Discrete & Computational Geometry)

   Abstract Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear. 

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2. NEUMANN PROBLEMS FOR p-HARMONIC FUNCTIONS, AND INDUCED NONLOCAL OPERATORS IN METRIC MEASURE SPACES

   Capogna, Luca; Kline, Joshua; Korte, Riikka; Shanmugalingam, Nageswari; Snipes, Marie (April 2022, Arxiv)

   Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p)θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non- homogenous non-local equation (−∆p)θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality. 

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3. Some local maximum principles along Ricci flows

   https://doi.org/10.4153/s0008414x20000772  

   Lee, Man-Chun; Tam, Luen-Fai (November 2020, Canadian Journal of Mathematics)

   null (Ed.)

   Abstract In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $$\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$$ for $t>0$ for some constant $${\alpha }>0$$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature . By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ . 

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4. O(2)-symmetry of 3D steady gradient Ricci solitons

   https://doi.org/10.2140/gt.2025.29.687  

   Lai, Yi (January 2025, Geometry & Topology)

   We prove that all 3D steady gradient Ricci solitons are O(2)-symmetric. The O(2)-symmetry is the most universal symmetry in Ricci flows with any type of symmetries. Our theorem is also the first instance of symmetry theorem for Ricci flows that are not rotationally symmetric. We also show that the Bryant soliton is the unique 3D steady gradient Ricci soliton with positive curvature that is asymptotic to a ray. 

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5. The stochastic Schwarz lemma on Kähler manifolds by couplings and its applications

   https://doi.org/10.1112/jlms.12849  

   Chae, Myeongju; Cho, Gunhee; Gordina, Maria; Yang, Guang (December 2023, Journal of the London Mathematical Society)

   Abstract We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall–Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete noncompact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau–Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a by‐product, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds. 

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