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1. Volume and macroscopic scalar curvature

   https://doi.org/10.1007/s00039-021-00588-y  

   Braun, Sabine; Sauer, Roman (December 2021, Geometric and Functional Analysis)

   Abstract We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. 

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2. 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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3. Curvature in the balance: the Weyl functional and scalar curvature of 4-manifolds

   https://doi.org/10.4310/PAMQ.2023.v19.n6.a5  

   LeBrun, Claude (January 2024, Pure and Applied Mathematics Quarterly)

   The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive- scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler 4-manifolds. 

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4. 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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5. Four-manifolds of pinched sectional curvature

   https://doi.org/10.1016/j.aim.2018.07.003  

   Cao, Xiaodong; Tran, Hung (January 2022, Pacific journal of mathematics)

   In this paper, we study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than 5 6 of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere, or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature. 

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