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1. Brownian Motions and Heat Kernel Lower Bounds on Kähler and Quaternion Kähler Manifolds

   https://doi.org/10.1093/imrn/rnaa199  

   Baudoin, Fabrice; Yang, Guang (August 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger–Yau-type lower bound for the heat kernels of such manifolds and also sharp Cheng’s type estimates for the Dirichlet eigenvalues of metric balls. 

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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. Complete Kähler–Einstein Metric on Stein Manifolds With Negative Curvature

   https://doi.org/10.1093/imrn/rnaa142  

   Lee, Man-Chun (June 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow. 

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4. Geometry of variational methods: dynamics of closed quantum systems

   https://doi.org/10.21468/SciPostPhys.9.4.048  

   Hackl, Lucas; Guaita, Tommaso; Shi, Tao; Haegeman, Jutho; Demler, Eugene; Cirac, Ignacio (January 2020, SciPost Physics)

   null (Ed.)

   We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states. 

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5. 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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