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1. 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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2. A lower bound for the Kähler-Einstein distance from the Diederich-Fornæss index

   https://doi.org/10.1090/proc/15335  

   Zimmer, Andrew ( April 2021 , Proceedings of the American Mathematical Society)

   In this paper we establish a lower bound for the distance induced by the Kähler-Einstein metric on pseudoconvex domains with positive hyperconvexity index (e.g. positive Diederich-Fornæss index). A key step is proving an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below. 

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3. Weighted Poincaré inequality and the Poisson Equation

   https://doi.org/10.1090/tran/8291  

   Munteanu, Ovidiu ; Sung, Chiung-Jue ; Wang, Jiaping ( March 2021 , Transactions of the American Mathematical Society)

   We develop Green’s function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity. 

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4. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

   Bamler, Richard H. ; Cifarelli, Charles ; Conlon, Ronan J. ; Deruelle, Alix ( June 2022 , ArXivorg)

   We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of C×P1 at one point. This completes the classification of such solitons in two complex dimensions. 

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5. 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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