More Like this

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. 

   more » « less
2. Ricci flow on Courant algebroids

   https://doi.org/10.1142/S0219199725500373  

   Streets, Jeffrey; Strickland-Constable, Charles; Valach, Fridrich (April 2025, Communications in Contemporary Mathematics)

   We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short-time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow. 

   more » « less
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. 

   more » « less
4. 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 $t>0$ . 

   more » « less
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. 

   more » « less