2023/07/21 by Andrea Loi, Loi, Andrea, Giovanni Placini +1
Health Professions · Mathematics · #32Q20 (Secondary) #32W20 (Primary) 32Q15 #53C42 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Therapeutic Uses of Natural Elements
paper · pdf · doi:10.48550/arxiv.2307.11500
openalex publication_date 2023/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler--Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., ρωk=λΩ. In particular, when k=1, under some condition on the maximal domain of definition of canonical coordinates, we show that λ is forced to be positive. Moreover, for arbitrary k, we prove two additional results. Namely, if ω and Ω are induced by a flat metric, then ω is Ricci-flat. Finally, if a Kähler-Ricci soliton Ω arises as Kähler--Ricci iteration of a metric ω induced by a complex space form, then the Kähler--Ricci soliton is forced to be trivial, that is, Kähler--Einstein. These three theorems extend well known results on Kähler--Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.