Title: On the Yau-Tian-Donaldson conjecture for weighted cscK metrics Show Chinese title

Title: 关于加权cscK度量的Yau-Tian-Donaldson猜想

Abstract: We establish a version, formulated in terms of non-Archimedean pluripotential theory, of the Yau-Tian-Donaldson conjecture for constant scalar curvature and, more generally, weighted extremal K\"ahler metrics with prescribed compact symmetry group on an arbitrary polarized projective manifold. This is accomplished by extending important previous work of Chi Li to the weighted case, and by establishing general slope formulas for the relevant weighted entropy and energy functionals. Among other things, our approach relies on key a priori estimates for cscK metrics due to Chen-Cheng (recently extended to the weighted case by Di Nezza-Jubert-Lahdili and Han-Liu), and on a crucial estimate for psh envelopes due to Berman-Demailly-Di Nezza-Trapani. Show Chinese abstract

Abstract: 我们在非阿基米德多势论的框架下，建立了常标量曲率的Yau-Tian-Donaldson猜想的一个版本，并且更一般地，对于任意极化射影流形上具有预定紧对称群的加权极值Kähler度量。 这是通过将Chi Li之前的重要工作扩展到加权情况，并建立相关加权熵和能量泛函的一般斜率公式来实现的。 其中，我们的方法依赖于Chen-Cheng关于cscK度量的关键先验估计（最近由Di Nezza-Jubert-Lahdili和Han-Liu扩展到加权情况），以及Berman-Demailly-Di Nezza-Trapani关于psh包络的关键估计。