In [3], X. Chen proposed a continuity path aiming to attack the existence problem of the constant scalar curvature Kähler(cscK) metric. He also proved the openness of the path at $t \in (0, 1)$ by the standard implicit function theorem on solutions of fourth order PDE. However, the openness at $t = 0$ is quite different in nature and it is in fact a deformation result from the solution of a second order PDE to the solution of a fourth order PDE. In this paper, we give a proof of the openness at $t = 0$, which asserts the existence of twisted cscK metrics for $t \gt 0$ sufficient small.

中文翻译：

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从给定的Kähler度量到扭曲的cscK度量的变形

在[3]中，X。Chen提出了一条连续路径，旨在解决恒定标量曲率Kähler（cscK）度量的存在问题。他还通过四阶PDE解的标准隐函数定理证明了$ t \ in（0，1）$处路径的开放性。但是，$ t = 0 $时的开放度本质上是完全不同的，并且实际上是从二阶PDE的解到四阶PDE的解的变形结果。在本文中，我们给出了$ t = 0 $时的开放度的证明，该结论断言对于$ t \ gt 0 $足够小的cscK度量存在。