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# Asian Journal of Mathematics

## Volume 23 (2019)

### Number 6

### Deformations from a given Kähler metric to a twisted cscK metric

Pages: 985 – 1000

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n6.a6

#### Author

#### Abstract

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.

#### Keywords

cscK metric, twisted cscK metric, cscK metric deformation

#### 2010 Mathematics Subject Classification

53-xx

Received 6 April 2016

Accepted 16 November 2018

Published 3 August 2020