This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains. We prove a “low energy” result in all dimensions, in the sense that if normalized energy in a large ball is small enough, then the normalized energy in any interior ball must also be small.
"Volume estimates for Kähler-Einstein metrics and rigidity of complex structures." J. Differential Geom. 93 (2) 191 - 201, February 2013. https://doi.org/10.4310/jdg/1361800865