The Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kähler cases.

In this talk, we discuss extremal Kähler versions of the Yau-Tian-Donaldson Conjecture by focussing on the difference between Fano cases and extremal cases. Important tools in our approach are the Chow norm and the geometry of test configurations.