Given a compact Kähler manifold $X$ polarized by some ample line bundle $L$, the Yau-Tian-Donaldson conjecture relates the existence of an extremal metric in the first Chern class of $L$ to a GIT stability notion of the pair $(X,L)$. This conjecture is valid for Fano varieties with the anti-canonical polarization but the general case is still open. In this talk, I will explain how the complex deformation theory of a projective manifold admitting an extremal metric gives an infinitesimal evidence for this conjecture. The deformation theory for extremal metrics will be illustrated with the examples of toric surfaces.