I presume you are referring to the Yamabe flow approach to the Yamabe problem, which in 2 dimensions reduces to a Ricci flow. Relevant references include

The Ricci flow on the 2-sphere (1991)

The Yamabe flow on locally conformally flat manifolds with positive ricci curvature (1992)

Global existence and convergence of Yamabe flow (1994)

I presume you are referring to the Yamabe flow approach to the Yamabe problem, which in 2 dimensions reduces to a Ricci flow. Relevant references include

• The Ricci flow on surfaces (1988)

• The Ricci flow on the 2-sphere (1991)

• The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature (1992)

• Global existence and convergence of Yamabe flow (1994)