I presume you are referring to the <A HREF="https://en.wikipedia.org/wiki/Yamabe_flow">Yamabe flow</A> approach to the <A HREF="https://en.wikipedia.org/wiki/Yamabe_problem">Yamabe problem</A>, which in 2 dimensions reduces to a Ricci flow. Relevant references include

<A HREF="https://projecteuclid.org/euclid.jdg/1214446319">The Ricci flow on the 2-sphere</A> (1991)

<A HREF="https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160450805">The Yamabe flow on locally conformally flat manifolds with positive ricci curvature</A> (1992)

<A HREF="https://projecteuclid.org/euclid.jdg/1214454674">Global existence and convergence of Yamabe flow</A> (1994)