In this note we study the conformal Ricci flow that Arthur Fischer introduced in 2004. We use DeTurck's trick to rewrite the conformal Ricci flow as a strong parabolic-elliptic partial differential equation. Then we prove short-time existence for the conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We show that the Yamabe constant is monotonically increasing along conformal Ricci flow on compact manifolds. We also show that the conformal Ricci flow is the gradient flow for the ADM mass on asymptotically flat manifolds.
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Lü et al. (2014) studied this question.