Existence of the $(\alpha ,\beta )$-Ricci-Yamabe flow on closed manifolds

Abstract

On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(\alpha ,\beta )$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(\alpha ,\beta )$-Ricci-Yamabe flow by estimating its curvatures.