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

TL;DR

This paper proves short and long-term existence of solutions to a generalized geometric flow on closed manifolds, extending Ricci and Yamabe flows, with curvature estimates ensuring stability over time.

Contribution

It introduces the $(oldsymbol{ extalpha,eta})$-Ricci-Yamabe flow and establishes foundational existence results, generalizing key geometric flows.

Findings

Short time existence of smooth solutions

Long time existence under curvature estimates

Extension of Ricci and Yamabe flows

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.