# On the Chern-Yamabe flow

## TL;DR

This paper investigates the convergence of a modified Chern-Yamabe flow on closed balanced and almost-Hermitian manifolds, establishing conditions under which solutions to the Chern-Yamabe problem exist.

## Contribution

It proves convergence of a modified flow under Sobolev norm smallness and existence of solutions when scalar curvature is close to constant in H"older norm.

## Key findings

- Flow converges under small Sobolev norm of scalar curvature.
- Solutions exist when scalar curvature is close to constant in H"older norm.
- Results apply to generic fundamental constants.

## Abstract

On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm then a slightly modified version of the Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Hermitian manifolds, is close enough to a constant function in a H\"older norm then the Chern-Yamabe problem has a solution for generic values of the fundamental constant.