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.
Categories: math.DG math.AP