Let (X,ω0) be a compact complex manifold of complex dimension n endowed with a Hermitian metric ω0. The Chern–Yamabe problem is to find a conformal metric of ω0 such that its Chern scalar curvature is constant. As a generalization of the Chern–Yamabe problem, we study the problem of prescribing Chern scalar curvature. We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of(X,ω0)
. On the other hand, we prove a version of conformal Schwarz lemma on (X,ω0). All these are achieved by using geometric flows related to the Chern–Yamabe flow. Finally, we prove the backwards uniqueness of the Chern–Yamabe flow.
