Results Related to the Chern–Yamabe Flow

Abstract

Let $$(X,\omega _0)$$ be a compact complex manifold of complex dimension n endowed with a Hermitian metric $$\omega _0$$ . The Chern–Yamabe problem is to find a conformal metric of $$\omega _0$$ 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 $$(X,\omega _0)$$ . On the other hand, we prove a version of conformal Schwarz lemma on $$(X,\omega _0)$$ . 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.