Prescribed Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures

In this paper, we investigate the problem of prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures, and establish some existence results. In particular, we obtain some sufficient conditions for the existence of a constant negative Chern scalar curvature metric in the conformal class.

💡 Research Summary

The paper studies the prescribed Chern‑scalar‑curvature problem on complete non‑compact Hermitian manifolds (Mⁿ, ω) of complex dimension n. For a smooth real function S on M, the goal is to find a conformal metric (\tilde\omega=e^{2nu}\omega) whose Chern scalar curvature satisfies (S_{Ch}(\tilde\omega)=S). In terms of the unknown function u, this is the nonlinear elliptic equation

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