RC-positivity and the generalized energy density I: Rigidity

In this paper, we introduce a new energy density function $\mathcal{Y}$ on the projective bundle $\mathbb{P}(T_M)M$ for a smooth map $f:(M,h)(N,g)$ between Riemannian manifolds $$\mathcal{Y}=g_{ij}{f}_{\alpha }^i{f}_{\beta }^j\frac{W^{\alpha }{W}^{\beta }}{\sum h_{\gamma \delta }{W}^{\gamma }{W}^{\delta }}\mathrm{.}$$ We get new Hessian estimates to this energy density and obtain various new Liouville type theorems for holomorphic maps, harmonic maps and pluri-harmonic maps. For instance, we show that there is no non-constant holomorphic map from a compact \emph{Hermitian manifold} with positive (resp. non-negative) holomorphic sectional curvature to a \emph{Hermitian manifold} with non-positive (resp. negative) holomorphic sectional curvature.