RC-positivity and the generalized energy density I: Rigidity

Abstract

$\def\Y{\mathscr{Y}}$In this paper, we introduce a new energy density function $\Y$ on the projective bundle $\mathbb{P}(T_M) \to M$ for a smooth map $f : (M,h) \to (N, g)$ between Riemannian manifolds\[\Y = g_{ij} f^i_\alpha f^j_\beta \dfrac{W^\alpha W^\beta}{\sum h_{\gamma\delta} W^\gamma W^\delta} \; \textrm{.}\]We establish new Hessian estimates to this energy density, which can be regarded as “average” versions of classical estimates. As applications, we 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 Hermitian manifold with positive (resp. nonnegative) holomorphic sectional curvature to a Hermitian manifold with non-positive (resp. negative) holomorphic sectional curvature.