RC-positivity and the generalized energy density I: Rigidity

arXiv1810.03276MaRDI QIDQ6307853

Xiao-Kui Yang

Publication date: 8 October 2018

Abstract: In this paper, we introduce a new energy density function $mathscrY$ on the projective bundle $mathbbP(T_M)>M$ for a smooth map $f:(M,h)>(N,g)$ between Riemannian manifolds ${\displaystyle }$mathscr Y=g_{ij}f^i_alpha f^j_�eta frac{W^alpha W^�eta}{sum h_{gammadelta} W^gamma W^delta}.${\displaystyle }$ 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.

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Vanishing theorems in algebraic geometry (14F17) Vanishing theorems (32L20)

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