RCpositivity and the generalized energy density I: Rigidity
Abstract
In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $$\mathscr Y=g_{ij}f^i_\alpha f^j_\beta \frac{W^\alpha W^\beta}{\sum h_{\gamma\delta} W^\gamma W^\delta}.$$ We get new Hessian estimates to this energy density and obtain various new Liouville type theorems for holomorphic maps, harmonic maps and pluriharmonic maps. For instance, we show that there is no nonconstant holomorphic map from a compact \emph{Hermitian manifold} with positive (resp. nonnegative) holomorphic sectional curvature to a \emph{Hermitian manifold} with nonpositive (resp. negative) holomorphic sectional curvature.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.03276
 arXiv:
 arXiv:1810.03276
 Bibcode:
 2018arXiv181003276Y
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 53C55;
 32L20;
 14F17
 EPrint:
 Preliminary version and comments are welcome