Stefano Pigola & Marco Rigoli 
Vanishing and Finiteness Results in Geometric Analysis [PDF ebook] 
A Generalization of the Bochner Technique

Harmonic, pluriharmonic, holomorphic maps and basic Hermitian and Kählerian geometry.- Comparison Results.- Review of spectral theory.- Vanishing results.- A finite-dimensionality result.- Applications to harmonic maps.- Some topological applications.- Constancy of holomorphic maps and the structure of complete Kähler manifolds.- Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality.