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$V$-HARMONIC MAPS AND THEIR APPLICATIONS
We establish a Schwarz lemma for 
$V$-harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.
We prove that the hypotheses in the Pigola–Rigoli–Setti version of the Omori–Yau maximum principle are logically equivalent to the assumption that the manifold carries a 
${C}^{2} $ proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori–Yau principle, formulated in terms of lower bounds for curvature.
