A Schwarz Lemma for -Harmonic Maps and Their Applications

Abstract

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.

A Schwarz–Pick lemma for minimal maps

“…However, such a good structure is not available in higher dimensions and codimensions. (b) There are various Schwarz-Pick type results for harmonic maps in the literature; see for instance [4,16,19]. On the other hand, a minimal map f between two Riemannian manifolds (M, g M ) and (N, g N ) becomes harmonic if we equip M with the graphical metric g = g M + f * g N .…”

A generalization of the Schwarz lemma for transversally harmonic maps

“…More specifically, he proved that every harmonic map with generalized dilatation by a constant β > 0 from a complete Riemannian manifold with Ricci curvature bounded below by a non-positive constant −k 1 to a Riemannian manifold with sectional curvature bounded above by a negative constant −k 2 is distance-decreasing up to the constant β 2 k 1 /k 2 . Recently, some Schwarz type lemmas have also established for generalized harmonic maps between Riemannian manifolds, see [CZ17], [CLQ22], etc.…”