On a diameter Schwarz lemma and its applications

We consider a holomorphic map from a complex Euclidean ball into a complete simply connected Kähler manifold with nonpositive Riemannian sectional curvature. Assuming the diameter bound of the image of such a map, we show that the operator norm of the holomorphic tangent map is bounded at the origin. Here, we do not need to assume the curvature of the target manifold is bounded above by a negative constant. A similar result has been obtained by S.-Y. Cheng in the context of harmonic maps. Our complex geometric approach produces sharp bounds in some special cases. The main ingredient is a generalization of a diameter Schwarz lemma in a single complex variable (due to Landau–Toeplitz) to higher-dimensional Euclidean spaces.