Convex hull properties of harmonic maps
In [Proc. Symp. Pure Math. 36, 147-151 (1980; Zbl 0455.58009)], \textit{S.-Y. Cheng} generalized the gradient estimate to harmonic maps from a manifold \(M\) with non-negative Ricci curvature to a Cartan-Hadamard manifold \(N\). In this case, the space of bounded harmonic functions \({\mathcal H}_0(M)\) is \(1\)-dimensional. The purpose of this paper is to study the image sets of harmonic maps \(u:M \to N\) from a manifold \(M\), whose space of bounded harmonic functions \({\mathcal H}_0(M)\) is finite dimensional, into a Cartan-Hadamard manifold \(N\). In fact, the manifolds \(M\) satisfying a mean value inequality \((\mathcal M)\) and a weak volume growth property \((\mathcal V)\) will have \(\dim{\mathcal H}_0(M)<{\infty}\) (see Section 3 for definitions).
scientific article; zbMATH DE number 1268527