Let $(M, g)$ be a complete, noncompact Riemannian $n$-dimensional manifold and let
$\phi \colon M \to \mathbb S^n$ be an harmonic map, where $\mathbb S^n$ is the euclidean $n$-dimensional sphere. 


What can we say about the image of $\phi$? Of course in general is not an open subset (constant maps are harmonic). But is it at least an open subset of a submanifold of $\mathbb{S}^n$? 


Is there a book with a gentle introduction to harmonic maps, harmonic map flow, and their basic properties? 


Any help will be highly apreciated! 



Properties of harmonic maps into spheres
Two very good introductions to the subject are:


1) Harmonic maps, conservation laws, and moving frames by Frédéric Hélein


2) Analysis Of Harmonic Maps And Their Heat Flows by Changyou Wang, Fanghua Lin


I think the answer to your question is probably no, since a special case is given by minimal surfaces and there exists some none proper examples, see <https://en.wikipedia.org/wiki/Nadirashvili_surface>, it is in $\mathbb{R}^3$ but it can be probably transpose to $\mathbb{S}^3$.


