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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!

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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 $\mathbb{S}^3$.

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