I have the following question: What are the angle preservation properties of harmonic maps?
Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the second best thing to an isometry.
Harmonic maps are closely related to conformal maps, but in general they distort both angles and lengths. Is there any precise mathematical statement that characterizes how much worse the harmonic maps in terms of preserving angles? And why they still preserve them up to a certain extent?