Talk:Harmonic map

I have disengaged the article "Harmonic map" from its redirection to "Harmonic function". As the new article shows, these two concepts are not just mere synonyms in different contexts, but harmonic maps form a mathematical field of investigation in its own right (harmonic functions being just one example among several others), and therefore deserves an article on its own. Mathanor 19:39, 23 February 2007 (UTC)

Despite the common word "harmonic", harmonic maps don't have much to do with harmonic analysis, so I've reverted Oleg's inclusion of the article into the "Harmonic analysis" category. Mathanor 08:10, 24 February 2007 (UTC)

Cool. I am not an expert, really. I just thought that was appropriate since harmonic functions are a particular case of harmonic maps, as you say yourself in the aritcle. Oleg Alexandrov (talk) 04:45, 1 March 2007 (UTC)

Some suggestions

First, the sentence "Harmonic maps are the least expanding maps in orthogonal directions" is quite mysterious by itself. Second, I think the marble-elastic analogy is a little confusing. Is there a reference for it? Jjauregui (talk) 01:22, 25 September 2008 (UTC)

Seiki Nishikawa. Variational Problems in Geometry, volume 205 of Translations of Mathematical Monographs. AMS, 2001. AnandJoshi USC (talk) 00:31, 6 March 2009 (UTC)

Unrelated, would an expert like to add mention of definitions for harmonic maps outside of the 'smooth' world of riemannian manifolds? I know they are used, but don't know enough of the original context to be able to say when they are 'the same' —Preceding unsigned comment added by 171.64.38.44 (talk) 23:49, 11 January 2009 (UTC)

References

Thank you to Bender235 (talk · contribs) for adding inline references. I should clarify the nature of the references I added, which I realize is a rather long list. I added them as a guide to future editors on what should be added to the article, as they are the most widely cited articles on harmonic maps. (In retrospect, perhaps I should have put them onto the talk page instead.)

I haven't yet found the time to put discussion of them into the article. If anyone else would like to do so, here is a brief guide through most of those references:

• Schoen-Uhlenbeck regularity theory. Schoen's article "Analytic aspects..." belongs with the two articles of Schoen and Uhlenbeck, as it is partly a survey on those papers. In it, he rederives the results of Eells-Sampson and Hamilton without using the heat flow. Giaquinta and Giusti's article is also a very close companion to Schoen-Uhlenbeck, using simpler methods to derive similar results. Later developments on the theme of Schoen-Uhlenbeck are the articles of Evans, Hélein, Bethuel. Simon's article is a renowned contribution which clarifies some of the nature of tangent harmonic maps, which are central to Schoen-Uhlenbeck's analysis.
• Gromov-Schoen coarse harmonic maps. Korevaar and Schoen's article develops the analytic methods of this paper, and it seems that they are usually cited together.
• Sacks-Uhlenbeck bubbling. Papers developing the bubbling theme in the heat flow setting are the two by Struwe, the article of Chen-Struwe, and of Ding-Tian. Schoen and Yau's 1979 article derives some of the same results as Sacks-Uhlenbeck by different methods.

I am not so familiar with Uhlenbeck's article "Harmonic maps into Lie groups", which I believe is strongly related to the physics literature. It is probably naturally paired with Dorfmeister-Pedit-Wu, which I am also not so familiar with.

I think each of the above topics should eventually have a section in the article, similar in depth to the current sections "The Bochner formula and rigidity" and "Eells and Sampson's theorem". Another major omission, which should eventually be fixed, is the lack of references to the books and surveys. Gumshoe2 (talk) 19:27, 25 August 2020 (UTC)