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Mar 12, 2021 at 14:09 comment added SubGeo You're right. I thought it was evident that I had meant the global problem, since the problem is conformally invariant and even the stereographic projection would do the job in the sphere minus a point. I'll consider giving more details next time.
Mar 9, 2021 at 20:25 comment added Robert Bryant @SubGeo: If you knew all this information and had divulged some of it in your question, you might have avoided the 'close' votes that it provoked. I'm afraid that several people have read your question and mistakenly concluded that it was either trivial or already answered in another question.
Mar 9, 2021 at 18:12 comment added SubGeo Thanks for your careful answer. I was aware of the results you mentioned, including Novikov's. I was really looking for a global smooth solution on the 3-sphere.
Mar 9, 2021 at 18:00 history answered Robert Bryant CC BY-SA 4.0