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Can a sequence of degree one maps on, say, the unit circle, converges to a constant map in $W^{1,2}$ norm?

If the answer is yes, would you please provide an explicit example?

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  • $\begingroup$ No, it cannot. For the same reason that a sequence of continuous maps of non-zero degree cannot converge uniformly to a constant map. $\endgroup$ Apr 11, 2019 at 22:08
  • $\begingroup$ Can you please elaborate a bit? Someone told me the answer is YES in fact, without giving me an example. $\endgroup$
    – JSCB
    Apr 11, 2019 at 23:15
  • $\begingroup$ I think they meant in higher dimensions where such a map need not be continuous. $\endgroup$
    – Deane Yang
    Apr 12, 2019 at 12:58
  • $\begingroup$ Re Deane: Oh, do you have an example for that? $\endgroup$
    – JSCB
    Apr 12, 2019 at 14:20

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Lift your map to the universal cover to obtain a function $f:R\to R$ satisfying $$f(x+1)=f(x)+1,$$ if the degree is $1$. If the Sobolev norm of $f-c$ is $<\epsilon$ then by Schwarz inequality $$(f(1)-f(0))^2\leq\int_0^1 (f'(t))^2dt<\epsilon,$$ which contradicts the first inequality if $\epsilon<1$.

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