Let $ f:M \rightarrow N $ be a minimal immersion (of arbitrary codimension or an hypersurface if it is necessary) and let $ |A| $ be the norm of its second fundametal form.If $ A $ is not identically zero is it true that the zeros of $ |A| $ are isolated?
If $ f:M \rightarrow R^3 $ is a minimal immersion then the conjecture above is true. Briefly this case follows from the fact that $ f $ can be locally represented as a conformal minimal immersion $ X: \Omega \subset R^2 \rightarrow R^3 $ and for conformal minimal immersions the statement above is true (see Osserman 'Minimal surfaces')
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