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Let $\Sigma$ be a minimal hypersurface of a smooth Riemannian manifold $(M,g)$ with second fundamental form $h$. What can one say about the set $\{p\in\Sigma:h(p)=0\}$? Is each point isolated? (I feel that it shouldn't be true in general, perhaps except if $(M,g)$ is $\mathbb{R}^3$.)

The question is because of Schoen-Simon-Yau's paper "Curvature estimates for minimal hypersurfaces" (https://projecteuclid.org/euclid.acta/1485889848); in the middle of page 280 they say that a Simons inequality for $|A|\Delta|A|$ holds distributionally "even if $|A|$ vanishes at various points." This seems correct to me when interpreted as $|A|$ vanishing at isolated points. So if the answer to my above question is "no" then I don't see why the theorems in Schoen-Simon-Yau's paper hold for general stable minimal hypersurfaces, without a conditional assumption on the vanishing set of $h$.

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    $\begingroup$ You can cross an example with $\mathbb{R}$ to see that the vanishing of the second fundamental form can occur on along a line. One way to interpret their argument is to consider $\sqrt{|A|^2 + \epsilon^2}$ in place of $|A|$ and think about the limit as $\epsilon\to 0$. You can see this argument worked out in J. Simons paper Minimal Varieties... Lemma 6.1.7. There might be some discussion of this in Colding--Minicozzi's book but I dont have it in front of me. $\endgroup$
    – Otis Chodosh
    Commented Nov 29, 2020 at 18:13

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