What is the current status on bad tangent cones at isolated singularities?

Question

Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.

Question. (i) Are the tangent cones at the origin regular, that is $\operatorname{sing} \mathbf{C} = \{ 0 \}?$ (ii) If one cone in $\operatorname{VarTan}(\lvert M \rvert,0)$ is regular, is there a 'direct' way of showing that they must all be regular? (By 'direct' I would mean without going via uniqueness of tangent cones à la Lojasiewicz–Simon.)

I believe (i) used to be an open question, but I am curious about its current status or partial progress. The Schoen–Simon regularity theory implies that the cones are either regular or cylindrical of the form $\mathbf{C}^7 \times \mathbf{R}$.

Correction. Otis Chodosh pointed out in his answer that this is incorrect. The aforementioned regularity theory of Schoen–Simon shows that the singular set of every tangent cone $\mathbf{C} \in \operatorname{VarTan}(\lvert M \rvert,0)$ is one-dimensional, but it can consist of several rays, one for each singularity in the link of the cone. It's the tangent cones taken at these rays that are cylindrical, of the form described above.

Answer 1

(i) This used to be a wide open area, but recently there has been some progress: Gabor Székelyhidi has constructed an example of an isolated singularity with a cylindrical tangent cone here: https://arxiv.org/pdf/2107.14786.pdf . If you are willing to change the metric from the flat metric to some other Riemannian metric, Leon Simon has constructed examples with cylindrical tangent cones but wild singular set https://arxiv.org/pdf/2101.06401.pdf (it's not known if this can be done in an analytic background metric).

(ii) I don't think there is any (known) way to prove that if one tangent cone is regular then they all are (without proving uniqueness of tangent cones). The basic problem is that the closure of the set of regular minimizing (or stable, etc) cones could contain singular cones. So it's not clear how to rule out a singular point having a "nearly" singular tangent cone AND a fully singular tangent cone (besides proving that the nearly singular cone is actually unique).

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Finally, I will note that

The Schoen–Simon regularity theory implies that the cones are either regular or cylindrical of the form $\mathbf{C}^7\times \mathbf{R}$

is not correct. In general, a tangent cone to a stable (or minimizing) hypersurface in $\mathbf{R}^9$ is $\mathbf{C}(\Sigma)$ where $\Sigma\subset\mathbf{S}^8$ is minimal with isolated singularities (this could all occur with multiplicity in the stable case, but Schoen--Simon basically says that this is not an issue). There's no reason that the singularities of $\Sigma$ have to be perfectly aligned along a spine, like with the link of (Simons cone) $\times \mathbf{R}$.

Note that in the usual "dimension reduction" arguement, you say: if $M^8\subset \mathbf{R}^9$ is stable minimal with singularity at $0$, then either (i) the tangent cone is smooth or (ii) you can take an ITERATED tangent cone (tangent cone to the tangent cone) and find something of the form $\mathbf{C}^7\times \mathbf{R}$. Note that there exists $x_i \in M$ and $\lambda_i\to\infty$ so that $\lambda_i(M-x_i)$ converges to this iterated tangent cone (so it's a blowup limit) but there need not be any actual tangent cone of this form (a priori, I am not sure that such an example has actually been constructed, but it seems very likely to exist).