Uniqueness of some cylindrical tangent cones (Q1903135)

It is known [the author, Ann. Math., II. Ser. 118, 525-571 (1983; Zbl 0549.35071)] that if a minimal submanifold \(M\) has a tangent cone \(\mathbb{C}\) at a singular point \(p\), then \(\mathbb{C}\) is the unique tangent cone of \(M\) at \(p\), and \(M\) approaches \(\mathbb{C}\) asymptotically in the appropriate smooth sense, provided that \(\text{sing } \mathbb{C}= \{0\}\) and \(\mathbb{C}\) has multiplicity 1. It has remained an open question whether or not this carries over to the case when \(\text{sing } \mathbb{C}\) consists of more than one point. Here we settle the question (in the affirmative) in certain of the cases when \(\mathbb{C}\) is an \(n\)-dimensional cylinder in \(\mathbb{R}^{n+1}\) with \((n-1)\)-dimensional cross section \(\mathbb{C}_0 \subset \mathbb{R}^n\) satisfying the same conditions as those mentioned above for \(\mathbb{C}\) (i.e., we treat certain cases when \(\mathbb{C}\) has the form \(\mathbb{C}= \mathbb{C}_0 \times \mathbb{R}\), with \(\text{sing } \mathbb{C}_0= \{0\}\) and multiplicity of \(\mathbb{C}_0=1\) at each point \(x\in \mathbb{C}_0 \smallsetminus \{0\}\)). In a suitable class of minimal hypersurfaces, we give conditions on the cross-section \(\mathbb{C}_0\) of \(\mathbb{C}\) which are sufficient to guarantee that \(\mathbb{C}\) is the unique tangent cone whenever it arises as a multiplicity one tangent cone at all.