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Uniqueness of some cylindrical tangent cones. (English) Zbl 0853.57031
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.

##### MSC:
 57R70 Critical points and critical submanifolds in differential topology
##### Keywords:
minimal submanifold; tangent cone; singular point
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