On Cylindrical singularities of minimal hypersurfaces

Abstract

In a neighborhood of the origin in ${\mathbb{R}}^9$, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone $C_{4,3}$, the cone over $S^4(\sqrt{4/7})\times S^3(\sqrt{3/7})$, whereas at the origin its unique tangent cone is the cone $C_{3,3}\times \mathbb{R}$. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean ${\mathbb{R}}^9$ with exactly one singular point at the origin, at which its unique tangent cone is $C_{4,3}$, and whose unique tangent cone at infinity is $C_{3,3}\times \mathbb{R}$.

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