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Uniqueness of certain cylindrical tangent cones

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arxiv 2012.02065 v1 pith:GEVFBLC7 submitted 2020-12-03 math.DG

classification math.DG
keywords coneconescylindricaltangenttimesmathbfsimonsimons
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abstract

We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_S= C(S^3\times S^3)$. Previously Simon proved a uniqueness result for cylindrical tangent cones that applies to a large class of cones $C$, however not to the Simons cone. The main new difficulty is that the cylindrical cone $C_S\times \mathbf{R}$ is not integrable, and we need to develop a suitable replacement for Simon's infinite dimensional Lojasiewicz inequality in the setting of tangent cones with non-isolated singularities.

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  1. Generic regularity for minimizing hypersurfaces in dimension 11

    math.DG 2025-06 conditional novelty 8.0 of 10

    Area-minimizing hypersurfaces are generically smooth in ambient dimension 11 after a C-infinity-small perturbation of the boundary or metric, and in dimensions 12 and up the singular set has dimension at most n-10-epsilon_n.

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