Uniqueness of certain cylindrical tangent cones

Uniqueness of certain cylindrical tangent cones

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Gábor Szekelyhidi, University of Notre Dame

Zoom linkhttps://princeton.zoom.us/j/594605776

Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form $C\times \mathbb{R}$, then this tangent cone is unique for a large class of minimal cones $C$. One of the hypotheses in this result is that $C\times \mathbb{R}$ is integrable and this excludes the case when $C$ is the Simons cone over $S^3\times S^3$. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in the non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.