One More Thing

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Zehua Cheng. “Generic Metrics on $S^{n+1}$ Preclude Linearly Stable Singular Tangent Cones of Area-Minimizing Boundaries.”.

Background: In geometry, “area-minimizing hypersurfaces” (think of soap films) are perfectly smooth in low dimensions, but in dimensions $n \ge 7$, they inevitably develop sharp points or “singularities”. The local shape of a singularity is called a “tangent cone”.

Main Theorem: I proved for a generic Riemannian metric on $S^{n+1}$ (specifically, a residual subset in the $C^3$ topology), no area-minimizing boundary will ever have a singular tangent cone that is linearly stable.

The Proof Strategy: The proof relies on three main pillars:

• A Perturbation Theorem: Showing you can make an arbitrarily small, explicit tweak to the space’s metric to eliminate a stable tangent cone.
• Openness: Proving that the set of metrics lacking a prescribed stable cone type is mathematically “open”.
• Baire Category Argument: Intersecting these open sets to prove this phenomenon holds true for a dense, generic set of metrics.