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Area-minimizing submanifolds are not generically smooth

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arxiv 2206.08315 v10 pith:PEI677QX submitted 2022-06-16 math.DG

Zhenhua Liu This is my paper · ORCID
classification math.DG
keywords area-minimizingsubmanifoldsbounddenotesdimensiongenericallylowersets
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abstract

We prove that area-minimizing submanifolds are not generically smooth, settling a conjecture of White that asks the generic smoothness of area-minimizing submanifolds. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is $\max\{d-5,d-c\},$ where $d$ denotes the dimension of the submanifold and $c$ denotes the codimension.

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  1. Non-persistence of strongly isolated singularities, and geometric applications

    math.DG 2024-11 accept novelty 8.0 of 10

    For generic metrics, stationary varifolds with only strongly isolated singularities either are smooth or have a more complicated singularity; in codimension one, only smooth or non-strongly-isolated objects persist.

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