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.
Area-minimizing submanifolds are not generically smooth
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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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math.DG 1years
2024 1verdicts
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Non-persistence of strongly isolated singularities, and geometric applications
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.