Area-minimizing submanifolds are not generically smooth

2022/06/16 by Zhenhua Liu, Liu, Zhenhua
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.08315

openalex publication_date 2022/06/16 · openalex created_date 2022/06/19 · openalex updated_date 2026/07/28

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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