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A sharp counterexample on the regularity of \(\Phi\)-minimizing hypersurfaces. (English) Zbl 0696.49074

The regularity results for area-minimizing hypersurfaces (integral currents) are shown to fail for hypersurfaces minimizing the integrals of certain other elliptic integrands \(\Phi\). In particular, the cone over \({\mathbb{S}}^ 1\times {\mathbb{S}}^ 1\) in \(R^ 4\) is \(\Phi\)-minimizing. This example shows the sharpness of the results of R. Schoen, L. Simon and F. J. Almgren [Acta Math. 139, 217-265 (1978; Zbl 0386.49030)], which guarantee regularity up through \({\mathbb{R}}^ 3\).
Reviewer: F.Morgan

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting

Citations:

Zbl 0386.49030
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References:

[1] R. Schoen, L. Simon, and F. J. Almgren Jr., Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II, Acta Math. 139 (1977), no. 3-4, 217 – 265. · Zbl 0386.49030 · doi:10.1007/BF02392238
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