On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities

Zhenhua Liu

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arXiv · 2110.13137

On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities

Zhenhua Liu

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Abstract

We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle a conjecture by Almgren and our answer is sharp dimensionwise. Furthermore, we can prescribe arbitrarily the strata in the Almgren stratification of the singular sets of our area-minimizing submanifolds, and our results hold in the category of integral currents, mod v currents and stable stationary varifolds.

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Zhenhua Liu. 2021-10-25. On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities. https://arxiv.org/abs/2110.13137

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