On a conjecture of Almgren II: area-minimizing submanifolds with fractal singular sets on almost any manifold

Zhenhua Liu

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arXiv · 2512.24859

On a conjecture of Almgren II: area-minimizing submanifolds with fractal singular sets on almost any manifold

Zhenhua Liu

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Abstract

This paper is the second in a two-part solution to Almgren's conjecture on the existence of area-minimizing submanifolds with fractal singular sets. In part one, we construct area-minimizing submanifolds with fractal singular sets on certain special manifolds. Here we continue our work and show that area-minimizing submanifolds with fractal singular sets exist on almost any smooth manifold.

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Zhenhua Liu. 2025-12-31. On a conjecture of Almgren II: area-minimizing submanifolds with fractal singular sets on almost any manifold. https://arxiv.org/abs/2512.24859

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