Annals of Mathematics· 2026Q1
On a conjecture of Almgren: area-minimizing submanifolds with fractal singular sets
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- Q1SCImago
- 2026year
Short summary
Zhenhua Liu has constructed area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds, settling Almgren's conjecture.
AI-generated from the title and abstract; the full text is not read.
Key points
- Area-minimizing submanifolds with fractal singular sets have been constructed.
- This construction settles Almgren's conjecture, with the result being dimensionally sharp.
- The strata within the Almgren stratification of the singular sets can be arbitrarily prescribed.
- The findings are valid for integral currents, mod v currents, and stable stationary varifolds.
AI-generated from the title and abstract; the full text is not read.
Abstract
We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle a conjecture of Almgren, and our answer is sharp dimension-wise. 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.
The authors' abstract, as published at the source. Annals of Mathematics, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics