Area minimising hypersurfaces mod $p$ do not admit immersed branch points

TL;DR

This paper proves that area minimising hypersurfaces mod p cannot have immersed branch points and are smoothly immersed outside small singular sets, advancing understanding of their regularity and singularity structure.

Contribution

It establishes the non-existence of immersed branch points in area minimising hypersurfaces mod p and characterizes their smooth immersion outside small singular sets.

Findings

No immersed branch points in area minimising hypersurfaces mod p.

Hypersurfaces are smoothly immersed outside a set of Hausdorff dimension at most n-3.

Results hold under stationarity and stability assumptions without relying on minimising property.

Abstract

We show that area minimising hypersurfaces mod $p$ do not admit immersed branch points, namely branch points about which all classical singularities are immersed. Furthermore, we show that if an $n$-dimensional area minimising hypersurface mod $p$ is smoothly immersed outside a $\mathcal{H}^{n-1}$-null set, then it is in fact smoothly immersed outside a closed set of Hausdorff dimension at most $n-3$. These results are consequences of a more general analysis of immersed stable minimal hypersurfaces with a certain `alternating' orientation. Indeed, our proof does not rely on the minimising property other than through stationarity, stability, and the verification of simple structural properties of the hypersurface.