A strong maximum principle for singular minimal hypersurfaces (Q1921334)

A geometric version of the strong maximum principle for scalar elliptic equations states that two minimal hypersurfaces of a Riemannian manifold one lying entirely on one side of the other must either coincide or be disjoint. This fact is extended to singular stationary hypersurfaces whose singular set has \((n-2)\)-dimensional Hausdorff measure zero. Here, \(n\) denotes the dimension of the hypersurfaces and ``stationary'' means that the first variation of area is zero. If such a hypersurface is in addition orientable and stable, then it is shown that its tangent cones have multiplicity one. Finally, a so-called ``extrinsic Harnack inequality'' is proven: if two disjoint, codimension-one, stationary integer rectifiable varifolds are close somewhere, then they are close everywhere.