A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces
In this paper, the author proves two theorems (Theorem 1.1, Theorem 1.2) settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of allowable singular sets. The proofs of both theorems rely on the author's recent regularity and compactness theory for stable minimal hypersurfaces and on earlier work of \textit{T. Ilmanen} [Calc. Var. Partial Differ. Equ. 4, No. 5, 443--467 (1996; Zbl 0863.49030)], \textit{L. Simon} [J. Differ. Geom. 26, 327--335 (1987; Zbl 0625.53052)], \textit{B. Solomon} and \textit{B. White} [Indiana Univ. Math. J. 38, No. 3, 683--691 (1989; Zbl 0711.49059)]. The author shows that the theorem of Solomon and White and that of Ilmanen can be sharpened into a single strong maximum principle (Theorem 1.1) which says that the only a priori regularity hypothesis needed to conclude that two \(n\)-dimensional stationary hypersurfaces must coincide is that the singular set of one of them has \((n-1)\)-dimensional Hausdorff measure zero. This condition is sharp in the sense that a larger singular set cannot be allowed. In the first Theorem (Theorem 1.1) the author uses the notion of ``possibly singular minimal submenifolds'' which is that of stationary integral varifolds. The author also proves a unique continuation result (Theorem 1.2) for stationary codimension 1 integral varifolds.
scientific article; zbMATH DE number 6369376
scientific article