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A strict maximum principle for area minimizing hypersurfaces. (English) Zbl 0625.53052
Suppose that \(M_ 1,M_ 2\) are smooth connected embedded minimal hypersurfaces in \(U\subset N\), where N is an \((n+1)\)-dimensional Riemannian manifold and U is open. The classical maximum principle then implies the following: If \((\bar M_ 1\sim M_ 1)\cup (\bar M_ 2\sim M_ 2)\subset \partial U\) and if \(M_ 1\) lies locally on one side of \(M_ 2\) in a neighbourhood of each common point, then either \(M_ 1=M_ 2\) or \(M_ 1\cap M_ 2=\emptyset\). If one weakens the hypothesis to \(H^{n-1}((\bar M_ 1\sim M_ j)\cap U)=0\), \(j=1,2\), then it still is true that either \(\bar M_ 1=\bar M_ 2\) or \(M_ 1\cap M_ 2=\emptyset\). The author shows that in the latter case we also have \(\bar M_ 1\cap \bar M_ 2\cap U=\emptyset\) provided \(M_ 1\) and \(M_ 2\) are area-minimizing. The theorem is formulated for mass-minimizing integer multiplicity currents. In this case automatically \(H^{n- 7+\alpha}((\bar M_ j\sim M_ j)\cap U)=0\), \(j=1,2\), for any \(\alpha >0\) (by the codimension one regularity theory).
The proof uses results of Bombieri and Giusti on the Harnack inequality for elliptic differential equations on minimal surfaces, as well as arguments from the codimension one regularity theory, in particular the Allard-De Giorgi theorem.
Reviewer: M.Grüter

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q20 Variational problems in a geometric measure-theoretic setting
49Q05 Minimal surfaces and optimization
35J60 Nonlinear elliptic equations
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