The barrier principle for minimal submanifolds of arbitrary codimension
The barrier principle for minimal hypersurfaces is a very useful tool in minimal surface theory. In this paper a corresponding result for minimal submanifolds of any codimension is proved. Recall that a hypersurface in a Riemannian manifold is \(k\)-mean convex if the sum of the \(k\) smallest principal curvatures is nonnegative. The main result of the paper is the following: Let \(M\) be a \(k\)-dimensional minimal submanifold of a Riemannian manifold \(N\), and \(B\subset N\) a \(k\)-mean convex hypersurface with unit normal \(\nu\). Assume that \(M\) and \(B\) are tangent at \(p\) and that \(M\) lies locally near \(p\) in the side of \(B\) where \(\nu\) points. Then \(M\subset B\). The authors construct topologically nontrivial domains in \(\mathbb{R}^n\) with \(2\)-mean convex boundaries and use their main result to prove a nonexistence condition for minimal surfaces with boundary in two parallel hyperplanes of \(\mathbb{R}^n\).
scientific article; zbMATH DE number 1973349