The maximum principle for minimal varieties of arbitrary codimension

We prove that an $m$-dimensional minimal variety in a Riemannianmanifold cannot touch the boundary at a point where the sum of thesmallest $m$ principal curvatures is greater than 0. We prove astronger maximum principle in case the variety is a hypersurface.We also prove analogous results for varieties with bounded mean curvature.

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Published 1 January 2010