The Maximum Principle for Minimal Varieties of Arbitrary Codimension

We prove that an m-dimensional minimal variety in a Riemannian manifold cannot touch the boundary at a point where the sum of the smallest m principal curvatures is greater than 0. We also prove an analogous result for varieties with bounded mean curvature. Let N be a smooth Riemannian manifold with boundary. In general, N need not be complete. Suppose X is a compactly supported C tangent vectorfield on N such that (1) X · νN ≥ 0 at all points of ∂N , where νN is the unit normal to ∂N that points into N . Then X generates a one-parameter family t ∈ [0,∞) 7→ φt of maps of N into itself such that φ0 is the identity map and such that d dt φt(·) = X(φt(·)). If V is a C submanifold of N with finite area, we let δV (X) denote the first variation of area of V with respect to X :