For a domain U on a certain k-dimensional minimal submanifold of S" or H", we introduce a "modified volume" M(U) of U and obtain an optimal isoperimetric inequality for 47rA <_ L ~ + KL 2,where K is the Gauss curvature of the sphere. In fact, in this form it is valid both for the sphere and for the plane, where K = 0. One might guess that it would hold equally for the hyperbolic plane H ~, where K -= -1. Schmidt [9] showed that this turns out to be the case.

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We prove that the area of a hypersurface which traps a given volume outside a convex domain C in Euclidean space R n is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when C has smooth boundary ∂C, we show that equality holds if and only if is a hemisphere which meets ∂C orthogonally.

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Let λ 1 be the first nontrivial eigenvalue of the Laplacian on a compact surface without boundary. We show that λ 1 = 2 on compact embedded minimal surfaces in S 3 which are invariant under a finite group of reflections and whose fundamental piece is simply connected and has less than six edges. In particular λ 1 =2 on compact embedded minimal surfaces in S 3 that are constructed by Lawson and by Karcher-Pinkall-Sterling.