I SOPERIMETRIC INEQUAL IT IES ON MIN IMAL SUBMANIFOLDS OF SPACE FORMS

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 nequality for U kkwk M ( D) k-I < Vol ( O D) k, where wk is the volume of the unit ball of R k. Also, we prove that if D is any domain on a minimal surface in S " (or H " respectively), then D satisfies an + isoperimetric inequality 27rA _< L 2 + A ~ (2~rA _< L ~- A 2 respectively). Moreover, we show that if U is a k-dimensional minimal submanifold of H", then (k- 1) VoZ(U) < _ Vol(OU). Let C be a simple closed curve in the plane, bounding tile domain D. Let the length of C be L and the area of D be A. Then tile classical isoperimetric inequality states that 47rA < L ~ with equality if and only if C is a circle. If D is a domain on the sphere, the sharp isoperimetric inequality for D was given by F. Bernstein [2]: 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