Relative isoperimetric inequality and linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality 2 pi Area (M) less than or equal to Length (partial derivative M - Gamma)(2) + kappa . Area (M)(2) for a 2-dimensional minimal surface M in the n-dimensional space form R-n(kappa) of nonpositive constant curvature kappa under the assumptions that M lies in the exterior of a convex domain K subset of R-n(kappa) and partial derivative M contains a subset Gamma which is contained in partial derivative K and along which M meets partial derivative X perpendicularly and that partial derivative M - Gamma is connected, or more generally radially-connected from a point in Gamma. Also we obtain an optimal version of linear isoperimetric inequalities for minimal submanifolds in a simply connected Riemannian manifolds with sectional curvatures bounded above by a nonpositive number. Moreover, we show the monotonicity property for the volume of a geodesic ball in such minimal submanifolds. We emphasize that in all the results of this paper minimal submanifolds M need not be area minimizing or even stable.