Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

Abstract:

We prove an optimal relative isoperimetric inequality

for a 2-dimensional minimal surface in the n-dimensional space form of nonpositive constant curvature κ under the assumptions that lies in the exterior of a convex domain and contains a subset Γ which is contained in and along which meets perpendicularly and that is connected, or more generally radially-connected from a point in Γ. 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 need not be area minimizing or even stable.