Isoperimetric inequalities for submanifolds. Jellett–Minkowski’s formula revisited

Abstract

n this paper, we provide an extension to the Jellett–Minkowski formula for immersed submanifolds within ambient manifolds which possess a pole and radial curvatures bounded from above or below. Using this generalized Jellett–Minkowski formula allows us to focus on several isoperimetric problems. Specifically, it becomes possible to concentrate on lower bounds for the isoperimetric quotients of any pre-compact domain with a smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a rotationally symmetric model space with strictly decreasing radial curvatures, an Aleksandrov-type theorem is provided.