Jellett-Minkowski's formula revisited. Isoperimetric inequalities for submanifolds in an ambient manifold with bounded curvature

TL;DR

This paper extends the Jellett-Minkowski formula to submanifolds in manifolds with bounded curvature, enabling new isoperimetric inequalities and an Aleksandrov type theorem for specific model spaces.

Contribution

It generalizes the Jellett-Minkowski formula to a broader class of ambient manifolds with curvature bounds, facilitating new isoperimetric results and geometric characterizations.

Findings

Derived generalized Jellett-Minkowski formula for curved ambient spaces

Established lower bounds for isoperimetric quotients of submanifolds

Proved an Aleksandrov type theorem in spaces with decreasing radial curvature

Abstract

In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with strictly decreasing radial curvatures, an Aleksandrov type theorem is provided.