Moving monotonicity formulae for minimal submanifolds in constant curvature

TL;DR

This paper introduces new monotonicity formulae for minimal submanifolds in space forms, leading to sharp area bounds and involving novel energy integrals over non-geodesic sets, extending previous Euclidean results.

Contribution

The paper presents new monotonicity formulae for minimal submanifolds in space forms, generalizing earlier Euclidean results and providing sharp area bounds via innovative energy integrals.

Findings

Derived monotonicity formulae for space forms.

Established sharp area bounds for minimal submanifolds.

Extended Euclidean moving-centre ball concepts to curved spaces.

Abstract

We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball. These monotonicity formulae involve an energy-like integral over sets which are, in general, not geodesic balls. In the Euclidean case, these sets reduce to the moving-centre balls introduced by the second author in [Zhu18].