# Moving-centre monotonicity formulae for minimal submanifolds and related equations

## TL;DR

This paper introduces a new moving-centre monotonicity formula for minimal submanifolds, extending classical results and resolving a conjecture, with applications to various geometric PDEs.

## Contribution

It develops a sharp moving-centre monotonicity formula for minimal submanifolds, generalizing classical results and applying to related geometric equations.

## Key findings

- Proves a sharp moving-centre monotonicity formula for minimal submanifolds.
- Derives similar formulas for stationary p-harmonic maps, mean curvature flow, and harmonic map heat flow.
- Establishes a new area bound for minimal submanifolds passing through arbitrary points.

## Abstract

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem. In this article we prove a sharp `moving-centre' monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary $p$-harmonic maps, mean curvature flow and the harmonic map heat flow.