On an inequality of Andrews, De Lellis and Topping

TL;DR

This paper extends the De Lellis-Topping method to establish almost Schur type inequalities, providing quantitative bounds on curvature deviations and exploring conditions for equality cases in Riemannian geometry.

Contribution

It introduces new almost Schur inequalities involving higher mean curvature and the Riemannian curvature tensor, with precise quantitative estimates and analysis of equality cases.

Findings

Quantitative bounds on higher mean curvature deviations

New sharp inequalities involving Riemannian curvature tensor

Characterization of equality cases in curvature inequalities

Abstract

Using the method of De Lellis-Topping, we prove some almost Schur type results. For example, one of our results gives a quantitative measure of how close the higher mean curvature of a submanifold is to its average value. We also derive another sharp Andrews-De Lellis-Topping type inequality involving the Riemannian curvature tensor and discuss its equality case.