On an Inequality of Andrews, De Lellis, and Topping

Using the method of De Lellis–Topping (Calculus of Variations and Partial Differential Equations, pp. 1–8, 2012 ), 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.