An almost-Schur type lemma for symmetric $(2,0)$ tensors and applications

TL;DR

This paper extends the almost-Schur lemma to symmetric (2,0)-tensors, providing new inequalities and applications to curvature measures of hypersurfaces and manifolds, broadening the scope of geometric analysis tools.

Contribution

It generalizes the almost-Schur lemma to symmetric (2,0)-tensors and applies it to curvature problems in hypersurfaces and conformally flat manifolds.

Findings

Established new inequalities for symmetric (2,0)-tensors.

Applied results to rth mean curvatures of hypersurfaces.

Derived bounds for scalar curvatures in conformally flat manifolds.

Abstract

In our previous paper in \cite{C}, we generalized the almost-Schur lemma of De Lellis and Topping for closed manifolds with nonnegative Rcci curvature to any closed manifolds. In this paper, we generalize the above results to symmetric $(2,0)$-tensors and give the applications including $r$th mean curvatures of closed hypersurfaces in a space form and $k$ scalar curvatures for closed locally conformally flat manifolds.