An Interpolating Curvature Condition Preserved By Ricci Flow

TL;DR

This paper introduces a new curvature condition interpolating between known invariants, proves a maximum principle for it, and establishes a rigidity result for manifolds with this curvature property under Ricci flow.

Contribution

It establishes an interpolating curvature invariance, derives a maximum principle for it, and proves a rigidity property for manifolds with this curvature condition under Ricci flow.

Findings

Interpolating curvature invariance between nonnegative and 2-non-negative curvature.

A strong maximum principle for $( ext{lambda}_1,  ext{lambda}_2)$-nonnegativity.

Rigidity of manifolds with $( ext{lambda}_1, ext{lambda}_2)$-nonnegative curvature operators.

Abstract

In this paper, we firstly establish an Interpolating curvature invariance between the well known nonnegative and 2-non-negative curvature invariant along the Ricci flow. Then a related strong maximum principle for the $(\lambda_1, \lambda_2)$-nonnegativity is also derived along Ricci flow. Based on these, finally we obtain a rigidity property of manifolds with $(\lambda_1,\lambda_2)$-nonnegative curvature operators.