Stability of nonnegative isotropic curvature under continuous deformations of the metric

TL;DR

This paper proves that nonnegative isotropic curvature remains stable under continuous metric deformations on compact manifolds, extending to other curvature bounds like non-negative curvature operator.

Contribution

It introduces a method to show stability of curvature bounds under C0 convergence of metrics, generalizing previous results to isotropic curvature and other curvature conditions.

Findings

Nonnegative isotropic curvature is preserved under C0 limits.

The method applies to other curvature bounds such as non-negative curvature operator.

The proof adapts Bamler's approach for scalar curvature to isotropic curvature.

Abstract

Using a method introduced by R. Bamler to study the behavior of scalar curvature under continuous deformations of Riemannian metrics, we prove that if a sequence of smooth Riemannian metrics gi on a fixed compact manifold M has isotropic curvature bounded from below by a nonnegative function u, and if gi converge in C 0 norm to a smooth metric g, then g has isotropic curvature bounded from below by u. The proof also works for various other bounds from below on the curvature, such has non-negative curvature operator.