The preservability of the curvature-adaptedness along the mean curvature flow

TL;DR

This paper studies whether the property of curvature-adaptedness, involving commutation of shape and normal Jacobi operators, is maintained during the mean curvature flow of hypersurfaces in symmetric spaces.

Contribution

It provides new insights into the preservation of curvature-adaptedness along the mean curvature flow in locally symmetric spaces.

Findings

Curvature-adaptedness is preserved under the mean curvature flow for certain hypersurfaces.

Conditions for preservation depend on the initial hypersurface's properties.

The results extend understanding of geometric evolution in symmetric spaces.

Abstract

In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.