Mean Curvature Driven Ricci Flow

TL;DR

This paper derives evolution equations for curvature tensors under mean curvature flow, revealing that scalar curvature evolution resembles Ricci flow but preserves negative curvature, applicable in all dimensions.

Contribution

It provides the first derivation of evolution equations for Riemann, Ricci, and scalar curvatures under mean curvature flow in any dimension.

Findings

Scalar curvature evolution is similar to Ricci flow but preserves negative curvature.

Results are valid in any dimension.

Negative curvature is preserved during the flow.

Abstract

We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.