On an extension of the H^{k} mean curvature flow

TL;DR

This paper extends the mean curvature flow theory to H^{k} mean curvature flow, establishing an extension theorem under new conditions and deriving related estimates, advancing understanding of geometric flow behaviors.

Contribution

It generalizes existing extension theorems to H^{k} mean curvature flow and introduces new estimates, addressing key analytical challenges.

Findings

Successful extension theorem for H^{k} mean curvature flow.

Development of a suitable Michael-Simon inequality for H^{k} flows.

Derivation of new estimates for the generalized mean curvature flow.

Abstract

In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H^{k} mean curvature flow, and to do a suitable Moser iteration process. These two problems are overcame by imposing some extra conditions which may be weakened or removed in our forthcoming paper [7]. On the other hand, we derive some estimates for the generalized mean curvature flow, which have their own interesting.