The blow-up of the conformal mean curvature flow

TL;DR

This paper introduces the conformal mean curvature flow for submanifolds in Euclidean space, proves finite-time blow-up of curvature, and demonstrates convergence to a round point under certain conditions.

Contribution

It establishes a blow-up theorem for the conformal mean curvature flow and applies it to prove convergence to a round point under pinched conditions.

Findings

Maximum of the second fundamental form tends to infinity in finite time.

Derived evolution formulas and inequalities for the flow.

Proved convergence to a round point under pinched conditions.

Abstract

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in $\bbr^n$, the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds in the Euclidean space, we also derive some more evolution formulas and inequalities which we believe to be useful in our further study of conformal mean curvature flow. Presently, these computations together with our main theorem are applied to provide a direct proof of a convergence theorem concluding that the external conformal forced mean curvature flow of a compact submanifold in $\bbr^n$ with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.