The blow-up of the conformal mean curvature flow

Abstract

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space R 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 R 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 R n with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.2000 Mathematics Subject Classification. Primary 53A30; Secondary 53B25. Key words and phrases. conformal mean curvature flow, conformal external force, blow-up of the curvature, round point.

On the mean curvature flow of submanifolds in the standard Gaussian space $^†$

“…To end the introduction, we shall brief the organization of this paper. In Section 2, by using the well-known trick of DeTurck, we reiterate the process similar to what we did in [20], showing the shorttime existence and uniqueness of the solution to the curvature flow (1.4) (see Theorem 2.2 in section 2). Section 3 contains only some direct computations that give us the necessary evolution equations for a number of basic quantities.…”

“…On the other hand, for any S ∈ Γ(⊗ r H * ⊗ N ), we have the following formula of commuting the Laplacian and gradient ( [20]):…”

“…Now come back to the curvature flow (1.3). We remark that the flow (1.3) turns out to be another special interesting case of the so-called "modified mean curvature flow with an external force " (see (1.1) in [20]). Besides the standard MCF and (1.3), this modified mean curvature flow with an external force also has some other important special cases that have been extensively studied by many authors.…”

“…Motivated by the major development of the classical mean curvature flow, we are very interested in and shall initiate the study of the mean curvature flow (MCF) of submanifolds in the Gaussian space (R m+p , e −|x| 2 /m g). Due to some of the idea proposed in our earlier study on the conformal MCF (see the statement in the introduction of [20]), instead of using the original mean curvature H directly, we would like to write analytically the MCF in (R m+p , e −|x| 2 /m g), using the "standard " mean curvature H, into the following form of initial problem of a degenerate parabolic partial differential equation:…”

“…Moreover, as done in [20], we always denote by ∇ g the Levi-Civita connection of any given Riemannian metric g on M m . Now arbitrarily fix a metric g and then define a metric-dependent vector field W ≡ W (g) on M by W (g) := tr g (∇ g − ∇ g ).…”