Xingxiao Li

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb{R}}^{m+p},e^{-|x{|}^2/m}\text{ol}g)$ where $({\mathbb{R}}^{m+p},\text{ol}g)$ is the standard Euclidean space and $x\in {\mathbb{R}}^{m+p}$ denotes the position vector. Note that, as a special Riemannian manifold, $({\mathbb{R}}^{m+p},e^{-|x{|}^2/m}\text{ol}g)$ has an unbounded curvature. Up to a family of diffeomorphisms on $M^m$, the mean curvature flow we considered here turns out to be equivalent to a special variation of the "{\em conformal mean curvature flow}\," which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to $m$, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval $[0,T)$ of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance.

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space ${\text{bbr}}^n$. 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 ${\text{bbr}}^n$, 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 ${\text{bbr}}^n$ with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.

$\xi$-submanifold in the Euclidean space ${\text{bbr}}^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in ${\text{bbr}}^{m+p}$. It is also a generalization of the $\lambda$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $\xi$-submanifolds are established. First, it is shown that a submanifold in ${\text{bbr}}^{m+p}$ is a $\xi$-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $({\text{bbr}}^{m+p},e^{-\text{fr}|x{|}^{2}m}\text{lagl}\cdot ,\cdot \text{ragl})$; Then, two weighted volume functionals $V_{\xi }$ and ${\bar{V}}_{\xi }$ are introduced and it is proved that $\xi$-submanifolds can be characterized as the critical points of these two functionals; Also, the corresponding second variation formulas are computed and the ($W$-)stability properties for $\xi$-submanifolds are systematically studied. In particular, it is proved that $m$-planes are the only properly immersed, complete $W$-stable $\xi$-submanifolds with flat normal bundle under a technical condition. It would be interesting if this additional restriction could be removed.