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

TL;DR

This paper investigates the behavior of mean curvature flow of submanifolds in Gaussian space, showing finite-time blow-up conditions and characterizing the flow's geometric evolution, including optimal lifespan bounds.

Contribution

It establishes finite-time blow-up criteria for submanifolds in Gaussian space and characterizes the flow's maximal existence interval with geometric conditions.

Findings

Submanifolds with position norm not equal to m blow up in finite time.

The maximal existence interval has an optimal upper bound.

Flow behavior is characterized by shrinking to origin or expanding to infinity.

Abstract

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ where $({\mathbb R}^{m+p},\ol g)$ is the standard Euclidean space and $x\in{\mathbb R}^{m+p}$ denotes the position vector. Note that, as a special Riemannian manifold, $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ 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.