Mean Curvature Flow Of Submanifolds Research Articles

On the Mean Curvature Flow of Submanifolds in the Standard Gaussian Space

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

The blow-up of the conformal mean curvature flow

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, we also prove that the external conformal forced mean curvature flow of a compact submanifold in $\bbr^n$ with the same pinched condition as Andrews-Bakers will be convergent to a round point in finite time.

Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$

We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time. The latter behavior is only possible if the dimension is even. These results generalize previous works by Huisken and Baker on the mean curvature flow of submanifolds of the sphere.

Mean curvature flow and Riemannian submersions

We prove that a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion is that the submersion has minimal fibers. We then lift some results taken from the literature (i.e., Andrews and Baker in J Differ Geom 85:357–395, 2010; Baker in The mean curvature flow of submanifolds of high codimension, 2011; Huisken in J Differ Geom 20:237–266, 1984; Math Z 195:205–219, 1987; Pipoli and Sinestrari in Mean curvature flow of pinched submanifolds of \({\mathbb {CP}}^{n}\)) to create new examples of evolution by mean curvature flow. In particular we consider the evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and of the tangent sphere bundle equipped with the Sasaki metric.

Mean Curvature Flow of Submanifolds With Small Traceless Second Fundamental Form

Consider a family of smooth immersions F(; t) : Mn ! Rn+k of submanifolds in Rn+k moving by mean curvature ow @F @t = ~H, where ~H is the mean curvature vector for the evolving submanifold. We prove that for any n 2 and k 1, the ow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in nite time, and the corresponding normalized ow converges exponentially in the C1-topology, to an n-sphere in some subspace Rn+1 of Rn+k.

The mean curvature flow of submanifolds of high codimension

The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study of the mean curvature flow of submanifolds of higher codimension has only recently started to receive attention. The mean curvature flow of submanifolds is the main object of investigation in this thesis, and indeed, the central results we obtain can be considered as high codimension analogues of some early hypersurface theorems. The result of Huisken's 1984 paper roughly says that convex hypersurfaces evolve under the mean curvature flow to round points in finite time. Here we obtain the result that if the ratio of the length of the second fundamental form to the length of the mean curvature vector is bounded (by some explicit constant depending on dimension but not codimension), then the submanifold will evolve under the mean curvature flow to a round point in finite time. We investigate evolutions in flat and curved backgrounds, and explore the singular behaviour of the flows as the first singular time is approached.