Global transversal stability of Euclidean planes under skew mean curvature flow evolutions

Abstract

In this paper, we prove that 2 dimensional transversal small perturbations of d-dimensional Euclidean planes under the skew mean curvature flow lead to global solutions which converge to the unperturbed planes in suitable norms. And we clarify the long time behaviors of the solutions in Sobolev spaces.

Appendix

The following is Hamilton’s interpolation inequality proved in [8, Section 12].

Lemma 6.1

Let T be any Tensor defined on manifold $\Sigma$. For $1\le j\le i-1$, there exists a constant C depending only on dimension of $\Sigma$ and i, which is independent of the metric g and connection such that

$$\begin{aligned} \int _{\Sigma } |\nabla ^{j} T|^{\frac{2i}{j}}d\mu \le C\max _{\Sigma }|T|^{2(\frac{i}{j}-1)}\int _{\Sigma } |\nabla ^{i} T|^{2}d\mu . \end{aligned}$$

(6.1)

The following is linear dispersive estimates.

Lemma 6.2

Let $1\le q\le 2$, and $f\in L^{q}({\mathbb {R}}^d)$. Then there exists a constant $C>0$ depending only on q, d such that

$$\begin{aligned} \Vert e^{i\Delta t}f\Vert _{L^{q'}_x}\le C t^{\frac{d}{2}(1-\frac{2}{q})}\Vert f\Vert _{L^q_x}, \end{aligned}$$

(6.2)

where $q'=\frac{q}{q-1}$.