Abstract
In this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere \({bf S}^{n}(\sqrt{2n})\) is the only complete embedded connected \(F\)-stable self-shrinker in \(\mathbf{R}^{n+k}\) with \(\mathbf{H}\ne 0\), polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in \(\mathbf{R}^4\) with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen–Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not \(F\)-stable.
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Acknowledgments
We wish to thank Professor Jiayu Li for interesting and helpful discussions.
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Appendices
Appendix A
In this appendix, we will prove the variations of normal vector field and mean curvature we need in Sect. 3. The proof is standard. When the variation vector \(\mathbf{V }\) is the mean curvature vector \(\mathbf{H }\), they are proved in [12]. We will follow the computations in [12].
We begin with fixing our notation. In a normal coordinate around some point in \(\Sigma \), the induced metric on \(\Sigma \) is given by
where \(\partial _i(i=1,\ldots ,n)\) are the partial derivatives with respect to the local coordinate. Here, \(\langle \cdot ,\cdot \rangle \) is the inner product of \(\mathbf{R }^{n+k}\).
We choose a local field of orthonormal frames \(e_1, \cdots , e_n, e_{n+1}, \cdots , e_{n+k}\) of \(\mathbf{R }^{n+k}\) along \(\Sigma _s\) such that \(e_1,\ldots , e_n\) are tangent vectors of \(\Sigma _s\) and \(e_{n+1},\ldots ,e_{n+k}\) are in the normal bundle over \(\Sigma _s\). From now on, we will agree on the following index ranges:
We can write
Let \(\mathbf{A }^{\alpha }=(h^{\alpha }_{ij})\), where \((h^{\alpha }_{ij})\) is a matrix. By the Weingarten equation, we have
where \(\bar{\nabla }\) is the Levi-Civita connection on \(\mathbf{R }^{n+k}\). Furthermore,
Suppose the variation vector filed is \(\mathbf{V }=V^{\alpha }e_{\alpha }\), i.e.,
satisfies
Then we have
Lemma 8.1
The induced metric satisfies
and
Proof
We prove it at a fixed point. We have
Here we have used the fact that
As at the fixed point \(p, g_{ij}(p)=\delta _{ij}\), we know that
\(\square \)
Lemma 8.2
Denote \(\langle \frac{\partial }{\partial s}e_{\alpha },e_{\beta } \rangle =\langle \bar{\nabla }_{\mathbf{V }}e_{\alpha },e_{\beta } \rangle =b^{\beta }_{\alpha }\), then \(b^{\beta }_{\alpha }=-b_{\beta }^{\alpha }\), and we have
Here, \(\nabla V^{\alpha }\) is the covariant differential for the induced metric on \(\Sigma _s\).
Proof
We have
\(\square \)
Lemma 8.3
The second fundamental form satisfies
Here, \(V^{\alpha }_{,ji}\) denotes the second covariant derivative for the connection on the normal bundle.
Proof
We compute at a fixed point \(p\in \Sigma \). We can choose a frame \(e_i\) so that \(\bar{\nabla }^T_{e_i}e_j(p)=0\), i.e., at \(p, \bar{\nabla }_{e_i}e_j=-h^{\beta }_{ij}e_{\beta }\). From
and the fact that \(\mathbf{R }^{n+k}\) is flat, we have
\(\square \)
Lemma 8.4
The mean curvature vector satisfies
Proof
By Lemma 8.1 and Lemma 8.3, we have
Combining with Lemma 8.2, we obtain
Note that
Thus we have
\(\square \)
Appendix B
In this appendix, we will give another two geometric identities satisfied on self-shrinkers with arbitrary dimension and codimension. These results generalized Theorem 5.2 and Lemma 10.8 of [8].
Suppose \(\Sigma ^n\subset \mathbf{R }^{n+k}\) is a self-shrinker. We choose a frame \(\{e_A\}_{A=1}^{n+k}\) on \(\mathbf{R }^{n+k}\) along \(\Sigma \) such that \(\{e_i\}_{i=1}^{n}\) are tangent to \(\Sigma \) and \(\{e_{\alpha }\}_{\alpha =n+1}^{n+p}\) are in the normal bundle. We will compute pointwise. So we will always choose the frame \(\{e_i\}_{i=1}^{n}\) such that \(\bar{\nabla }_{e_i}^Te_j(p)=0\), i.e., at \(p, \bar{\nabla }_{e_i}e_j=-h^{\alpha }_{ij}e_{\alpha }\).
Lemma 9.1
Let \(L\) be defined by (3.9). Suppose \(\mathbf{w }\in \mathbf{R }^{n+k}\) is a fixed vector. Then on a self-shrinker \(\Sigma ^n\) in \(\mathbf{R }^{n+k}\), we have
Proof
By definition,
where
Then computing at \(p\) using the above chosen frame, we have
and
Here, we have used (6.23) and Codazzi equation. Taking trace of (9.4) and using (6.25), we obtain
Putting (9.6) into (9.5), we obtain
By definition of the operator \(L\), this is equivalent to (6.24). \(\square \)
The following result needs “flat normal bundle” assumption on the self-shrinker.
Lemma 9.2
If we extend the operator \(L\) to tensors, then on a self-shrinker \(\Sigma ^n\) in \({{\varvec{R}}}^{n+k}\) with flat normal bundle, we have
Proof
We will show that
In general, we have the following Simons’ equality for the second fundamental form [12, 26]:
Combining with (9.1), we have
By Ricci equation,
The last equality follows from our assumption that the normal curvature is zero. Thus we obtain (9.9) from (9.11), This proves the lemma. \(\square \)
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Arezzo, C., Sun, J. Self-shrinkers for the mean curvature flow in arbitrary codimension. Math. Z. 274, 993–1027 (2013). https://doi.org/10.1007/s00209-012-1104-y
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DOI: https://doi.org/10.1007/s00209-012-1104-y