Closed lagrangian self-shrinkers in $\R^{4}$ symmetric with respect to a hyperplane

A self-shrinker of the mean curvature flow is defined to be an immersed submanifold in the Euclidean space, $F:M^{n}\to\R^{n+k}$, which satisfies the quasilinear elliptic system

Here, $\vec{H}$ is the mean curvature vector given by the trace of the second fundamental form and $\perp$ means the projection onto the normal bundle of $M^{n}$.

It is well known that the blow-up limit at the Type I singularity of the mean curvature flow is a self-shrinker. Self-shrinkers themselves also provide self-similar solutions of the mean curvature flow that shrink to the origin. Furthermore, if we consider the Euclidean space as a weighted Riemannian manifold with the Gaussian density $e^{-\frac{|x|^{2}}{2}}$, the solution of (1.1) corresponds to a minimal submanifold in a weighted sense. For these reasons, it is important to understand the geometry of self-shrinkers.

Abresch and Langer [1] completely determined closed self-shrinking planar curves. These curves are called Abresch-Langer curves and note that they have reflection symmetry. Moreover, there is a positive constant associated with each curve so that if two Abresch-Langer curves have the same constants, then they are the same up to a rigid motion (see Lemma 2.1). However, unlike the case of curves, the situation becomes more complicated, so it is hard to expect a complete classification in higher dimensions.

On the other hand, the mean curvature flow preserves the Lagrangian condition in Kähler-Einstein manifolds, including the Euclidean space (see [11]). Since the mean curvature flow corresponds to the negative gradient flow for a volume functional, it provides a potential way to obtain a volume minimizer. For instance, special Lagrangian submanifolds are volume minimizing in a Calabi-Yau manifold, so the Lagrangian mean curvature flow plays a significant role in the construction of these examples. However, the development of finite-time singularities is the main difficulty of this method (see for instance [9]). Therefore, in order to manage possible singularities, it is necessary to study Lagrangian self-shrinkers.

In this paper, we consider Lagrangian self-shrinkers in $\R^{4}\simeq\mathbb{C}^{2}$. Since Lagrangian self-shrinking sphere cannot exist by the theorem of Smoczyk [12], the simplest example is the surface of genus 1. Many immersed Lagrangian self-shrinking tori were constructed: products of Abresch-Langer curves, Anciaux’s tori (see [2]), and Lee-Wang’s tori (see [6], [7]).

One interesting observation we would like to emphasize is that all known embedded examples of closed Lagrangian self-shrinkers in $\mathbb{C}^{2}$ become the Clifford torus. This is not true in $\mathbb{C}^{n}$($n\geq 3$) since two or more different embedded Lagrangian self-shrinkers can be found in Anciaux’s examples (see [2]). Moreover, as the normal and tangent bundles of a Lagrangian submanifold are diffeomorphic and the self-intersection number is given by the Euler characteristic of the normal bundle, one can conclude that embedded Lagrangian surfaces in $\mathbb{C}^{2}$ should have genus 1. Therefore, it is natural to ask whether the Clifford torus is the only embedded example in $\mathbb{C}^{2}$:

This question is analogous to Lawson’s conjecture that whether the Clifford torus is the only embedded minimal torus in $S^{3}$. The Lawson conjecture was proved by Brendle [3], by clever use of the maximum principle for two-point functions on a given surface. However, due to the increased codimension, it is not easy to apply the maximum principle in our case. But, assuming a particular symmetry condition, we obtain a positive answer for Question 1.1.

Before we state the main results, we recall some rigidity results on the Clifford torus as a Lagrangian self-shrinker. Castro and Lerma [4] characterized Lee-Wang’s tori as compact Hamiltonian stationary Lagrangian self-shrinkers in $\mathbb{C}^{2}$ and the Clifford torus was characterized as the only embedded example among them. In [5], they proved that if a compact Lagrangian self-shrinker satisfies either $|\vec{H}|^{2}=const$ or $|\vec{H}|^{2}\leq 2$ or $|\vec{H}|^{2}\geq 2$, then it is the Clifford torus. They also proved that a compact Lagrangian self-shrinker without change of sign on the Gauss curvature and satisfying $|\sigma|^{2}\leq 2$ is the Clifford torus. Here, $\sigma$ denotes the second fundamental form. Then Li and Wang [8] generalized previous results into two directions: the Clifford torus is the unique compact Lagrangian self-shrinker with $|\sigma|^{2}\leq 2$, and if a compact Lagrangian self-shrinker has no sign change on the Gauss curvature, then it is one of the products of Abresch-Langer curves.

Now we recall the result of Ros related to Lawson’s conjecture, which states that the Clifford torus is the unique embedded minimal torus in $S^{3}$ symmetric with respect to four pairwise orthogonal hyperplanes in $\R^{4}$ (see Theorem 6 in [10]). Since the almost complex structure does not commute with reflections in general, we could expect that there would be a non-trivial restriction on a Lagrangian submanifold if we assume the reflection symmetry. Motivated by this, we study Lagrangian self-shrinkers symmetric with respect to a hyperplane in $\R^{4}$.

Then we have the following theorem.

We remark that it requires only one hyperplane of symmetry in Theorem 1.2, while Ros’ theorem assumed four orthogonal hyperplanes of symmetry. For the proof of Theorem 1.2, we were not able to apply the method of Ros [10] due to the high codimension. Instead, we observe that each Lagrangian self-shrinker with reflection symmetry satisfies global relations which are similar to transcendental relations (2.4) on Abresch-Langer curves (see Proposition 4.3). By using such relations, we prove that $\Sigma^{2}$ is flat. Then, by the result of Li and Wang (see Proposition 2.2), we could finish the proof. We also show that global relations on a product of Abresch-Langer curves coincide with the transcendental relation on each curve.

Since the Clifford torus is the only embedded self-shrinker among the products of Abresch-Langer curves, we obtain a new geometric characterization of the Clifford torus as a Lagrangian self-shrinker:

The paper is organized as follows. In Section 2 we provide some basic definitions and show that a hyperplane of symmetry can be assumed to be the coordinate hyperplane $\{x_{1}=0\}$ in a way that preserves the Lagrangian condition. We also recall two previous results that play an important role in this paper. In the next section, we compute local equations for Lagrangian self-shrinkers with reflection symmetry in terms of isothermal coordinates. Then we obtain global relations in Section 4. Finally, we prove the main results in Section 5.

The author would like to express his gratitude to Jaigyoung Choe for helpful discussions and thoughtful encouragement. This work was supported in part by NRF-2018R1A2B6004262.

Identify $\R^{4}$ with $\mathbb{C}^{2}$ via $\left(x_{1},x_{2},x_{3},x_{4}\right)\leftrightarrow\left(x_{1}+ix_{2},x_{3}+ix_{4}\right)$, where $i=\sqrt{-1}$. We use the almost complex structure $J$ on $\R^{4}$ which corresponds to the multiplication of $i$ in $\mathbb{C}^{2}$. Let $\left\langle\ ,\ \right\rangle$ be the standard hermitian product in $\mathbb{C}^{2}$. Then the Euclidean inner product in $\R^{4}$ is given by the real part of $\left\langle\ ,\ \right\rangle$ under the identification.

The Kähler form $\omega$ on $\left(\R^{4},J\right)$ is given by

where $x_{i}$’s are coordinate functions of $\R^{4}$. An immersion $F:\Sigma^{2}\to\R^{4}$ is called a Lagrangian if

and we call $F:\Sigma^{2}\to\R^{4}$ a Lagrangian self-shrinker if it satisfies (1.1) and the Lagrangian condition (2.1).

For arbitrary $p\in\Sigma$, one can always find local isothermal coordinates $x$, $y$ in a neighborhood of $p$ such that

where the lower indices denote partial derivatives. In terms of the isothermal coordinates, the Lagrangian condition (2.1) is equivalent to

Now let $F:\Sigma^{2}\to\R^{4}$ be a Lagrangian self-shrinker symmetric with respect to a hyperplane $P$. First, we claim that $P$ contains the origin $O$. Indeed, by the reflection symmetry, $\Sigma$ and its reflected surface shrink to the same point along the mean curvature flow. As the surface shrinks to the origin, $O$ should be invariant under the reflection, which implies that $O\in P$.

Let $\nu\in\R^{4}\simeq\mathbb{C}^{2}$ be the unit normal vector of $P$. Then there exists a unitary matrix $G\in U(2)$ such that

The unitary group $U(2)$ can be identified as a subgroup of $O(4)$ consisting of elements which commute with $J$, so there is an orthogonal transformation $\tilde{G}\in O(4)$ corresponding to $G$. Then $\tilde{G}$ sends $P$ to the coordinate hyperplane $\{x_{1}=0\}$. Moreover, it preserves Lagrangian submanifolds since it commutes with $J$. Thus, $\tilde{G}\circ F:\Sigma^{2}\to\R^{4}$ is a Lagrangian self-shrinker symmetric to $\{x_{1}=0\}$. Therefore it suffices to consider Lagrangian self-shrinkers $F:\Sigma^{2}\to\R^{4}$ symmetric with respect to $\{x_{1}=0\}$. This family of self-shrinkers will be denoted by $\mathscr{F}$.

Lagrangian self-shrinkers with reflection symmetry can be easily found in the products of Abresch-Langer curves. For later use, we recall two results on Abresch-Langer curves and their product self-shrinkers. The first result is about the transcendental relation on those curves (see Theorem A in [1] and Lemma 5.3 in [13]).

The second one is about the characterization of products of Abresch-Langer curves as Lagrangian self-shrinkers with constant Gauss curvature in $\R^{4}$ (see Proposition 5.6 in [8]).

Throughout this paper, the gradient and Laplacian on a given surface will be denoted by $\nabla$ and $\Delta$, respectively.

Let $F:\Sigma^{2}\to\R^{4}\simeq\mathbb{C}^{2}$ be a closed Lagrangian self-shrinker symmetric with respect to $\{x_{1}=0\}$, i.e., $F\in\mathscr{F}$. Using isothermal coordinates, we obtain the following local equations:

In this section, we prove that Lagrangian self-shrinkers in $\mathscr{F}$ have a special property analogous to the transcendental relation on Abresch-Langer curves as in Lemma 2.1. From local equations in Proposition 3.1, we obtain the following lemma:

Next, we prove that the constant in the previous lemma cannot be zero.

Finally, we obtain the following proposition.

Next, we compute explicit constants on products of Abresch-Langer curves. All self-shrinkers in this family have reflection symmetry so that Proposition 4.3 can be applied. We observe that each constant coincides with the one that appeared in Lemma 2.1. The precise computation would be done as follows.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be the Abresch-Langer curves associated with the constants $c_{\Gamma_{1}}$ and $c_{\Gamma_{2}}$ as in Lemma 2.1. Suppose $\Gamma_{1}$ and $\Gamma_{2}$ are given by $\gamma_{1}(s)$ and $\gamma_{2}(t)$, respectively, where $s$ and $t$ are arc-length parametrizations. Consider the Lagrangian self-shrinker $\Gamma_{1}\times\Gamma_{2}\subset\R^{2}\times\R^{2}=\R^{4}$. If we write $\gamma_{1}(s)=r_{1}(s)e^{i\theta_{1}(s)}$, then

where the upper dot denotes the derivative with respect to $s$. As $\Gamma_{1}\times\Gamma_{2}\subset\R^{4}$ is given by the product, we have

so that

where we have used (4.6) in the last step. A direct computation gives

and then with (4.6), we obtain

We deduce from (4.7) and (4.8) that

and similarly $C_{2}=c_{\Gamma_{2}}^{2}$.

In this section, we prove the main results of this paper.