A rigidity result for $2$-dimensional $\lambda$-translators

1.   Introduction

A $\lambda$-translating soliton (or, simply, a $\lambda$-translator) to the mean curvature flow is an immersed hypersurface $x: M^{n} \rightarrow \mathbb{R}^{n+1}$ in $\mathbb{R}^{n+1}$ satisfying the equation

Here, and in what follows, $H$ is the mean curvature, $\nu$ is a non-zero constant unit vector and $e_{n+1}$ denotes the unit normal vector field.

A special case of (1.1) is when $\lambda = 0$. In such a case the immersion $x$ is called a translating soliton of the mean curvature flow, or, simply, a translator ^[22]. Translators play an important role in the study of mean curvature flow. On one hand, a translating soliton is a solution of the mean curvature flow that evolves purely by translations along the direction $T$. On the other hand, they arise as blow-up solutions for a mean convex flow under type II singularities ^[9,11]. For instance, Huisken and Sinestrari ^[10] proved that, under the condition of a type II singularity of an MCF (i.e., an MCF with a mean convex solution), there exists a blow-up solution which is a convex translating solution. As we know, translating solitons have been widely studied and various interesting results have been obtained in recent years. For more information about translating solitons, please refer to the literature (^{[1,4,6,7,12,16,8,17,18,19,23,25]}).

In 2018, López classified, in ^[14], all $\lambda$-translators in $\mathbb{R}^{3}$ that are invariant by a one-parameter group of translations and a one-parameter group of rotations. He also studied the shape of a compact $\lambda$-translator of $\mathbb{R}^{3}$ in terms of its boundary in ^[15]. Inspired by the work of Cheng and Wei ^[3], Li et al. ^[13] classified $2$-dimensional complete $\lambda$-translators in the Euclidean space $\mathbb{R}^{3}$ and the Minkowski space $\mathbb{R}^{3}_{1}$ with constant squared norm $S$ of the second fundamental form, which use the generalized maximum principle from ^[2]. Recently, Yang et al. ^[26] developed a new technique to study the rigidity of self-shrinkers. In this paper, we will use a similar method to study the classification theorem for $2$-dimensional complete $\lambda$-translators with the non-zero constant Gauss curvature. The specific conclusions are as follows:

Theorem 1.1. There are no complete $\lambda$-translators with the non-zero constant Gauss curvature in $\mathbb R^{3}$.

2.   Preliminaries

Let $x: M^{2} \rightarrow \mathbb{R}^{3}$ be an isometric immersion of a surface of the $3$-dimensional Euclidean space $\mathbb{R}^{3}$. Denote the Levi-Civita connections of $M^{2}$ and $\mathbb{R}^{3}$ by $\nabla$ and $D$, respectively. Around each point of $M^{2}$, we choose a local orthonormal frame field $\{e_{A}\}_{A = 1}^{3}$ in $\mathbb{R}^{3}$ with the dual coframe field $\{\omega^{A}\}_{A = 1}^{3}$ such that, restricted to $M^{2}$, $e_{1}, e_{2}$ are tangent on $M^{2}$ and $e_{3}$ is a normal vector field. The gauss and weingarten formulae are given, respectively, by

where $h$ and $A$ are the second fundamental form and the shape operator, respectively. As we know, the second fundamental form $h$ and the shape operator $A$ are related by

Let

be the mean curvature and the squared norm of the second fundamental form. For those points $p\in M^{2}$, near which we could take a principal frame $\{e_{1}, e_{2}\}$ such that

the mean curvature $H$, the squared norm of the second fundamental form $S$ and the Gauss curvature $K$ are given, respectively, by

For any fixed $i, j, k$, since $\langle e_{j}, e_{j}\rangle = 1$ and $\langle e_{i}, e_{j}\rangle = 0\ (i\neq j)$, we have

for some smooth functions $\Gamma^{k}_{ji}$ near $p$. Then,

It follows from $A(e_{k}, e_{k}) = \lambda_{k}$ and $A(e_{k}, e_{m}) = 0\ (k\neq m)$ that

and

Thus,

That is,

Since covariant differentiation is torsion free, by calculating the Lie bracket $[e_{1}, e_{2}]$, it can be obtained that

Suppose that the given hypersurface $x: M^{n}\rightarrow \mathbb{R}^{3}$ is a $\lambda$-translator with a translating vector $\nu$. For a tangent $C^{1}$-vector field $V$ on $M^{2}$, define a differential operator

where $\Delta$ and $\nabla$ denote the Laplacian and the gradient operator, respectively. Under the local orthonormal frame field $\{e_{1}, e_{2}\}$, from the definition (1.1) of the translator in $\mathbb{R}^{3}$, by a direct computation, we have the following basic formulas:

and

where $V = \nu^{\top}$, i.e., the tangent component of the translating vector $\nu$ when restricted to $M^{2}$.

3.   The proof of Theorem 1.1

In this section, by differentiating the eigenvalues of the shape operator on the set of umbilic points (cf. ^[5,20,21,24]), we are able to prove that the mean curvature $H$ and the principal curvature are each constant. First, we assume that $\lambda_{1}$ is not constant on an open set $\Theta\subset M^{2}$ which is composed of non-umbilic points. That is, $\lambda^{2}_{1}-K\neq 0$ on $\Theta$. Next, we will prove the following several important propositions.

Proposition 3.1. For the function $\lambda_{1}$ defined on $\Theta\subset M^{2}$, we have

Proof. For the convenience of calculation, assume that $a = \langle\nu, e_{1}\rangle$ and $b = \langle\nu, e_{2}\rangle$. Recall that $\lambda_{2} = \frac{K}{\lambda_{1}}$ and $H = \lambda_{1}+\frac{K}{\lambda_{1}}$; by the fifth and sixth equations of (2.5), we have

Thus,

We will use (2.3) to get

Substituting (3.5) and (3.6) into (2.5), we obtain (3.1)–(3.4). □

Proposition 3.2. There exists a point $p\in \Theta\subset M^{2}$ such that $e_{1}(\lambda_{1})\neq 0$ at $p$.

Proof. Assume that $e_{1}(\lambda_{1}) = 0$ on $\Theta$. Since $\lambda_{1}$ is not constant on $\Theta$, then there is a point $p\in\Theta$ such that $e_{2}(\lambda_{1})\neq0$.

It follows from (3.1) and (3.4) that

and

Differentiating (3.7) with respect to $e_{2}$ yields

Substituting (3.7) and (3.9) into (3.8), we know

It is obvious that $\lambda_{1}$ is a constant function on $\Theta$ which contradicts the fact that there is a point $p\in\Theta$ such that $e_{2}(\lambda_{1})\neq0$. □

Proposition 3.3. For the non-constant function $\lambda_{1}$ on $\Theta\subset M^{2}$, the following two differential equations hold

Proof. In this part, we will consider the differential problems on a neighborhood $U$ of $p$ such that $e_{1}(\lambda_{1})\neq 0$.

It follows from (3.1)–(3.4) and $\lambda^{2}_{1}-K\neq 0$ that

Differentiating (3.1) with respect to $e_{1}$ leads to

Substituting the first and third equations of (3.12) into (3.13) gives

Especially,

Differentiating (3.4) with respect to $e_{1}$ yields

Substituting the third equation of (3.12) into (3.14), we have

It follows from (2.4), (3.6) and the third equation of (3.12) that

Differentiating (3.3) with respect to $e_{2}$ leads to

Substituting the second and third equations of (3.12) into (3.17), we know that

Combining (3.16) with (3.18), we obtain

And, by (3.15), we get

Substituting the first and fourth equations of (3.12) into (3.19) gives

□

Proposition 3.4. For a $2$-dimensional complete $\lambda$-translator $x:M^{2}\rightarrow \mathbb{R}^{3}$ with the non-zero constant Gauss curvature $K$, the mean curvature $H$ is constant.

Proof. Since the Gauss curvature $K$ is a non-zero constant on $M^{2}$, we know that $\lambda_{1}\neq0$ and $\lambda_{2}\neq0$ on the whole of $M^{2}$. We hereby declare that $\lambda_{1}$ is a constant on $M^{2}$. Then, $\lambda_{2}$ and the mean curvature $H$ must be a constant. In fact, let us assume that there is an open subset $\Theta\subset M^{2}$ such that $\lambda_{1}$ is non-constant and $\lambda_{1}\neq \lambda_{2}$. This implies that $\lambda^{2}_{1}-K\neq0$. For the convenience of calculation, take $e_{1}(\lambda_{1}) = \lambda_{1}^{\prime}$, $e_{1}\cdot e_{1}(\lambda_{1}) = \lambda_{1}^{\prime\prime}$ and $e_{1}\cdot e_{1}\cdot e_{1}(\lambda_{1}) = \lambda_{1}^{\prime\prime\prime}$. Differentiating $(3.11)$ with respect to $e_{1}$, we obtain

It follows from $(3.10)$ and $(3.20)$ that

Making use of $(3.11)$ and $(3.21)$, we obtain

Thus,

since $\lambda_{1}^{\prime}\neq 0$.

Finally, using $(3.11)$ and $(3.22)$ again, it can be obtained that

It is obvious that $\lambda_{1}$ is a constant function. It is a contradiction. □

The proof of Theorem 1.1. It follows from Proposition 3.4 that the mean curvature $H$ is constant and each principal curvature is constant. Since the Gauss curvature $K$ is a non-zero constant, it follows from (1.1) and (2.6) that

So, the non-zero constant vector $\nu = \nu^{\top}$ is tangent to $x(M^{2})$ at each point of $M^{2}$. It is obvious that $x: M^{2}\to \mathbb{R}^{3}$ is, locally, a plane or a cylinder. This is impossible since the Gauss curvature $K$ is a non-zero constant. Theorem 1.1 is proved.

4.   Conclusions

One concern is that $h_{ij} = A(e_{i}, e_{j})$ might not be differentiable in a local eigen frame if some positive principal curvatures repeat. However, in this article, we mainly study the principal curvature eigenvalues of the second fundamental form in the locally open set $\Theta$ composed of non-umbilic points. let $\{e_{1}, e_{2}\}$ be the adapted moving frame around a point $p$ in $\Theta$. Then, for any eigenvalue $\lambda_{i} \; (i = 1, 2)$ of multiplicity one and at the point $p$, it follows that principal curvatures are differentiable. By differentiating the eigenvalues of the shape operator on the set of umbilic points, we have that the mean curvature $H$ and each principal curvature are constant.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

We would like to thank the anonymous referees for many valuable suggestions. The first author was supported by 2023 Teacher Education Curriculum Reform in Henan Province (Grant No. 2023-JSJYYB-023).

Conflict of interest

The authors declare no conflict of interest.