A characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_{1}$

In literature, the Heisenberg group and its sub-Laplacian are active in many fields of analysis and sub-Riemannian geometry, control theory, semiclassical analysis of quantum mechanics, etc. (cf. [18, 19, 20, 21, 22]). It also has applications in signal analysis and geometric optics [31, 32, 33]. Research on the sub-Riemannian geometry and its analytical consequences, in particular geodesics, has been studied widely and extensively in the past ten years (cf. [19, 23, 24, 25, 26, 27, 29, 28, 30]). In this paper, the Heisenberg group is studied as a pseudo-hermitian manifold. Like Euclidean geometry, it is a branch of Klein geometries, and the corresponding Cartan geometry is pseudo-hermitian geometry.

Recall that the Heisenberg group $H_{1}$ is the space $\mathbb{R}^{3}$ associated with the group multiplication

which is a $3$-dimensional Lie group. The space of all left invariant vector fields is spanned by the following three vector fields:

The standard contact bundle on $H_{1}$ is the subbundle $\xi$ of the tangent bundle $TH_{1}$ which is spanned by $\mathring{e}_{1}$ and $\mathring{e}_{2}$. It is also defined to be the kernel of the contact form

The CR structure on $H_{1}$ is the endomorphism $J:\xi\to\xi$ defined by

One can view $H_{1}$ as a pseudo-hermitian manifold with $(J,\Theta)$ as the standard pseudo-hermitian structure. There is a natural associated connection $\nabla$ if we regard all these left invariant vector fields $\mathring{e}_{1},\mathring{e}_{2}$ and $T$ as parallel vector fields. A naturally associated metric on $H_{1}$ is the adapted metric $g_{\Theta}$, which is defined by $g_{\Theta}=d\Theta(\cdot,J\cdot)+\Theta^{2}$. It is equivalent to define the metric by regarding $\mathring{e}_{1},\mathring{e}_{2}$ and $T$ as an orthonormal frame field. We sometimes use $<\cdot,\cdot>$ to denote the adapted metric. In this paper, we use the adapted metric to measure the lengths and angles of vectors, and so on.

A pseudo-hermitian transformation (or a Heisenberg rigid motion) in $H_{1}$ is a diffeomorphism in $H_{1}$ which preserves the standard pseudo-hermitian structure $(J,\Theta)$. We let $PSH(1)$ be the group of Heisenberg rigid motions, that is, the group of all pseudo-hermitian transformations in $H_{1}$. For details of this group, we refer readers to [4, 6], in which the fundamental theorem in the Heisenberg groups has been studied. We say that two surfaces are congruent if they differ by an action of a Heisenberg rigid motion.

The concept of minimal surfaces or constant mean curvature surfaces plays an important role in differential geometry to study the basic properties of manifolds. There is an analogous concept in pseudo-hermitian manifolds, which are called $p$-minimal surfaces. In this paper, we focus on studying such kinds of surfaces in the Heisenberg group $H_{1}$. Throughout this article, all objects we discuss are assumed to be $C^{\infty}$ smooth, unless we specify otherwise. Suppose $\Sigma$ is a surface in the Heisenberg group $H_{1}$. There is a one-form $I$ on $\Sigma$ which is induced from the adapted metric $g_{\Theta}$. This induced metric is defined on the whole surface $\Sigma$ and is called the first fundamental form of $\Sigma$. The intersection $T\Sigma\cap\xi$ is integrated to be a singular foliation on $\Sigma$ called the characteristic foliation. Each leaf is called a characteristic curve. A point $p\in\Sigma$ is called a singular point if at which the tangent plane $T_{p}\Sigma$ coincides with the contact plane $\xi_{p}$; otherwise, $p$ is called a regular (or non-singular) point. Generically, a point $p\in\Sigma$ is a regular point, and the set of all regular points is called the regular part of $\Sigma$. On the regular part, we can choose a unit vector field $e_{1}$ such that $e_{1}$ defines the characteristic foliation. The vector $e_{1}$ is determined up to a sign. Let $e_{2}=Je_{1}$. Then $\{e_{1},e_{2}\}$ forms an orthonormal frame field of the contact bundle $\xi$. We usually call the vector field $e_{2}$ a horizontal vector field. Then the $p$-mean curvature $H$ of the surface $\Sigma$ is defined by

The $p$-mean curvature $H$ is only defined on the regular part of $\Sigma$. There are two more equivalent ways to define the $p$-mean curvature from the point of view of variation and a level surface (see [3, 34]). We remark that this notion of mean curvature was proposed by J.-H. Cheng, J.-F. Hwang, A. Malchiodi, and P. Yang from the geometric point of view to generalize the one introduced first by S. Pauls in $H_{1}$ for graphs over the $xy$-plane [35]. Also, in [36], M. Ritoré and C. Rosales exposed another method to compute the mean curvature of a hypersurface. If $H=0$ on the whole regular part, we call the surface a $p$-minimal surface. The $p$-mean curvature is the line curvature of a characteristic curve, and hence the characteristic curves are straight lines (for the detail, see [3]). There also exists a function $\alpha$ defined on the regular part such that $\alpha e_{2}+T$ is tangent to the surface $\Sigma$. We call this function the $\alpha$-function of $\Sigma$. It is uniquely determined up to a sign, which depends on the choice of the characteristic direction $e_{1}$. Define $\hat{e}_{1}=e_{1}$ and $\hat{e}_{2}=\frac{\alpha e_{2}+T}{\sqrt{1+\alpha^{2}}}$, then $\{\hat{e}_{1},\hat{e}_{2}\}$ forms an orthonormal frame field of the tangent bundle $T\Sigma$. Notice that $\hat{e}_{2}$ is uniquely determined and independent of the choice of the characteristic direction $e_{1}$. In [6], it was shown that these four invariants,

form a complete set of invariants for surfaces in $H_{1}$. We remark that all the results provided in [6] still hold in the $C^{2}$-category. For each regular point $p$, we can choose suitable coordinates around $p$ to study the local geometry of such surfaces. There always exists a coordinate system $(x,y)$ of $p$ such that

We call such coordinates $(x,y)$ a compatible coordinate system. It is determined up to a transformation in (2.19). Notice that the compatible coordinate systems are dependent on the characteristic direction $e_{1}$.

Let $\Sigma\subset H_{1}$ be a constant $p$-mean curvature surface with $H=c$. In terms of a compatible coordinate system $(U;x,y)$, the $\alpha$-function satisfies the following equation

which first appeared as a Codazzi-like equation $(1.12)$ in [4] with $D=-1/{\alpha}$.

It is a nonlinear ordinary differential equation and is an example of the so-called Liénard equations, named after the French physicist Alfred-Marie Liénard. The Liénard equations were intensely studied as they can be used to model oscillating circuits. Conversely, in this paper, we show that if there exists a solution $\alpha(x,y)$ to the Liénard equation (1.2), we are able to construct a constant $p$-mean curvature surface with $H=c$ and this given solution $\alpha$ as its $\alpha$-function. This result is summarized as Theorem 1.1. One motivation of this theorem comes from the famous Sine-Gordon Equation (SGE),

which is considerably older than the Korteweg de Vries Equation (KdV). It was discovered in the late eighteenth century in the study of pseudospherical surfaces, that is, surfaces of Gaussian curvature $K=-1$ immersed in $\operatorname{\mathbb{R}}^{3}$, and it was intensively studied for this reason. It arises from the Gauss-Codazzi equations for pseudospherical surfaces in $\operatorname{\mathbb{R}}^{3}$ and is known as an integrable equation [13]. In addition, it can also be viewed as a continuum limit [14]. Its solutions and solitons have been widely discussed by the Inverse Scattering Transform and other approaches.

There is a bijective relation between solutions $u$ of the SGE with $\Im(u)\subset(0,\frac{\pi}{2})$ and the classes of pseudospherical surfaces in $\operatorname{\mathbb{R}}^{3}$ up to rigid motion. If $u:\operatorname{\mathbb{R}}^{2}\rightarrow\operatorname{\mathbb{R}}$ is a solution such that $\sin u\cos u$ is zero at a point $u_{0}$, then the immersed pseudoshperical surface has cusp singularities. For example, the pseudospherical surfaces corresponding to the 1-soliton solutions of SGE are the so-called Dini surfaces and have a helix of singularities.

The study of line congruences gives rise to the concept of Bäcklund transformations. A line congruence $L:M\rightarrow M^{*}$ is called a Bäcklund transformation with the constant angle $\theta$ between the normal to $M$ at $p$ and the normal to $M^{*}$ at $p^{*}=L(p)$ and the distance between $p$ and $p^{*}$ is $\sin\theta$ for all $p\in M$. The classical Bäcklund transformation for the Sine-Gordon equation was constructed in the nineteenth century by Swedish differential geometer Albert Bäcklund by means of a geometric construction [15, 16, 17]. We then are motivated to present analogous theorems for Heisenberg groups.

In this article, we sometimes call the Liénard equation (1.2) as the Codazzi-like equation from the geometrical point of view [4, 6]. We would also like to specify that a graph $(x,y,u(x,y))$ is $p$-minimal if it satisfies the $p$-minimal equation (see [3])

This is a degenerate hyperbolic and elliptic partial differential equation.

Theorem 1.1 is the fundamental theorem for surfaces in $H_{1}$. After we make a detailed investigation of the original version of the integrability conditions (2.1), (1.3) is more useful than the previous one in some sense (for the origin version, we refer readers to [6] or Theorem 2.1 of this paper). It also appeared as Theorem H in [4] with a slightly different formulation as the authors of [4] did not prescribe the metric.

Theorem 1.1 follows from our detailed study on the integrability condition (see (2.1)) of the fundamental theorem (Theorem 2.1) for surfaces in $H_{1}$. Actually, if we let $\hat{\omega}_{1}{}^{2}$ be the Levi-Civita connection associated to the induced metric with respect to the orthonormal frame field $\{\hat{e}_{1},\hat{e}_{2}\}$, as specified in Theorem 1.1, then (1.3) means that $\hat{\omega}_{1}{}^{2},\alpha$, and $H$ satisfy the integrability condition (2.1). This is equivalent to saying that $a,b,\alpha$ and $H$ satisfy the integrability condition (2.13) (see Subsection 2.2), which is another version of (2.1). We then have Theorem 1.1.

Given a function $\alpha(x,y)$ in a coordinate neighborhood $(U;x,y)\subset\operatorname{\mathbb{R}}^{2}$ which satisfies the Codazzi-like equation (1.2), we are able to construct a family of constant $p$-mean curvature surfaces. Therefore, it suggests that a good strategy is to investigate constant $p$-mean curvature surfaces by means of the Codazzi-like equation (1.2); in particular, $p$-minimal surfaces. In this paper, we will focus on the theory of $p$-minimal surfaces. Strategically, we first study the equation (1.2) with $c=0$, that is,

For nonlinear ordinary differential equations, it is known that it is rarely possible to find explicit solutions in closed form, even in power series. Fortunately, we indeed obtain a complete set of solutions to (1.5) in a simple form (see Section 4), stated as follows.

In Subsection 5.1, we are able to use the types of the solutions in Theorem 1.2 to divide the $p$-minimal surfaces into several classes, which are vertical, special type I, special type II and general type (see Definition 5.1 and 5.2). Each type of these $p$-minimal surfaces is open and contains no singular points. Generically, each $p$-minimal surface is a union of these types of surfaces, and ”type” can be shown to be invariant under an action of a Heisenberg rigid motion. Now for each type, whether it is special or general, if a function $\alpha$ is given, then Proposition 5.3, 5.4 and 5.5 express the formula for the induced metric $a,b$ (see (5.5), (5.6) and (5.7)), which is a representation of $I$, on the $p$-minimal surfaces with this given $\alpha$ as $\alpha$-function. From these formulae, we see that such constructed $p$-minimal surfaces depend upon two functions $k(y)$ and $h(y)$ for each given $\alpha$. Nonetheless, in Section 6, we proceed to normalize these invariants to the following normal forms in terms of an orthogonal coordinate system $(x,y)$, which is a coordinate system such that $a=0$. Such a coordinate system is determined up to a translation on $(x,y)$, thus we call it a normal coordinate system.

Therefore, $\zeta_{1}(y)$ constitutes a complete set of invariants for $p$-minimal surfaces of special type I (or of special type II), whereas both $\zeta_{1}(y)$ and $\zeta_{2}(y)$ constitute a complete set of invariants for $p$-minimal surfaces of general type. We hence give a representation for $p$-minimal surfaces (see Section 6). From Theorem 1.3, together with Theorem 1.1, the following version of the fundamental theorem holds immediately for $p$-minimal surfaces in $H_{1}$.

Analogs for constant $p$-mean curvature surfaces are also included in Subsection 5.2. We first derive the general form of $\alpha$ to the Codazzi-like equation (1.2) for $c\neq 0$ ( stated as Theorem 1.4), and Proposition 5.6 provides the formula for the induced metric $a$ and $b$.

Since each function of $\sin(x)$ and $\cos(x)$ differs by a sign if we replace the angle $x$ by $x+\frac{\pi}{2\lambda}$, we can assume, without loss of generality, that $c_{2}\geq 0$. We use $c_{2}$ to divide constant $p$-mean curvature surfaces into two classes. One class is those surfaces with $c_{2}>1$, and the other is that with $0\leq c_{2}\leq 1$. It is worth our attention that, for surfaces with $c_{2}>1$, the denominator of the formula for $\alpha$ is never zero. That means that the surfaces won’t extend to one with singular points. Moreover, if the surface is closed, it will be a closed constant $p$-mean curvature surface without singularity, which means this surface is of type of torus. It indicates that it is possible to find a Wente-type torus in the class of these surfaces. We can also normalize these invariants to the normal forms. However, Proposition 5.6 implies that the normalization for constant $p$-mean curvature surfaces needs to be modified since $a$ must be nonzero. Therefore, the transformation laws help us obtain the normal coordinates $(\tilde{x},\tilde{y})$ in Subsection 6.5 and hence we have the complete set of invariants $\{\zeta_{1}(y),\zeta_{2}(y)\}$ for constant $p$-mean curvature surfaces (see Theorem 6.3).

Due to Theorem 1.5, for each pair of function $\zeta_{1}(y)$ and $\zeta_{2}(y)$, we define in Subsection 6.3 eight maximal $p$-minimal surfaces in the sense specified in Theorem 1.6. Roughly speaking, it says that any connected $p$-minimal surface with type is a part of one of these eight classes. One notices that generically, a $p$-minimal surface is a union of those $p$-minimal surfaces with type.

As applications of this theory, in Section 8, we give a complete description of the structures of the singular sets of $p$-minimal surfaces in the Heisenberg group $H_{1}$.

The result in Theorem 1.7 is just a special case of Theorem 3.3 in [3]. However, we give a computable proof of this result for $p$-minimal surfaces. We also have the description of how a characteristic leaf goes through a singular curve, which is called a ”go through” theorem in [3].

Due to Theorem 1.8, we have the following result.

Theorem 1.9 plays a key point to enable us to recover the Bernstein-type theorem (see Section 8), which was first shown in the original paper [3] (or see [1, 7, 8]), and says that

for some constants $A,B,C\in\operatorname{\mathbb{R}}$, and

where $A,B$ are constants such that $A^{2}+B^{2}=1$ and $g\in C^{\infty}(\operatorname{\mathbb{R}})$, are the only two classes of entire smooth solutions to the $p$-minimal graph equation (1.4). In addition, in Section 7, we present some basic examples which, in particular, help us figure out the Bernstein-type theorem.

Finally, in Section 9, depending on a parametrized curve $\mathcal{C}(\theta)=(x(\theta),y(\theta),z(\theta))$ for $\theta\in\operatorname{\mathbb{R}}$, we deform the graph $u=0$ in some way to construct $p$-minimal surfaces with parametrization

for $r\in\operatorname{\mathbb{R}}$. It is easy to checked that $Y$ is an immersion if and only if either $\Theta(\mathcal{C}^{\prime}(\theta))-(y^{\prime}(\theta)\cos{\theta}-x^{\prime}(\theta)\sin{\theta})^{2}\neq 0\ \textrm{or}\ r+(y^{\prime}(\theta)\cos{\theta}-x^{\prime}(\theta)\sin{\theta})\neq 0$ for all $\theta$ (see Remark 9.2). In particular, the surface $Y$ defines a $p$-minimal surface of special type I if the curve $\mathcal{C}$ satisfies

for all $\theta$, or equivalently

In addition, the corresponding $\zeta_{1}$-invariant reads

where $\int\big{[}x^{\prime}(\theta)\cos{\theta}+y^{\prime}(\theta)\sin{\theta}\big{]}d\theta$ is an anti-derivative of the function $x^{\prime}(\theta)\cos{\theta}+y^{\prime}(\theta)\sin{\theta}$.

Similarly, the surface $Y$ defines a $p$-minimal surface of general type if the curve $\mathcal{C}$ satisfies

for all $\theta$. In addition, the corresponding $\zeta_{1}$- and $\zeta_{2}$-invariant read

From this construction, together with Theorem 7.8 which gives parametrizations for $p$-minimal surfaces of special type II, we conclude that we have generically provided a parametrization for any given $p$-minimal surface (see the argument in Section 9).

Acknowledgments. The first author’s research was supported in part by NCTS and in part by MOST 109-2115-M-007-004 -MY3. The second author’s research was supported in part by MOST 108-2115-M-032-008-MY2 and 110-2115-M-032-005-MY2.

In this section, we first review the fundamental theorem for surfaces in the Heisenberg group $H_{1}$ (Theorem 2.1). For the details, we refer the reader to [6]. Next, we give another version (Theorem 1.1) of this theorem in terms of compatible coordinate systems.