2.1 The phase plane

The phase plane is defined as the set $\Theta :=(0,\,\infty )\times (0,\,2\pi )-\mathcal {S}$, where

(2.5)\begin{equation} \mathcal{S}:=\left\lbrace(x,\theta):\ x>0,\ \theta\in(0,2\pi),\ x=\mathcal{S}(\theta):=\frac{-b\sin\theta}{a}\right\rbrace. \end{equation}

The orbits $\gamma (s)=(x(s),\,\theta (s))$ are the solutions of system (2.4). Next, we summarize some of the main properties of the phase plane that follow from its definition:

1. 1. We define $\Theta _1=\Theta \cap \{\theta <\pi \}$ and $\Theta _2=\Theta \cap \{\theta \in (\pi,\,2\pi )\}$. Since $x>0$, $\mathcal {S}$ is a compact arc in $\Theta _1$ (resp. in $\Theta _2$) joining the points $(0,\,0)$ and $(0,\,\pi )$ (resp. $(0,\,\pi )$ and $(0,\,2\pi )$) if and only if $ab<0$ (resp. if $ab>0$).

2. 2. The points lying on $\mathcal {S}$ are singular for system (2.4). Therefore, an orbit cannot have a finite endpoint at $\mathcal {S}$, although it can converge to a point in $\mathcal {S}$ at finite time. Any such limit point in $\mathcal {S}$ corresponds to a circle of singular points of the corresponding $\Phi$-surface.

3. 3. By solving $\theta '=0$ in (2.4), we deduce that the curve $\Gamma :=\Theta \cap \{x=\Gamma (\theta )\}$, where

   \[ x=\Gamma(\theta):=\frac{a\sin\theta}{\phi(\cos\theta)},\quad \theta\in(0,2\pi),\ \phi(\cos\theta)\neq 0, \]
   corresponds to points of the profile curve $\alpha$ with vanishing curvature. This curve exists in $\Theta$ wherever $a\sin \theta \phi (\cos \theta )>0$.

4. 4. If $\phi (0)\neq 0$, the point

   \[ e_0=\left(\frac{a}{\phi(0)},\frac{\pi}{2}\right),\ \mathrm{if}\ a\phi(0)>0,\hspace{.5cm} \left(\frac{-a}{\phi(0)},\frac{3\pi}{2}\right),\ \mathrm{if}\ a\phi(0)<0, \]
   called the equilibrium, is a constant orbit that trivially solves (2.4). The $\Phi$-surface generated by this orbit is the circular flat cylinder of vertical rulings and radius $|a/\phi (0)|$.

5. 5. The lines $\theta =\pi /2,\,3\pi /2$ and the curves $\Gamma,\,\mathcal {S}$ divide $\Theta$ into connected components where the coordinate functions of an orbit are monotonous.

Throughout this paper we will always assume that the prescribed function $\Phi \in C^{1}(\mathbb {S}^{2})$ is positive and even, that is $\Phi (X)=\Phi (-X)$, for every $X\in \mathbb {S}^{2}$. In particular, $\phi$ given by (2.1) in terms of $\Phi$ satisfies $\phi (y)=\phi (-y)$ for every $y\in [-1,\,1]$. The even condition has the following geometric property on the phase plane: if $(x(s),\,\theta (s))$ is a solution of (2.4), then $(x(-s),\,\pi -\theta (-s))$ is also a solution of (2.4) whenever $\theta \in (0,\,\pi )$. When $\theta \in (\pi,\,2\pi )$, it follows that $(x(-s),\,3\pi -\theta (-s))$ is a solution to (2.4). Geometrically, the subsets $\Theta _1$ and $\Theta _2$ of $\Theta$ are symmetric with respect to the lines $\theta =\pi /2,\,\ \theta =3\pi /2$, respectively.

2.2 The local character of the PDE

Locally, a surface satisfying (1.1) can be expressed as the graph of a function $u(x,\,y)$ that is a solution of the PDE (1.2), which can be written as the solution of a function $\mathfrak {F}(u_x,\,u_y,\,u_{xx},\,u_{yy},\,u_{xy})=0$. Note that the variable $u$ does not appear, since (1.2) is invariant under Euclidean translations. A straightforward computation shows that the discriminant of this PDE is

\[ \mathfrak{F}_{u_{xx}}\mathfrak{F}_{u_{yy}}-\frac{1}{4}\mathfrak{F}_{u_{xy}}^{2}=(1+u_x^{2}+u_y^{2})^{2}(a^{2}+b\Phi). \]

The character of this PDE inspires the following definition.

For instance, if $\Sigma$ is a $\Phi$-surface of elliptic type then the maximum principle for fully nonlinear PDEs applies. Consequently, classic techniques originally developed for surfaces of constant mean curvature and positive constant Gauss curvature can be adapted to this case, see e.g. [Reference Rosenberg and Sa Earp27]. The parabolic case leads to $\Phi$ being a constant function, and an explicit description of the rotational parabolic linear Weingarten surfaces can be found in [Reference Bueno and López5].

2.3 Existence of radial solutions

In this section we discuss the existence of radial solutions, $u=u(r)$, of (1.2) that intersect orthogonally the axis of rotation. This is equivalent to establish a classical solution of

(2.6)\begin{equation} \left\{\begin{array}{ll}a\left(\dfrac{u''}{(1+u'^{2})^{3/2}}+\dfrac{u'}{r\sqrt{1+u'^{2}}}\right)+b\dfrac{u''u'}{r(1+u'^{2})^{2}}=\phi\left(\dfrac{1}{\sqrt{1+u'^{2}}}\right), & \mbox{in } (0,\,\delta)\\ u'(0)=0. \end{array}\right. \end{equation}

Any solution of this initial value problem generates an orbit of system (2.4) having the point $(0,\,0)$ as endpoint. However, (2.6) is singular at $r=0$, as well as (2.4) at $(0,\,0)$, hence standard theory cannot be invoked in order to ensure the existence of such an orbit. Nevertheless, by the work of the first author and López, for every $a,\,b,\,\Phi$ such that the elliptic relation $a^{2}+b\Phi >0$ holds, there exists $\delta >0$ small enough and $u\in C^{2}([0,\,\delta ]),\,\ u>0,$ that solves (2.6). We give a hint of the method used to prove it, see theorem 4.1 in [Reference Bueno and López5] for further details.

Assume that $u=u(r)$ is a radial solution of (1.2). After multiplying by $r$ and dividing by $a$, the equation in (2.6) transforms into

\[ \left(\frac{ru'}{\sqrt{1+u'^{2}}}\right)'+\frac{b}{2a}\left(\frac{u'^{2}}{1+u'^{2}}\right)'=\frac{r}{a}\phi\left(\frac{1}{\sqrt{1+u'^{2}}}\right). \]

If we define the functions $f,\,g\colon \mathbb {R} \to \mathbb {R}$ by

\[ f(y)=\frac{y}{\sqrt{1+y^{2}}}, \quad g(y)=\frac{1}{a}\phi\left(\frac{1}{\sqrt{1+y^{2}}}\right), \]

and after integration, we get

\[ rf(u')+\frac{b}{2a}f(u')^{2}=\int_0^{r} tg(u'(t))\, dt. \]

After solving $f(u')$ and eventually $u'$ by taking $f^{-1}$ and integrating, we define the operator

\[ (\mathsf{T}u)(r)=\int_0^{r}f^{{-}1}\left(\frac{2a}{b}\left({-}s+\sqrt{s^{2}+\frac{b}{a}\int_0^{s}tg(u'(t))\, dt}\right)\right)\, ds. \]

Note that $u(r)$ is a solution of (2.6) if and only if it is a fixed point of $\mathsf {T}$. It was proved in theorem 4.1 in [Reference Bueno and López5] that $\mathsf {T}$ is a contraction in the space $C^{1}([0,\,\delta ])$ for $\delta >0$ small enough, hence the existence of a fixed point $u\in C^{1}([0,\,\delta ])\cap C^{2}((0,\,\delta ])$ that solves (2.6) is ensured. The fact that $u(r)$ has $C^{2}$-regularity at $r=0$ follows by taking limits in (2.6) and applying the L'Hôpital rule.

By the even condition on $\Phi$, we can reflect $u$ with respect to a horizontal plane and obtain a downwards oriented graph intersecting orthogonally the axis of rotation. In particular, we derive the following consequence on the phase plane.

3.1 The case $a>0,\,b>0$

The classification result obtained in this section is the following:

Proof. Let $\phi \in C^{1}([-1,\,1])$ the 1-dimensional function defined by (2.1) in terms of $\Phi$. In particular, $\phi$ is positive and even, and since $a,\,b>0$, the structure of the phase plane is as follows (see fig. 1):

1. 1. The curve $\mathcal {S}$ lies entirely in $\Theta _2$ and is a compact arc joining the points $(0,\,\pi )$ and $(0,\,2\pi )$. Such a curve does not exist in $\Theta _1$

2. 2. The curve $\Gamma$ lies entirely in $\Theta _1$ and is a compact arc joining the points $(0,\,0)$ and$(0,\,\pi )$. Such a curve does not exist in $\Theta _2$.

FIGURE 1.

The phase plane for the case $a>0,\,b>0$ and the orbit of the rotational sphere in blue.

We prove first the existence of the complete examples. Since $\Gamma (\pi /2)=a/\phi (0)$, the equilibrium $e_0=(a/\phi (0),\,\pi /2)$ exists in $\Theta _1$ and generates a circular, flat cylinder of constant mean curvature $\phi (0)/(2a)$, obtaining the first example in the classification.

To prove the existence of a strictly convex sphere, consider the orbit $\gamma _+(s)=(x_+(s),\,\theta _+(s))$ having $(0,\,0)$ as endpoint, given by lemma 2.2, and the corresponding rotational $\Phi$-surface, $\Sigma _+$. First, we prove that $x_+(s)\rightarrow \infty,\,\theta _+(s)\rightarrow \theta _0\in (0,\,\pi /2]$ cannot happen.

Arguing by contradiction, suppose that $x_+(s)\rightarrow \infty$ and $\theta _+(s)\rightarrow \theta _0\in (0,\,\pi /2]$, which implies that $\gamma _+$ is strictly contained in the monotonicity region $\{x>\Gamma (\theta ),\,\ \theta <\pi /2\}$. In particular, the function $\theta _+(s)$ can be seen as a function of $x_+(s)$ and the chain rule yields

\[ \theta_+'(s)=\frac{d\theta_+}{ds}=\frac{d\theta_+}{dx_+}\frac{dx_+}{ds}=\theta_+'(x)\cos\theta_+(x). \]

Therefore, $\theta _+(x)\rightarrow \theta _0\in (0,\,\pi /2]$ as $x\rightarrow \infty$, and so $\theta _+'(x)\rightarrow 0$ as $x\rightarrow \infty$. But since $\theta _+'(x)\cos \theta _+(x)=\theta _+'(s)$, we have

\[ \theta_+'(x)\cos\theta_+(x)=\frac{x\phi(\cos\theta_+(x))-a\sin\theta_+(x)}{ax+b\sin\theta_+(x)}, \]

and taking limits as $x\rightarrow \infty$ we get $0=\phi (\theta _0)/a$, a contradiction.

So, $\gamma _+$ starts at $(0,\,0)$ and cannot stay forever in $\{x>\Gamma (\theta ),\,\ \theta <\pi /2\}$. Next, we see that $\gamma _+$ cannot converge to $e_0$. In fact, we prove that the orbits close enough to $e_0$ are ellipses enclosing $e_0$ in their inner regions. The linearized system of (2.4) around $e_0$ is given by

\[ \left(\begin{matrix} u\\ v \end{matrix}\right)'= \left(\begin{matrix} 0 & -1\\ \frac{a^{2}+b\phi(0)}{(b+\frac{a^{2}}{\phi(0)})^{2}} & -\frac{a\phi'(0)}{a^{2}+b\phi(0)} \end{matrix}\right)\left(\begin{matrix} u\\ v \end{matrix}\right). \]

Noting that $\phi '(0)=0$ since $\phi$ is even, we check that the orbits of the linearized are ellipses around the origin. By classical theory of non-linear autonomous systems, there are two possible configurations for the orbits of (2.4) around $e_0$: either all such orbits are closed curves having $e_0$ in their inner regions; or they spiral around $e_0$. However, the latter possibility cannot happen since the orbits of (2.4) are symmetric with respect to the line $\theta =\pi /2$. In particular, all the orbits stay at a positive distance to $e_0$.

Since $\gamma _+$ cannot stay forever in the monotonicity region $\{x>\Gamma (\theta ),\,\ \theta <\pi /2\}$, nor converge to $e_0$, the only possibility is that it intersects the line $\theta =\pi /2$ at some finite point $(x_+,\,\pi /2),\,\ x_+>a/\phi (0)$. By symmetry of the phase plane, $\gamma _+$ has $(0,\,\pi )$ as endpoint, i.e. it agrees with the orbit $\gamma _-$ described in lemma 2.2, and is a compact arc joining $(0,\,0)$ and $(0,\,\pi )$. See fig. 1, the orbit in blue. In conclusion, $\Sigma _+$ is a rotational, strictly convex $\Phi$-sphere.

Next, we prove the existence of a 1-parameter family of unduloid-type $\Phi$-surfaces. For that, fix some $x_0\in (0,\,a/\phi (0))$ and let $\gamma _{x_0}(s)$ be the orbit in $\Theta _1$ passing through $(x_0,\,\pi /2)$ at $s=0$. For $s>0$, $\gamma _{x_0}(s)$ lies in the monotonicity region $\{x<\Gamma (\theta ), \theta <\pi /2\}$ and it intersects the curve $\Gamma$, where the $\theta$-coordinate reaches a minimum. Then, $\gamma _{x_0}$ lies in $\{x>\Gamma (\theta ),\,\ \theta <\pi /2\}$, and since it cannot intersect $\gamma _+$ nor converge to $e_0$, has to reach the line $\theta =\pi /2$ at some point $(\widehat {x}_0,\,\pi /2)$, with $a/\phi (0)<\widehat {x}_0< x_+$. By symmetry, $\gamma _{x_0}$ is a closed orbit having $e_0$ in its inner region. See fig. 2, the orbit in orange. This orbit generates a properly embedded (since $\theta \in (0,\,\pi )$ and hence $z'(s)=\sin \theta >0$) $\Phi$-surface, $\Sigma _{x_0}$, that is periodic and has an unduloid-type behaviour. The parameter $x_0$ defining this family is the neck-size, i.e. the minimum distance of $\Sigma _{x_0}$ to the axis of rotation.

FIGURE 2.

Left: the phase plane for $a>0,\,b>0$ and a prescribed function $\Phi$ under the hypotheses of theorem 3.1. The orbits correspond to the complete examples. Right: the profile curve of the $\Phi$-surface corresponding to each orbit.

Finally, we prove the existence of a 1-parameter family of nodoid-type $\Phi$-surfaces. Fix some $x_1>b/a$, and consider the orbit $\gamma _{x_1}(s)$ passing through $(x_1,\,3\pi /2)$ at $s=0$. From the monotonicity properties of $\Theta _2$, $\gamma _{x_1}(s)$ lies in $\{x>\mathcal {S}(\theta ),\, \theta \in (3\pi /2,\,2\pi )\}$ for $s>0$ (see (2.5) where $\mathcal {S}$ was defined) and has increasing both $x$ and $\theta$-coordinates until reaching the line $\theta =2\pi$ at some $(\overline {x}_1,\,2\pi )$ with $\overline {x}_1>x_1$. Similarly, and by symmetry, $\gamma _{x_1}(s)$ lies in $\{x>\mathcal {S}(\theta ),\, \theta \in (\pi,\,3\pi /2)\}$ for $s<0$ and has $(\overline {x}_1,\,\pi )$ as endpoint.

At this point, when $s$ further decreases, $\gamma _{x_1}$ lies in the monotonicity region $\{x>\Gamma (\theta ),\,\ \theta \in (\pi /2,\,\pi )\}$ of $\Theta _1$ until intersecting the line $\theta =\pi /2$ at some $(\widehat {x}_1,\,\pi /2)$ with $\widehat {x}_1>x_+$. Again by symmetry, $\gamma _{x_1}$ has the point $(\overline {x}_1,\,0)$ as endpoint. See fig. 2, the orbit in purple. Note that in particular, $\gamma _{x_1}$ does not intersect $\mathcal {S}$, hence the corresponding surface $\Sigma _{x_1}$ has no singularities. Moreover, it is properly immersed, periodic and has self-intersections, sharing the same properties as the constant mean curvature nodoids. The parameter $x_1$ defining this family is again the neck-size of each surface.

This completes the classification result in the complete case. Now we describe the non-complete examples.

First, fix some $\theta _0\in (0,\,\pi /2]$. At the point $(0,\,\theta _0)$, system (2.4) has existence and uniqueness; here the fact that $b\neq 0$ is paramount. Consequently, there exists an orbit $\gamma _{\theta _0}$ such that $\gamma _{\theta _0}(0)=(0,\,\theta _0)$ and $\gamma _{\theta _0}(s)$ lies in $\{x<\Gamma (\theta ),\,\ \theta <\pi /2\}$ for $s>0$ small enough. By uniqueness, $\gamma _{\theta _0}$ cannot intersect any $\gamma _{x_0}$ corresponding to the unduloids, nor $\gamma _+$ corresponding to the sphere. So, $\gamma _{\theta _0}$ intersects the line $\theta =\pi /2$ at some finite point $(x_{\theta _0},\,\pi /2)$. By symmetry, $\gamma _{\theta _0}$ is a compact arc having the point $(0,\,\pi -\theta _0)$ as its other endpoint. See Fig. 3 left, the orbit in orange.

FIGURE 3.

Left: the orbits corresponding to the rotational $\Phi$-surfaces of type $\mathit {1.5,\,1.6}$ of theorem 3.1. Right: the profile curve of the $\Phi$-surface corresponding to each orbit.

This orbit generates a $\Phi$-surface, $\Sigma _{\theta _0}$, that intersects the axis of rotation at a bottom point with angle function $\cos \theta _0$ and at a top point with angle function $-\cos (\theta _0)$. In particular, $\Sigma _{\theta _0}$ is singular at these cusp points where it fails to be $C^{1}$, hence is non-complete. Recall that $\gamma _{\theta _0}$ lies in both $\{x<\Gamma (\theta )\}$ and $\{x>\Gamma (\theta )\}$, hence the Gauss curvature of $\Sigma _{\theta _0}$ changes of sign and vanishes at two parallels, which correspond to the points of $\gamma _{\theta _0}$ intersecting $\Gamma$. In particular, none of these surfaces has strictly monotonous angle function. See fig. 3 right, the profile curve in orange. This proves the existence of surfaces of item $\mathit {1.8}$ of theorem 3.1.

Now, fix some $\theta _1\in (3\pi /2,\,2\pi )$. Again, at the point $(0,\,\theta _1)$ system (2.4) has existence and uniqueness. Therefore, there exists an orbit $\gamma _{\theta _1}$ such that $\gamma _{\theta _1}(0)=(0,\,\theta _1)$ and $\gamma _{\theta _1}(s)$ lies in $\{x<\mathcal {S}(\theta ),\,\ \theta \in (3\pi /2,\,2\pi )\}$ for $s>0$ small enough. We prove next that $\gamma _{\theta _1}$ cannot converge to a point in $\mathcal {S}$.

Proof of the Claim: We begin by highlighting that a first integral of system (2.4) for the particular choice $\phi =c\in \mathbb {R}$ is

(3.1)\begin{equation} a(x(s)\sin\theta(s)-x(s_0)\sin(\theta_0))+\frac{b}{2}(\sin^{2}\theta(s)-\sin^{2}\theta(s_0))=\frac{c}{2}(x(s)^{2}-x(s_0)^{2}). \end{equation}

In particular, if $c=0$ and we fix the initial condition $x(s_0)=0,\,\ \theta (s_0)=\theta _0\in (3\pi /2,\,2\pi )$ the first integral transforms into

(3.2)\begin{equation} ax(s)\sin\theta(s)+\frac{b}{2}(\sin^{2}\theta(s)-\sin^{2}\theta_0)=0. \end{equation}

Let $\sigma _{\theta _0}(t)=(x_{\theta _0}(t),\,\theta _{\theta _0}(t))$Footnote ¹ be an orbit of system (2.4) for $\phi =0$ and the initial condition $x(s_0)=0,\, \theta (t_0)=\theta _0\in (3\pi /2,\,2\pi )$. If $\sigma _{\theta _0}(t)\rightarrow \mathcal {S}$ then $\theta _{\theta _0}\rightarrow \theta _*$ and $x_{\theta _0}\rightarrow -b\sin \theta _*/a$, for some $\theta _*\in (3\pi /2,\,\theta _0)$. Substituting in (3.2) yields

\[ -\frac{b}{2}\sin^{2}\theta_*-\frac{b}{2}\sin^{2}\theta_0=0, \]

which is a contradiction since $b>0$. Therefore, a simple substitution in (3.2) shows that $\sigma _{\theta _0}(t)$ reaches the line $\theta =3\pi /2$ at $(x_{\theta _0},\,3\pi /2)$ with $x_{\theta _0}=b/(2a)\cos ^{2}\theta _0$.

Let us consider now the orbit $\gamma _{\theta _1}=(x_{\theta _1}(s),\,\theta _{\theta _1}(s))$, and suppose that there exist two instants $s_1,\,t_1$ such that $\gamma _{\theta _1}(s_1)=\sigma _{\theta _0}(t_1)$. Comparing system (2.4) for the prescribed function $\phi >0$ and the particular case $\phi =0$, respectively, yields

\[ \theta_{\theta_1}'(s_1)<\theta_{\theta_0}'(t_1). \]

Now we stand in position to prove claim 3.2. Arguing by contradiction, suppose that $\gamma _{\theta _1}$ converges to $\mathcal {S}$. Consider some $\theta _0\in (\theta _1,\,2\pi )$ and let $\sigma _{\theta _0}$ be the orbit of system (2.4) for $\phi =0$ and the initial condition $\sigma _{\theta _0}(t_0)=(0,\,\theta _0)$. We have already proven that this orbit has $(b/(2a)\cos ^{2}\theta _0,\,3\pi /2)$ as finite endpoint. Also, $\sigma _{\theta _0}(t)$ for $t>t_0$ small enough lies above $\gamma _{\theta _1}$; see fig. 4.

FIGURE 4.

The diagram of the contradiction in the proof of claim 3.2.

If $\gamma _{\theta _1}$ converges to $\mathcal {S}$, then by continuity $\gamma _{\theta _1}(s_1)=\sigma _{\theta _0}(t_1)$ with $s_1>s_0,\,\ t_1>t_0$. But at these instants we have $\theta _{\theta _1}'(s_1)>\theta _{\theta _0}'(t_1)$, arriving to a contradiction. Thus, $\gamma _{\theta _1}$ intersects the line $\theta =3\pi /2$ at some finite point $(x_{\theta _1},\,3\pi /2)$, with $x_{\theta _1}< x_{\theta _0} < b/(2a)$. $ \Box $

Consequently, $\gamma _{\theta _1}$ passes through some $(x_{\theta _1},\,3\pi /2),\,\ x_{\theta _1}< b/(2a)$, and by symmetry $\gamma _{\theta _1}$ closes at the point $(0,\,3\pi /2-\theta _1)$; see fig. 3 left, the orbit in purple. The $\Phi$-surface $\Sigma _{\theta _1}$ generated by $\gamma _{\theta _1}$, is compact and intersects the axis of rotation at two singular cusp points, where $\Sigma _{\theta _1}$ fails to be $C^{1}$. Moreover, $\Sigma _{\theta _1}$ has strictly positive Gauss curvature. See fig. 3 right, the profile curve in purple. This proves the existence of surfaces of item $\mathit {1.5}$ of theorem 3.1.

We prove the existence of the last non-complete examples. First, note that the definition of $\widehat {x}_1$ in the construction of the complete, nodoidal $\Phi$-surfaces, maps the interval $(b/a,\,\infty )$ to $(x_1^{\infty },\,\infty )$, for some $x_1^{\infty }\geq x_+$Footnote ². We show that $x_1^{\infty }\neq x_+$.

Arguing by contradiction, suppose that $x_1^{\infty }=x_+$ and consider the orbit $\gamma$ passing through $(\overline {x},\,\pi )$, with $0<\overline {x}< b/a$. Then, for $s<0$, $\gamma (s)$ lies in $\{x>\Gamma (\theta ), \theta \in (\pi /2,\,\pi )\}$ and stays there for $s$ decreasing until converging to the line $\theta =\pi /2$. However, $\gamma$ cannot reach any $(\widehat {x},\,\pi /2)$ since necessarily $\widehat {x}\geq x_+$ and we already described the orbits passing through these points as the complete nodoids and the sphere. In particular, by the way it was defined, the orbit $\gamma _{x_1^{\infty }}$ passing through $(x_1^{\infty },\,\pi /2)$ has to converge to the point $(b/a,\,3\pi /2)$ after intersecting the line $\theta =\pi$ at some $(\overline {x}_1^{\infty },\,\pi )$. By symmetry, there exists an orbit in $\{x>\mathcal {S}(\theta ),\,\theta \in (3\pi /2,\,2\pi )\}$ having $(\overline {x}_1^{\infty },\,2\pi )$ as endpoint and converging to the point $(b/a,\,3\pi /2)$. By analogy, we name this orbit again as $\gamma _{x_1^{\infty }}$; see fig. 5 left, the orbit in purple.

FIGURE 5.

Left: the orbits corresponding to the rotational $\Phi$-surfaces of type $\mathit {1.7,\,1.8}$ of theorem 3.1. Right: the profile curve of the $\Phi$-surface corresponding to each orbit.

The $\Phi$-surface corresponding to the orbit $\gamma _{x_1^{\infty }}$ has a certain nodoidal shape but with a parallel removed consisting of singular points, hence is non-complete, and this behaviour is repeated after a vertical translation. See fig. 5 right, the profile curve in purple.

Now fix some $\widehat {x}_2\in (x_+,\,x_1^{\infty })$ and consider $\gamma _{\widehat {x}_2}(s)$ the orbit passing through $(\widehat {x}_2,\,\pi /2)$ at $s=0$. For $s>0$, $\gamma _{\widehat {x}_2}$ intersects the line $\theta =\pi$ at some $(\overline {x}_2,\,\pi ),\,\ \overline {x}_2<\widehat {x}_2$, and by symmetry it has $(\overline {x}_2,\,0)$ as endpoint for $s<0$. When $s>0$ further increases, $\gamma _{\widehat {x}_2}$ enters the region $\{x>\mathcal {S}(\theta ),\,\ \theta \in (\pi,\,3\pi /2)\}$ and since $\gamma _{\widehat {x}_2}$ cannot intersect any other orbit, it ends up converging to some $(\mathcal {S}(\theta _2),\,\theta _2)$ with $\theta _2\in (\pi,\,3\pi /2)$. By symmetry, there is a symmetric orbit, which we will name again $\gamma _{\widehat {x}_2}$ by analogy, that has the point $(\mathcal {S}(\theta _2),\,3\pi -\theta _2)$ as endpoint and converges to $(\overline {x}_2,\,2\pi )$ as $s$ increases. See fig. 5 left, the orbit in orange.

The corresponding $\Phi$-surface has strictly positive Gauss curvature, is homeomorphic to an annulus and has non-monotonous height function. See fig. 5 right, the profile curve in orange. This proves the existence of the surfaces of item $\mathit {1.7}$ in theorem 3.1.

Finally, since the orbits of the phase plane foliate it, there exists an orbit in the region $\{x<\mathcal {S}(\theta )\}$ whose extremes converge to the points $(\mathcal {S}(\theta _2),\,\theta _2)$ and $(\mathcal {S}(\theta _2),\,3\pi -\theta _2)$, and that intersects the line $\theta =3\pi /2$ at some $(x_2,\,3\pi /2)$. See fig. 5 left, the orbit in brown. The corresponding $\Phi$-surface has all the properties stated in item $\mathit {1.6}$ of theorem 3.1, concluding its proof. $\Box$

3.2 The case $a>0,\,b<0$

The classification result obtained in this section is the following:

Proof. We define $\phi$ in terms of $\Phi$ by (2.1) as usual. In this case, the structure of the phase plane is as shown in fig. 6: both curves $\mathcal {S}$ and $\Gamma$ lie entirely in $\Theta _1$ and are compact arcs joining the points $(0,\,0)$ and $(0,\,\pi )$.

FIGURE 6.

The structure of the phase plane in the elliptic case for $a>0,\,b<0$, and the orbits corresponding to the complete rotational $\Phi$-surfaces.

The existence of the complete examples carries over verbatim as in the case $a>0,\,b>0$ exhibited in theorem 3.1, and the orbits of these complete examples are shown in fig. 6. The description of the surfaces of type $\mathit {1.5}$ and $\mathit {1.6}$ is also the same as in the case $a>0,\,b>0$; see fig. 7, the orbits and profile curves in black and brown. Details are skipped at this point.

FIGURE 7.

Left: the orbits corresponding to the rotational $\Phi$-surfaces of type $\mathit {1.5-1.8}$ of theorem 3.3. Right: the profile curve of the $\Phi$-surface corresponding to each orbit.

For the existence of the examples of type $\mathit {1.7}$, fix some $\theta _0\in (\pi,\,3\pi /2]$ and let $\gamma _{\theta _0}$ be the orbit having $(0,\,\theta _0)$ as endpoint at the instant $s=0$. By symmetry of $\Theta _2$, $\gamma _{\theta _0}$ has the point $3\pi -\theta _0$ as endpoint, and ends up at some $(x_1,\,2\pi )$. For $s<0$, $\gamma _{\theta _0}$ intersects the line $x=\pi$ at $(x_1,\,\pi )$ and then enters the region $\Theta _1$. Since $\gamma _{\theta _0}$ cannot intersect the orbit corresponding to the sphere, nor any orbit of the nodoids, $\gamma _{\theta _0}$ intersects the line $x=\pi /2$. By symmetry, $\gamma _{\theta _0}$ intersects the line $x=0$ at $(x_1,\,0)$. See fig 7, the orbit and profile curve in purple.

The $\Phi$-surface generated by this orbit intersects the axis of rotation at two cusp points, its height function is non-monotonous, and its Gauss curvature is everywhere positive. This proves the existence of the rotational examples of type $\mathit {1.7}$.

Finally, we prove the existence of the examples of type $\mathit {1.8}$. Name $\mathcal {W}_0$ to the inner region bounded by the orbit $\gamma _0$ of the rotational $\Phi$-sphere and the curve $\mathcal {S}$ of singular points. Fix some $(x_0,\,\theta _0)\in \mathcal {W}_0-\Gamma$ and let $\gamma$ be the orbit passing through such point. Assume moreover that $\gamma$ does not correspond to an unduloid. It is clear that the only possibility for $\gamma$ is to be a bi-graph over the line $x=\pi /2$, intersecting it at a single point, having two endpoints located at the curve $\mathcal {S}$ and intersecting the curve $\Gamma$ twice. See fig 7, the orbit and profile curve in orange.

The $\Phi$-surface generated by this orbit has the topology of an annulus, has strictly increasing height function (since $\gamma$ is strictly contained in $\Theta _1$) and its Gauss curvature changes sign. This exhibits the existence of the rotational examples of type $\mathit {1.8}$ and proves theorem 3.3.