Correction to: Equilibrium of Surfaces in a Vertical Force Field

Correction to: Mediterr. J. Math. (2022) 19:3 https://doi.org/10.1007/s00009-021-01877-4

In the previous paper [2, Section 3.2], we describe the family of titled $[\varphi ,{\vec {e}}_{3}]$-catenary cylinders as surfaces obtained from a $[\varphi ,{\vec {e}}_{3}]$-catenary cylinder $\Sigma$, by rotation of angle $\theta \in ]0,\pi /2[$ about the x-axis and dilation by scale factor $1/\cos \theta$. The authors state that the resulting surface, ${\widetilde{\Sigma }}$, is always a $[\varphi ,{\vec {e}}_{3}]$-minimal surface, but this is only true when the starting $[\varphi ,{\vec {e}}_{3}]$-catenary cylinder is a grim reaper translating soliton, which follows directly from the relationship between their mean curvatures. In fact, following the same notation as in [2, Section 3.2], $\widetilde{H} =\cos \theta \, H$, and ${\widetilde{\Sigma }}$ will be a $[\varphi ,{\vec {e}}_{3}]$-minimal surface if and only if

$$\begin{aligned} {\dot{\varphi }}(u) = {\dot{\varphi }}(u+y\ \sin \theta )\quad \text {for any } y\in {\mathbb {R}}, \end{aligned}$$

that is, if and only if $\Sigma$ is a grim reaper translating soliton. In this case, ${\widetilde{\psi }}$ will be a tilted grim reaper cylinder.

The following result updates the classification of complete flat $[\varphi ,{\vec {e}}_{3}]$-minimal surfaces in ${\mathbb {R}}^3$ [2, Theorem 3.7]:

FormalPara Theorem 1

Let $\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a diffeomorphism and $\Sigma$ be a complete flat $[\varphi ,{\vec {e}}_{3}]$-minimal surface in ${\mathbb {R}}^{3}.$ Then,  one of the following statements holds

• $\Sigma$ is a vertical plane.

• $\Sigma$ is a grim reaper cylinder (maybe tilted).

• $\Sigma$ is a $[\varphi ,{\vec {e}}_{3}]$-catenary cylinder.

FormalPara Proof

From basic differential geometry, $\Sigma =\gamma \times \Pi ^\perp$ is a ruled surface and its Gauss map is constant along the rules, where $\gamma$ is a complete regular curve in a plane $\Pi \subset {\mathbb {R}}^3$. Thus, $\Sigma$ can be parametrized by $\psi (s,t)=\gamma (s)+t{\vec {v}}$, with $\gamma (s)$ a complete regular curve contained in a plane $\Pi$ and ${\vec {v}}$ a unit vector orthogonal to $\Pi$. We may assume that $(s,t)\in {\mathbb {R}}^2$ and $\vert \gamma '\vert =1$, then, the Gauss map N of $\psi$ and its mean curvature H are given by

$$\begin{aligned} N(s,t)=\gamma '(s)\wedge {\vec {v}},\quad H(s,t)=\kappa _{\gamma }(s), \end{aligned}$$

where $\kappa _{\gamma }$ is the curvature of $\gamma$. Hence, $\Sigma$ is $[\varphi ,{\vec {e}}_{3}]$-minimal if and only if the following relation holds

$$\begin{aligned} \kappa _{\gamma }(s)=-{\dot{\varphi }}(\langle \gamma (s)+t{\vec {v}},{\vec {e}}_{3}\rangle )\langle \gamma '(s)\wedge {\vec {v}},{\vec {e}}_{3}\rangle . \end{aligned}$$

Differentiating with respect to t in the above expression, we obtain that

$$\begin{aligned} 0=\ddot{\varphi }(\langle \gamma (s)+t{\vec {v}},{\vec {e}}_{3}\rangle )\langle {\vec {v}},{\vec {e}}_{3}\rangle \langle \gamma '(s)\wedge {\vec {v}},{\vec {e}}_{3}\rangle . \end{aligned}$$

If $\langle {\vec {v}},{\vec {e}}_{3}\rangle = 0$, arguing as in [2, Theorem 3.7], for any horizontal rule ${\mathcal {L}}$ of $\Sigma$ there exists a $[\varphi ,{\vec {e}}_{3}]$-catenary cylinder ${\mathcal {C}}$ containing ${\mathcal {L}}$ and tangent to $\Sigma$ along ${\mathcal {L}}$. Thus, from standard theory of uniqueness of solution for an ODE, up to horizontal translation, $\Sigma$ must coincide with ${\mathcal {C}}$.

If $\langle {\vec {v}},{\vec {e}}_{3}\rangle \ne 0$,

$$\begin{aligned} 0=\ddot{\varphi }(\langle \gamma (s)+t{\vec {v}},{\vec {e}}_{3}\rangle )\langle \gamma '(s)\wedge {\vec {v}},{\vec {e}}_{3}\rangle \end{aligned}$$

and we can assume that $\ddot{\varphi }$ does not vanish everywhere otherwise, from [1], $\varphi$ is a linear function and $\Sigma$ is either a vertical plane or a grim reaper cylinder (maybe tilted). Thus, we have that $\gamma '\wedge {\vec {v}}$ is orthogonal to ${\vec {e}}_{3}$ and $\Sigma$ must be a vertical plane. $\square$

Now, the Corollary 3.8 in [2] updates to

FormalPara Corollary 2

Let $\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a increasing diffeomorphism with $\dddot{\varphi }\le 0$ and $\Sigma$ be a complete locally convex $[\varphi ,{\vec {e}}_{3}]$-minimal surface in ${\mathbb {R}}^{3}$. If the Gauss curvature vanishes anywhere, then one of the following statements holds

• $\Sigma$ is a vertical plane.

• $\Sigma$ is a grim reaper cylinder (maybe tilted).

• $\Sigma$ is a $[\varphi ,{\vec {e}}_{3}]$-catenary cylinder.

FormalPara Remark 3

Although it does not affect any of the results shown throughout the paper, the Eqs. (6) and (7) in [2, Lemma 2.1] must be change to

$$\begin{aligned}{} & {} \nabla ^{2}H =-\eta \nabla ^2{\dot{\varphi }} - (\nabla \mathcal {A})(\nabla \varphi ,\,\cdot ,\, \cdot \,)- H \mathcal {A}^{[2]}- {{\mathcal {B}}} \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \Delta \mathcal {A}+ (\nabla \mathcal {A})(\nabla \varphi ,\, \cdot ,\, \cdot \,)+\eta \nabla ^2{\dot{\varphi }} + |\mathcal {A}|^2\mathcal {A}+ {\mathcal {B}}=0, \end{aligned}$$

(7)