Invariant singular minimal surfaces
Let \(\alpha\) denote a real number, and for a unit vector \({\mathbf a} \in {\mathbb R}^3\), let \({\mathbb R}^3_+({\mathbf a}) := \{p\in {\mathbb R}^3\,:\, \langle p, {\mathbf a}\rangle >0\}\) be the half-space determined by \({\mathbf a}\). For a smooth immersion \(\phi: M \to {\mathbb R}^3_+({\mathbf a})\) of an oriented connected compact surface \(M\), the potential \(\alpha\)-energy of \(\phi\) in the direction \({\mathbf a}\) is defined by \[ E(\phi) = \int_M \langle p, {\mathbf a}\rangle^\alpha\, dM. \] It is well known that the immersion \(\phi\) is a critical point of the energy \(E\) for all compactly supported variations if and only if \[ 2H(p) = \alpha \frac{\langle N(p), {\mathbf a}\rangle}{\langle \phi(p), {\mathbf a}\rangle} \quad (p\in M),\tag{1} \] where \(N\) and \(H\) denote the Gauss map and the mean curvature of \(\phi\). \(M\) is said to be a \textit{singular minimal surface} or \(\alpha\)-\textit{minimal surface} if the mean curvature \(H\) satisfies (1). \(M\) is a minimal surface when \(\alpha = 0\), and from now on, we will assume \(\alpha \neq 0\) unless stated otherwise. In this paper, the author classifies all singular minimal surfaces in Euclidean space that are invariant by a uniparametric group of translations and rotations. Among them, the author proves that the only cylindrical singular minimal surfaces are planes parallel to the vector \({\mathbf a}\) or \(\alpha\)-catenary cylinder where the rulings are orthogonal to \({\mathbf a}\). An \(\alpha\)-catenary curve \(\gamma : I \to {\mathbb R}^2\) can be parametrized by a positive function \(f\) in the form \(f(s)f''(s) = \alpha (1+f'(s)^2)\), and it gives an example of \(\alpha\)-minimal surface \(M\) by considering the generalized cylinder with basis \(\gamma\). In case of \({\mathbf a} = (0,0,1)\), such \(M\) is called an \(\alpha\)-catenary cylinder. The second main result of the paper is the following. If an \(\alpha\)-minimal surface \(M\) with respect to the direction \({\mathbf a}\) is a rotational surface about the axis \(L\), then either the axis \(L\) is parallel to \({\mathbf a}\) or the axis \(L\) lies contained in the plane \({\mathbb R}^3_{0}({\mathbf a}) := \{p\in {\mathbb R}^3\,:\, \langle p, {\mathbf a}\rangle =0\}\). In this case, the author also gives explicit parametrizations for \(M\). The author also investigates the generalizations of rotational singular minimal surfaces by considering the family of surfaces foliated by circles and helicoidal surfaces. Finally, using the maximum principle of the singular minimal surface equation (1), the author obtains some control of the size of \(\alpha\)-minimal surfaces when \(\alpha \) is negative. In particualr the author proves that there do not exist \(\alpha\)-minimal entire graphs if \(\alpha <0\).
scientific article; zbMATH DE number 6894770
scientific article