On minimal surfaces in the real special linear group \(SL(2,\mathbb{R})\) (Q1382839)

The author studies minimal surfaces in the real special linear group \(SL(2,\mathbb{R})\) endowed with a canonical left-invariant metric. In particular, the author defines rotational surfaces in \(SL(2,\mathbb{R})\) as those which are invariant under the right translation of the subgroup \(SO(2)\), and classifies minimal rotational surfaces in \(SL(2,\mathbb{R})\) and, more generally, rotational surfaces of constant mean curvature. The author also defines a conoid in \(SL(2,\mathbb{R})\) analogous to that in \(\mathbb{R}^3\) and classifies minimal conoids. Finally, the author deals with the stability of these surfaces, showing that any minimal rotational surface is stable, a rotational surface of constant mean curvature \(H\) is stable if and only if \(| H|\leq 1\), and a minimal conoid is stable if the pitch \(C\geq 0\).