On minimal annuli in a slab
Let \(P_ t= \{(x, y, z)\): \(z=t\} \subset \mathbb{R}^ 3\) and let \(S\subset \mathbb{R}^ 3\) be a slab in the Euclidean space \(\mathbb{R}^ 3\) given by \[ S= S(-1, 1)=\{ (x,y, z);\;-1\leq z\leq 1\}. \] The author proves the following theorem. Suppose that \(A\subset S(-1, 1)\) is a proper minimal annulus in a slab \(S\), and that \(A(1)= A\cap P_ 1\), \(A(-1)= A\cap P_{-1}\) are straight lines or circles. (1) If both \(A(1)\) and \(A(-1)\) are circles, then \(A_ t= A\cap P_ 1\) is a circle for \(-1< t<1\). In particular, \(A\) is embedded; (2) If \(A(1)\) or \(A(-1)\) is a straight line, and the other one is a circle, and \(A\) is embedded, then \(A_ t\) is a circle, for \(-1<t <1\). (3) If \(A(1)\) or \(A(-1)\) is a straight line, and \(A\) is embedded, then \(A(t)\) is a circle for \(-1<t <1\). Part (1) is a classical theorem of M. Shiffman. Part (3) implies a result of Hoffman-Karcher-Rosenberg which states the same conclusion, assuming that \(A(1)\) and \(A(-1)\) are parallel straight lines; it implies also a result of Toubiana that no such minimal annulus \(A\) exists if \(A(-1)\) and \(A(1)\) are nonparallel straight lines. All these results, together with part (2) which is new, are given a unified proof.
scientific article; zbMATH DE number 718589