We examine surfaces of the type proved to be minimizing under a connectivity condition by Dorff et al. We determine which of these surfaces are stable soap films. The connectivity condition is shown to be very restrictive; few of these surfaces are stable (locally minimizing) without it. 2. The surfaces of Dorff et al. Equitent surfaces are constructed via the union of sections of spheres and planes. Starting with a cone over a wire-frame polyhedron, the center of the cone is then MSC2010: primary 49Q10; secondary 49Q05, 53A10.

Over a one-dimensional Gorenstein local domain R, let E be the endomorphism ring of the maximal of R, viewed as a subring of the integral closure R. If there exist finitely generated R-modules M and N , neither of them free, whose tensor product is torsion-free, we show that E must be local with the same residue field as R.

Following our previous work with Dorff and Lawlor, we extend results for the so-called equitent problem of fixed boundary and fixed volume. We define sufficient conditions, which in ‫ޒ‬ 2 and ‫ޒ‬ 3 are also necessary, for local minima to be piecewise spherical, and we show that these are areaminimizing in their homotopy class. We also give new examples of these surfaces in ‫ޒ‬ 2 and ‫ޒ‬ 3 .