We show that the only compact and connected subsets (i.e. continua) X of the plane ${\mathbb{R}}^2$ which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X, are, up to homeomorphism, the circle ${\mathbb{S}}^1$, the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain any arcs. It follows that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set.

The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps $f,g:C\to X$ such that $f(C)\subset g(C)$ there exists $c_0\in C$ so that f(c₀) = g(c₀). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero.

For every genus g, we prove that ${\mathbf{S}}^2\times \mathbf{R}$ contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the ${\mathbf{S}}^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in ${\mathbf{R}}^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in ${\mathbf{R}}^3$ of every prescribed genus occur as such limits of examples in ${\mathbf{S}}^2\times \mathbf{R}$.

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.