Maximal metric surfaces and the Sobolev-to-Lipschitz property

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger, which satisfies a related maximality condition.

A note on spacelike surfaces in Minkowski 3-space

We characterize and classify spacelike surfaces endowed with a canonical principal direction in Minkowski 3-space E13. Under the maximality condition, a new characterization for the catenoid of the 1st kind is obtained.

Analytic and coanalytic families of almost disjoint functions

If is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable , no member of which is covered by finitely many functions from , there is such that for all there are infinitely many integers k such that f(k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality condition.