We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable structure the measure need no longer be pointwise doubling.

Abstract. We show that the Heisenberg group H n contains a measure zero set N such that every Lipschitz function f : H n → R is Pansu differentiable at a point of N . The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of N and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.

We characterize those mappings from a compact subset of R into the Heisenberg group H n which can be extended to a C m horizontal curve in H n . The characterization combines the classical Whitney conditions with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates.

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We propose a notion of domain with boundary of positive mean curvature and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here least gradient is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the boundary trace of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer [27] to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of continuous solutions can fail, even in the weighted Euclidean setting with Lipschitz weights. * The authors thank Estibalitz Durand-Cartagena, Marie Snipes and Manuel Ritoré for fruitful discussions about the subject of the paper.

Abstract. We show that every Carnot group G of step 2 admits a Hausdorff dimension one 'universal differentiability set' N such that every Lipschitz map f : G → R is Pansu differentiable at some point of N . This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.