Equidimensional isometric maps

In Gromov's treatise Partial Differential Relations (volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 1986), a continuous map between Riemannian manifolds is called isometric if it preserves the length of rectifiable curves. In this note we develop a method using the Baire category theorem for constructing such isometries. We show that a typical $1$-Lipschitz map is isometric in canonically formulated extension and restriction problems

On the gradient set of Lipschitz maps

We prove that the essential range of the gradient of planar Lipschitz maps has a connected rank-one convex hull. As a corollary, in combination with the results in [Faraco, D., and Székelyhidi, Jr., L., Tartar's conjecture and localization of the quasiconvex hull in R2x2, Acta Math., to appear.] we obtain a complete characterization of incompatible sets of gradients for planar maps in terms of rank-one convexit

Regularity of quasiconvex envelopes

We prove that the quasiconvex envelope of a differentiable function which satisfies natural growth conditions at infinity is a $C^1$ function. Without the growth conditions the result fails in general. We also obtain results on higher regularity (in the sense of $C^{1,\alpha}_{\rm loc}$) and similar results for other types of envelopes, including polyconvex and rank-1 convex envelopes

Universal singular sets in the calculus of variations

For regular one-dimensional variational problems, Ball and Nadirashvilli introduced the notion of the universal singular set of a Lagrangian $L$ and established its topological negligibility. This set is defined to be the set of all points in the plane through which the graph of some absolutely continuous $L$-minimizer passes with infinite derivative. Motivated by Tonelli's partial regularity results, the question of the size of the universal singular set in measure naturally arises. Here we show that universal singular sets are characterized by being essentially purely unrectifiable --- that is, they intersect most Lipschitz curves in sets of zero length and that any compact purely unrectifiable set is contained within the universal singular set of some smooth Lagrangian with given superlinear growth. This gives examples of universal singular sets of Hausdorff dimension two, filling the gap between previously known one-dimensional examples and Sychev's result that universal singular sets are Lebesgue null. We show that some smoothness of the Lagrangian is necessary for the topological size estimate, and investigate the relationship between growth of the Lagrangian and the existence of (pathological) rectifiable pieces in the universal singular set. We also show that Tonelli's partial regularity result is stable in that the energy of a `near' minimizer $u$ over the set where it has large derivative is controlled by how far $u$ is from being a minimizer

Uniformly distributed measures in Euclidean spaces

of Marstrand [6] according to which the existence of non-trivial s-dimensional densities (of a measure in R n ) implies that s is an integer. Our results, however, do not seem to be strong enough to show the main result of [9] that measures having s-dimensional density are s rectifiable, because they do not give any information about the behaviour of X at infinity. In addition to the analyticity result mentioned above, we also show that X is an algebraic variety provided that it is bounded and obtain more precise results in the special cases n = 1 and n = 2. For various reasons, including the political changes in Europe, this note has not been published for many years, although our main approach has become known and used in the literature. We would like to thank the Max Planck Institute for bringing us together and thus