Two Counterexamples in Low-dimensional Length Geometry

One of the key properties of the length of a curve is its lower semicontinuity : if a sequence of curves γi converges to a curve γ, then length(γ) ≤ lim inf length(γi). Here the weakest type of pointwise convergence suffices. There are higher-dimensional analogs of this semicontinuity for Riemannian (and even Finsler) metrics. For instance, the Besicovitch inequality (see, e.g., [1] and [4]) implies that if a sequence of Riemannian metrics di on a manifold M uniformly converges to a Riemannian metric d, then Vol(M,d) ≤ lim inf Vol(M,di). Furthermore, the same is true if the limit metric is Finsler (where any “reasonable” notion of volume for Finsler manifolds can be used); the proof, though, is more involved (see [2, 7]). However, we shall give an example of an increasing sequence of Riemannian metrics di on a 2-dimensional disk D that uniformly converge to a length metric d on D such that Area(D, di) < 1 10 and Area(D, d) > 1 (where by Area(D, d) we mean the 2-dimensional Hausdorff measure). Furthermore, the metrics di and d can be realized by a uniformly converging sequence of embeddings of D into R. Our motivation for studying the semicontinuity of the surface area functional came from [3], where a more sophisticated Besicovitch-type inequality for Finsler metrics was shown. The proof is essentially Finsler, even though the inequality makes sense for general length spaces. The counterexample undermines a natural approach to proving length-area inequalities for length spaces by means of approximations by Riemannian (more generally, Finsler) metrics. Similar considerations lead to the following question: can every intrinsic metric on a disk be approximated by an increasing sequence of Finsler metrics? There is some evidence suggesting that the answer is likely to be affirmative in dimension two. However, we shall give an example of an intrinsic metric on a 3-dimensional ball such that no neighborhood of the origin admits a Lipschitz bijection to a Euclidean region. In this elementary exposition we present both counterexamples. Unfortunately, people often choose not to publish the results of research that led to counterexamples rather than proofs of desired theorems; as such, even published counterexamples tend to be forgotten. Hence we cannot be confident in the complete newness of the results. At the very least, we use this paper to raise open problems and embed these problems into a new context. The paper is organized as follows. In the rest of the Introduction we give rigorous formulations of the results and outline the proofs. §§2 and 3 contain proofs of Theorems