Two counterexamples in low-dimensional length geometry

Abstract

One of the key properties of the length of a curve is its lower semi-continuity: if a sequence of curves γi converges to a curve γ, then length(γ) ≤ lim inf length(γi). Here the weakest type of point-wise convergence suffices. There are higher-dimensional analogs of this semi-continuity for Riemannian (and even Finsler) metrics. For instance, the Besicovitch inequality (see, for instance, [1] and [4]) implies that if a sequence of Riemannian metrics di on a manifold M uniformly converges to a Riemannian metric d, then V ol(M,d) ≤ lim inf V ol(M,di). Furthermore, the same is true if the limit metric is Finsler (where one can use any “reasonable” notion of volume for Finsler manifolds); the proof, though, is more involved (see [2], [7]). However, we will give an example of an increasing sequence of Riemannian metrics di on a 2-dimensional disc D, which uniformly converge to a length metric d on D such that Area(D, di) 1 (where by Area(D, d) we mean the 2-dimensional Hausdorff measure). Furthermore, metrics di and d can be realized by a uniformly converging sequence of embeddings of D into R. Our motivation for studying the semi-continuity of the surface area functional came from [3], where a more sophisticated Besicovitch-type inequality for Finsler metrics is shown. The proof is essentially Finsler, even though the inequality makes sense for general length spaces. The counter example 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 disc be approximated by an increasing sequence of Finsler metrics? There is some evidence suggesting that the answer is likely affirmative in dimension two. However, we will 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 counter-examples. 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 complete novelty 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. Sections 2 and 3 contain proofs